Understanding the interfacial structure–property relationship of complex fluid–fluid interfaces is increasingly important for guiding the formulation of systems with targeted interfacial properties, such as those found in multiphase complex fluids, biological systems, biopharmaceuticals formulations, and many consumer products. Mixed interfacial flow fields, typical of classical Langmuir trough experiments, introduce a complex interfacial flow history that complicates the study of interfacial properties of complex fluid interfaces. In this article, we describe the design, implementation, and validation of a new instrument capable of independent application of controlled interfacial dilation and shear kinematics on fluid interfaces. Combining the Quadrotrough with both in situ Brewster angle microscopy and neutron reflectometry provides detailed structural measurements of the interface at the mesoscale and nanoscale in relationship to interfacial material properties under controlled interfacial deformation histories.
Complex fluid–fluid interfaces comprise of surface-active materials that reside at air–liquid or liquid–liquid interfaces and display nontrivial rheological responses. Such interfaces can include both soluble and insoluble interfacial components that may form complex interfacial monolayer or multilayer phases. Examples of such interfacial systems include asphaltene nanoaggregates,1,2 food emulsifiers,3 microgels,4 interfacial protein aggregates,5,6 and tear film stabilizers7 with applications across broad fields of technology, such as oil recovery, food formulations, and biotherapeutics. The interdependence between the macroscopic deformation and interfacial microstructure formation often dictates the overall material behavior such that the development of constitutive structure–property relationships can aid the design of desirable material properties under various conditions of processing and use. It is also relevant that the presence of an interface can also act to destabilize a system (e.g., protein interfacial aggregation in biopharmaceutics8,9) such that the instrument presented here has diagnostic value as well.
The development of interfacial structure–property relationships for complex interfaces is hindered by the limited instrumentation capable of the simultaneous structure and rheological characterizations. A few commonly used approaches enable microstructure visualization of liquid interfaces. For example, both atomic force microscopy and ellipsometry give an in-plane spatial image of the interface, but both require interfacial material transfer to a solid surface resulting in ex situ measurements. Fluorescence microscopy requires a mixture of interfacial material with fluorescent markers, which can alter the overall structure of the interface due to steric hinderance and unknown intermolecular interactions and has limited spatial resolution. Alternatively, Brewster angle microscopy is an appealing noninvasive mesoscale technique, which requires a refractive index difference between a clean air–liquid interface and one with interfacial material. X-ray and neutron reflectometry are also common in situ techniques to determine the out-of-plane microstructure and molecular composition. While x-ray reflectometry can span a wider range of length scales to probe smaller features, it provides low contrast for soft material interfaces and their surrounding subphases. On the other hand, neutron reflectometry has the unique ability to create high scattering contrast between the subphase and interface through selective deuteration without significantly altering material interactions. Most of these techniques have been coupled with an interfacial compression technique, such as a Langmuir57 trough,10–12 to study interfacial structures under dilatational/compressional behavior; however, using such methods to interpret rheological results, specifically dilatational rheology, is nontrivial and the results are often misinterpreted.13–15 For example, Langmuir trough measurements have caused an underestimation of the dilatational modulus of complex interfaces at small surface areas15 and uniaxially compressed interfaces can consequently create unidirectional wrinkling of elastic interfaces.16
Current approaches to studying dilatational rheology with well-controlled kinematic conditions are still sparse and often not readily available commercially. The measurements from pendant drop technique prove difficult to interpret due to anisotropic stress complexities that deviate from the Young–Laplace equation used for fitting drop-shape.17,18 Commercial Langmuir–Pockels troughs that are utilized as rheometers also show the same complications due to the mixed flow fields upon uniaxial compression; dilatational and shear strains are often not trivial to decouple for certain complex fluid interfaces.13–15 The resulting anisotropic surface pressure response of the interface is often detected using a Wilhelmy plate that is oriented perpendicular or parallel to the moving barriers responsible for the imposed strain field. To prevent shear strains in the trough technique and to better understand dilatational behavior, a new radial trough was created with the capability of pure dilatational deformation that also resulted in the development of a new finite strain constitutive model for elastic interfaces past the infinitesimal strain limit.15,19 With the intent of elucidating interfacial morphology under these pure dilatational deformations, a miniaturized radial trough was built and implemented under a conventional inverted microscope to investigate mesoscale interfacial structures while also measuring its dilatational rheology.20 These are some of the first known advancements toward understanding the true structure and dilatational properties of interfaces.21
Along the same path of innovation, there has been progress toward hybridizing interfacial shear rheology with trough compression techniques to allow more seamless measurements as a function of surface concentration due to interfacial compressibility. Additionally, these combined trough and shear rheometry instruments are modified to accommodate microscopes for structural visualization. A few interfacial shear rheometry techniques exist commercially, while others are built in-lab. Both bicone22 and double wall ring23 are geometry attachments to traditional rheometers. The magnetic rod interfacial stress rheometer24 shears the interface using a magnetic needle that oscillates between two Helmholtz coils. A laboratory “microbutton” probe designed by Choi et al.25 is used for microrheometry for localized interfacial rheology. Most of these techniques have been implemented with the Langmuir trough for probing shear rheology, and sometimes microstructure, of interfaces under various compressed states.26–28 To our knowledge, there is no known technique that is capable of implementing both shear and dilatational strain deformation with microstructural characterization for interfacial systems, which would help improve studies of complex interfaces such as thixotropic and/or viscoelastic materials that are sensitive to deformation history.
In this paper, we introduce a new trough instrument, termed the Quadrotrough, with the capabilities of performing both pure shear and dilatational strains at the same air–liquid interface with deformation control. Instrument performance is validated by surface pressure measurements on model inviscid and viscoelastic interfaces, which demonstrates the importance of deconvoluting the strain kinematics for compressible, viscoelastic interfaces. The Quadrotrough can function as a standalone instrument to directly measure the surface pressure of the interface under a broad range of well-defined shear, dilation, and mixed interfacial deformations. Particle tracking validates the interfacial deformation kinematics. In situ application of Brewster angle microscopy can resolve the in-plane interfacial microstructure and in situ application of neutron reflectometry can resolve the out-of-plane interfacial nanostructure. This unique combination facilitates the development of robust structure–property relationships for complex fluid interfaces.
II. INSTRUMENT DESIGN AND OPERATION
A. Quadrotrough description
The Quadrotrough was designed to strain a planar interface in two fundamental kinematic modes, viz., dilatation/compression15 and pure shear29,30 [Fig. 1(a)], as well as their varied combinations. Separation of interfacial kinematics is unattainable in the conventional Langmuir trough13,14 [Fig. 1(b)], which creates complex, strain-dependent deformation kinematics.13,31 This is especially important when considering complex interfaces that are inherently dependent upon the details of their deformation history through their interfacial microstructure, both in-plane and out-of-plane.
1. Strain directionality
Rheological dilation response is associated with isotropic interfacial kinematics, similar to thermodynamic bulk compressibility, while rheological pure shear response is associated with a deformation that introduces directionality characterized in terms of primary axes of compression and extension and maintains constant area. Interfacial stress, or interfacial tension as denoted for simple interfaces, is often measured with a Wilhelmy plate and plate orientation relative to the deformation axes is an important consideration. For pure dilation, defined as a positive strain (+θ), or pure compression, defined as a negative strain (−θ), the Wilhelmy plate orientation should be irrelevant for detecting interfacial stress as the strain is isotropic.13,31
On the other hand, the rheological shear response of an interface is anisotropic; therefore, the Wilhelmy plate detects different stress components based on its orientation relative to the principal axes of deformation. Under consideration here is planar shear at constant interfacial area (i.e., no dilation or compression). For reference as seen in Fig. 1(a), when the Wilhelmy plate is perpendicular to the extensional axes (case: shear forward A, shear backward B), we consider this to be a positive strain (+γ). When the Wilhelmy plate is parallel to the expansional axes (case: shear forward B, shear backward A), we consider this to be a negative strain (−γ). It should be noted that shear forward and shear backward pairings are directionally reverse of each other. As seen in Fig. 1(a), there are a total of two directional shear strain pairings (A and B) of shear forward and shear backward relative to the Wilhelmy plate orientation. Both pure shear deformations shaded in yellow are considered +γ, while both those in white are considered −γ. In this paper, we adopt the default orientation of mode A where the Wilhelmy plate is oriented perpendicular to the extensional axes during shear forward (+γ) and oriented parallel to extensional axes during shear backward (−γ) when reverting to the isotropic state, unless otherwise specified [Fig. 1(a)].
It should be noted that for Langmuir trough experiments [Fig. 1(b)], apparent compression and expansion are coupled with undesirable complex shear strains, as demonstrated elsewhere.13,31 Apparent compression and expansion imposed in the Langmuir trough still follow the same strain directionality, where positive and negative dilatational strains represent expansion and compression, respectively. However, Wilhelmy plate orientation is no longer irrelevant as the strain is no longer isotropic [Fig. 1(b)]. Additionally, when the Wilhelmy plate is oriented perpendicular or parallel to the barrier motion, it is not directly oriented along the principal shear stretch axes and the shear strain directionality is not trivial. The consequence of coupling both shear and dilatational strains under these apparent motions on the stress–strain relationships is discussed in later sections.
2. Mechanical design
The schematic of the Quadrotrough setup as shown in Fig. 2 is mainly comprised of a custom-made Teflon trough, an elastic barrier, and a force balance (Nanoscience Instruments, KSV NIMA). The Teflon trough here is of the same design as the radial trough and design specifications are listed elsewhere, including the aluminum heat exchanger.15 The only modifications made were two cutouts to the top section of the Teflon lip to allow the neutron beam path to reach the air–liquid interface at the meniscus divot,15,23 as shown in Fig. 2(c). The elastic barrier is a 10 mm width poly(styrene–butadiene–styrene, SBS) band (Vreeberg BV) that can achieve a maximum compression ratio of Amax/Amin = 3 and a maximum nominal shear strain . Additionally, a “neutron invisible” fluoroelastomer band is developed with Amax/Amin = 2 and , and more material details are specified elsewhere.2 The elastic barrier is stretched by the stepper motor controlled fingers through the two modes of strain, as shown in Fig. 1(a). The whole instrument setup sits inside a custom-made plexiglass chamber [Fig. 2(a)] and is humidity controlled by an external humidifier to prevent fluid evaporation during experiments (not shown).
The Quadrotrough instrument uses commercial hardware for surface tensiometry. The surface tensiometer force balance (KSV NIMA) is independently controlled by the KSV NIMA software and surface tension is measured through a standard platinum Wilhelmy plate (wetted perimeter: 39.23 mm, KSV NIMA). Interfacial kinematics are controlled using a mechanical design employed previously for the radial trough.15 The elastic barrier is held by four aluminum fingers that are controlled by stepper motors (Nanotec Electronics) and move along individual linear axes (MOVTEC) using another step motor at linear velocities. Each aluminum finger can also move up and down by a servo motor (HITEC) to control the placement of the elastic barrier at the air–water interface [Fig. 2(b)]. These motors are programmed through LabVIEW (National Instruments™) for elastic barrier movement. As each of the four linear translation motors can be addressed independently, a wide range of interfacial deformations are possible. Most interesting for the work presented here, the elastic barrier can be deformed in two pure kinematic strain modes: dilation/compression or pure shear (nominal constant area). During experiments, step motors travel at a linear velocity of 1 mm/min, which translates to a variable area compression of 6.1–3.6 mm2/s and variable shear rate ranging from 3 × 10−4 to 8 × 10−4 s−1.
3. Particle image velocimetry measurements
Macroscopic particle image velocimetry (PIV) was set up on the Quadrotrough to validate the interfacial flow field within the region bounded by the elastomer barrier. PIV measurements were made by seeding 250 µm polystyrene (PS) tracer particles (Phosphorex, Inc.) on a monolayer of poly(N-isopropylacrylamide), or PNIPAM microgels (hydrodynamic radius ∼ 45 nm) that were spread on water to form a uniform low viscoelastic monolayer at the air–water interface. PIVlab32–34 was used to analyze the velocity fields with frames taken every 30 s, and subsequently, velocity gradients and flow type parameter were determined using Paraview.35 Velocity fields were analyzed within the central region of the trough, or region of interest, which was the same sampling location for surface pressure measurements, microscopy, and neutron reflectometry. Select velocity vector distributions are provided in the supplementary material for dilatational and shear strain conditions to show normality (Fig. S3-4).
4. Brewster angle microscopy measurements
The commercial Brewster angle microscope (MicroBAM, KSV NIMA) is modified to fit on the Quadrotrough to visualize interfacial mesoscale structures while the Quadrotrough mechanically deforms the interface. Due to the large footprint of the microscope head, the force balance cannot be set up to center the Wilhelmy plate in tandem to the microscope (i.e., microscopy and surface tension are measured via separate, replicate experiments).
B. Rheo-MAGIK: Quadrotrough with neutron reflectometer
The Quadrotrough is implemented as a sample environment on a neutron reflectometer, specifically MAGIK, at the NIST Center for Neutron Research (NCNR).36 The neutron reflectometer was horizontally modified to accommodate the Quadrotrough for air–liquid measurements, as described elsewhere.37 The Q-range for this sample environment is measured from 0.01 to 0.16 Å−1, which took ∼8 h for each experiment. Data were reduced using Reductus, the online data reduction program, and details explaining reduction protocol are provided in the supplementary material.38,39 The horizontally modified reflectometry beamline MAGIK is still under development. The current setup operates with a beam footprint larger than the enclosed region of the elastomer band, where mechanical deformation of the interface is controlled by the Quadrotrough; this, in turn, limits the actual Q-range probing the region of interest. Detailed explanations of the current setup and future improvements to the horizontal MAGIK setup are reported in the supplementary material.
C. Interfacial mechanics
1. Kinematics of deformed interface
The following development is restricted to sharp, geometrically flat interfaces, while a more general development for interfaces with curvature can be found in the work of Jaensson et al.40 The rate-of-deformation tensor for a 2D interface is defined as and the deviatoric part of the rate-of-deformation tensor is defined as .40 The first invariant of Ds (trace) is the dilatational rate of change, or area rate of change, defined as . The second invariant of the deviatoric is defined as , also known as the surface determinant, which is proportional to the shear rate.40 As such, the rate-of-deformation tensor can be split into its dilatational and area-preserving (shear) components as follows:
To further evaluate the relative rate-of-deformation components, a flow type parameter, Λ*, is defined such that
where is the magnitude of the rate-of-expansion tensor and is the magnitude of the shear strain rate tensor. Since these magnitudes are proportional to the area rate of change, , and shear rate, , throughout the paper, another shorthand for and will be used along with their tensorial counterparts and as the magnitude of deformation (see the supplementary material for details). The limiting value of Λ* = 1 represents pure dilation/compression and Λ* = −1 represents planar shear stretch. The instrument’s ability to create these limiting flow types is validated using particle imaging velocimetry of model interfaces seeded with polystyrene particles as presented in Sec. IV.
For two-dimensional fluids, the finite interfacial deformation gradient tensor is defined as F = ∂xs/∂Xs, where xs and Xs are the position vectors of the interface in the deformed and reference configuration, respectively. For material-frame indifferent models, the deformation gradient tensor can be defined by a principal stretch ratio ; for a two-dimensional interface, two in-plane principal stretch ratios exist. For irrotational deformations, the interfacial deformation gradient can be written in terms of the left Cauchy–Green interfacial strain tensor, where C = FFT. The Jacobian determinant defined as J = det(C) is used to describe area change, as the area is J1/2. The three primary deformation modes of the Quadrotrough can then be described by their respective deformation gradient tensors as follows:
a. Pure dilatational/compressional strain.
For pure dilation and compression, the deformation gradient tensor, F, and Cauchy–Green deformation tensor, C, are defined as
where λ = L/L0 is the isotropic ratio of principal stretches in the deformed and reference states.31 The square root of the Jacobian determinant indicates an area change upon dilatational strain of λ2 = A/A0 as expected.
b. Planar shear stretch.
For planar shear stretch,29,30 the deformation gradient tensor and with the Cauchy–Green deformation tensor are defined as
where λ = L/L0 is the principal stretch ratio along the major principal axis in the deformed and reference states; the minor principal stretch ratio is related to the major principal stretch ratio based on the constant area constraint.29,31 The Jacobian determinant for pure shear stretch is 1, which indicates a constant area deformation. However, shear deformation contributes to the stress tensor for systems with a shear modulus and this can affect the measurement of surface pressure.
c. Convoluted shear and dilatational strain.
Mixed deformation fields can be described by combining the aforementioned deformation tensors and Cauchy–Green tensors such that
where λd and λs are used here to distinguish the principal stretches from dilation/compression and shear stretch, respectively. The Jacobian here is more complex and indicative of a nonconstant area.
2. Interfacial stresses
For interfacial systems, the total surface stress tensor, σs, has isotropic and deviatoric components,
where σαβ comprises the static surface tension of the interface formed between bulk phases α and β with surface excess Γ and σe is the extra stress term dependent on interfacial deformation (e.g., shear or dilatational strain). Both are dependent on surface concentration, Γ, which can be specified by the initial conditions for an insoluble monolayer or given by the Gibbs surface excess in equilibrium with a bulk phase. The former stress can produce an elasticity, known as Gibbs elasticity or Kgibbs, arising from a surface concentration gradient such that Kgibbs = dσαβ/d ln Γ, whereas the latter stress is responsible for rheological properties dependent on deformation. Gibbs elasticity is a thermodynamic property that is not tied to rheological properties dependent on strain or strain rates. However, complex interfaces with memory may retain nonequilibrium structures even at quiescent states, which would arise as an apparent Gibbs elasticity in the static term σαβ. Thus, for these complex interfaces, interfacial deformation history can play a role in true rheological properties (σe) while seemingly exist in static states (σαβ) unless the interface is at equilibrium. It is common to refer interfacial stress of a sample to being relative to a clean interface, , known as surface pressure, Π, such that
or alternatively written as
where Παβ is the static surface pressure. It should be noted that traditional surface pressure measurements using methods such as the Wilhelmy plate used here generally assume Παβ represents the equilibrium interfacial pressure and thereby neglect the influence of any deviatoric stress that may be due to the rheological response of the interface.
Examination of Eq. (9) suggests that a measurement of a surface pressure–area isotherm is analogous to a bulk pressure–volume isotherm measurement. However, a few distinctions must be made because common isotherm measurements for complex fluid interfaces contain rheological stress components due to microstructural memory. The term “equilibrium” isotherm may be considered a misnomer for complex fluid interfaces with memory as such measurements often include extra stresses other than the purely thermodynamic contribution.40,41 For example, for viscoelastic interfaces that display high elasticity, the measured surface pressure during continuous compression and expansion can contain both the equilibrium stress compressibility and extra rheological stress components at any finite, continuous compression speed. It should be noted that complicated mixed flow fields introduced in traditional Langmuir troughs include shear flow, which may manifest itself as an additional contribution in the measured surface pressure for these interfaces.13,31 Throughout this paper, we will refer to the isotherms as “apparent” in terms of it being experimentally observable when using the same protocol. This apparent isotherm may intrinsically include deviatoric stresses, depending on interfacial properties, which often manifest as a time-dependent behavior. A detailed discussion of these issues can be found in Verwijlen et al.31
3. Constitutive equations for finite elasticity
Interfacial rheological properties are expressed via constitutive equations that resolve the stress–strain relationship of fluid interfaces, which have been derived for viscous,42 viscoelastic,15,31 and elastoplastic19 interfaces as summarized elsewhere.40 To demonstrate how stress–strain relationships can be evaluated using the Quadrotrough instrument, we focus on the finite deformations of an elastic interface using the Hencky model while noting that the development of constitutive models for rheologically complex interfaces is an active area of research.19,40 To illustrate the basic use of the instrument in the following sections, we provide the working equations for the basic elastic model written in terms of Hencky strain in this section, and the reader is referred to a detailed discussion of this model and its limitations by Verwijlen et al.31
The finite Hencky strain constitutive model, , is defined as31,43,44
where rheological properties K and G are defined as the surface dilatational modulus and surface shear modulus in the model, respectively. Constitutive equations can be presented for each mode of deformation in the Quadrotrough to assess the rheological properties of the fluid interface using surface pressure measurements. As discussed by Carrozza et al.,19 alternative approaches to modeling elastoplastic interfaces include additional terms that compensate for geometric dilution effects for interfaces without connected networks (i.e., division of each term by the area change, ).
a. Pure dilation–stress constitutive equation.
where K is the surface dilatational modulus and λ2 = A/A0, or ratio of deformed and undeformed area. Equation (9) expresses the result that for pure dilation, both the rheological and equilibrium surface tension components are isotropic, which indicates that surface pressure measurements are Wilhelmy plate orientation independent. Moreover, note that the surface dilatational modulus K is assumed to be a constant material property but, in practice, may itself depend on the surface coverage and can exhibit strong hysteresis, which for an insoluble monolayer depends on the applied deformation.19,40
b. Planar shear stretch–stress constitutive equation.
where G is the surface shear modulus. The same considerations concerning the nature of the material properties and its dependence on deformation are shown in Eq. (14) for shear stretch as that considered for the dilatational modulus in Eq. (13).19,40 Importantly, shear breaks the symmetry of the interface such that when the Wilhelmy plate is oriented perpendicular to the major extensional axis, the measured force acting on the plate results in
while when the Wilhelmy plate is oriented parallel to the major extensional axis, the measured surface pressure is
This constitutive equation for planar shear stretch predicts that the measured surface pressure of an elastic interface depends on the Wilhelmy plate orientation.
c. Mixed strain–stress constitutive equation.
Mixed flow fields, such as those observed in a more traditional Langmuir trough, can affect surface pressure measurements due to rheological responses from both flow fields. Following the development in Verwijlen et al., for the specific case of a trough with motion of two symmetric barriers, the working equations become31
This result becomes the basis for a method to extract information about the bulk and shear moduli of an elastic interface from comparison of two Langmuir trough measurements with parallel and perpendicular plate orientations.13,31
III. INSTRUMENT CHARACTERIZATION
In this section, the Quadrotrough flow field as a function of the stress and strain rate are demonstrated using an inviscid microgel system to validate the interfacial deformation field generated by the instrument. Then, performance examples using the Quadrotrough are introduced to demonstrate how stress–strain relationships can be determined for both simple and more complex interfacial samples using a d-stearic acid model insoluble interface.
A. Interfacial flow field
Microgels are nano- or micro-sized gel particles made of cross-linked polymers that can be solvent-swollen and behave like soft colloids. These microgels demonstrate high interfacial activity and have displayed low shear viscoelasticity of G ∼ O(10−4) N/m and no detectable dilatational elasticity below the volume phase transition (VPTT = 31 °C).45,46 Poly(N-isopropylacrylamide) (PNIPAM) with 5% N,N′-methylenebis(acrylamide) (MBA) microgels (ρ ≈ 1000 kg m−3, hydrodynamic radius ∼ 45 nm) suspended in a chloroform solution were spread at the air–water interface by solvent evaporation to dampen capillary attraction between seeded polystyrene particles when measuring interfacial flow. Measurements were performed at 22 °C, where the microgels are in their swollen state. To study the compressional flow field, PNIPAM microgels were spread at 0.46 µm2/particle and compressed to 0.33 µm2/particle. To study the shear flow field, the microgels were sheared at 0.33 µm2/particle.
Expansion/compression mode was performed by moving the barrier fingers linearly at a speed of 1 mm/min such that the interfacial area ranged from 86 to 120 cm2. Experimental velocity fields show minimal particle motion near the center of the trough and high particle velocity closer to the barriers (Fig. 3). The calculated flow type parameter Λ* ∼ 1 shows that the trough generates spatially uniform dilatational and compressional interfacial flow (Fig. 3). The theoretical compression rate is nominally 3.3 × 10−4 s−1 and PIV measurements show similar rates from spatial averaging (Fig. 4). Further details of the calculation are provided in the supplementary material. We also examined the degree of shear and rotational flow in the trough and observe the noise to be an order of magnitude lower than the dilatational flow (Fig. 4). These results validate that the dilatational flow fields of a near inviscid interface during an experiment agree with the expectation values from mechanical motion of the motors.
Using the same methods, the planar shear stretch kinematics are reported in Fig. 5, where the compressional barrier finger speed was set to 1 mm/min and the extensional barrier finger speed was set to a speed to maintain a constant area inside the boundary of the barriers. In the first set of experiments, the interface was strained to at a nearly constant area and strained back to the original isotropic shape. Λ* ∼ −1 demonstrates that shear flow is dominant at the interface upon shearing forward and shearing back with a spatially uniform strain rate. The measured shear rate was consistent with the theoretically determined 3.7 × 10−4 s−1, as shown in Fig. 6. Additionally, the rotational and dilatational flows detected are an order of magnitude lower than the dominant shear flow and are considered noise. To further characterize the kinematic motion due to mechanical motors, planar shear stretch was imposed in the opposite direction by switching the extensional and compressional motor axes, as shown in Fig. 6. The resulting shear rates demonstrate the uniform control of all four motors to manipulate the elastic barrier equally. The kinematics of the trough at a shear strain , a higher strain, were also validated to show that the experimental shear rate matched the theoretical shear rate of 5 × 10−4 s−1 (Fig. S5). The measured shear rate was also demonstrated to be an order of magnitude higher than the rotational and dilatational noise. In summary, PIV measurements of tracer particle motion at a near inviscid interface validate the imposed dilation/compression and shear flow kinematics can be accurately achieved in the Quadrotrough.
The interface and bulk fluids are hydrodynamically coupled and the degree of transport coupling between the two systems depends on their separate rheological properties. Because the bulk subphase acts as a sink for interfacial momentum, the exact flow field at the interface is not easily isolated and its deformation in the given geometry must be evaluated. Two dimensionless parameters, viz., fluid Boussinesq and Reynolds numbers, are utilized to evaluate subphase effects.
A macroscopic Boussinesq number, Bq, is defined as47,48
where ηs and η are the interfacial and bulk viscosities, respectively, V is a characteristic velocity, Ls and Lb are characteristic length scales over which the velocity decays at the interface and bulk subphase, respectively, Ps is the contact perimeter based on the rheological probe at the interface, and Ab is the contact area between the probe and the bulk subphase. A characteristic length scale, a = LsAb/(LbPs), is defined to simplify parameters dependent on probe geometry. In other words, the Boussinesq number defines the interfacial drag relative to the subphase drag. For the Quadrotrough, this characteristic length scale a is estimated as the average length from the barrier to the center of the trough and is a function of area, where the maximum length is amax ≈ 154 mm and amin ≈ 89 mm. For example, in the case of a PNIPAM microgel insoluble monolayer considered here, the minimum Bq ≈ 6 for estimated property values of ηs ≈ 10−4 Pa s m and η ≈ 10−3 Pa s indicates a finite coupling of interfacial with bulk flows. Ideally, Bq ∼ O(102) is desired to completely neglect the effects of bulk flow.49
The interfacial Reynolds number, Res, enables evaluating the subphase convection effects to the interface and is defined as
where ρs is the density of the interfacial material (kg m−3), V is the velocity at the interface proportional to barrier velocity, a is a characteristic length scale dependent on geometry, and η is the bulk viscosity. Again for the PNIPAM interface, assuming a density similar to that of water, ρs = 1000 kg m−3, at very low barrier speeds of 1 mm min−1 Res ∼ 1–2 such that the assumption of low Reynolds number holds true, no subphase turbulence is expected in experiments and subphase convection is minimal at low barrier speeds.
IV. PERFORMANCE EXAMPLES
The Quadrotrough enables surface rheological characterization and further information about the structure of the interface can be obtained by its implementation in situ on a neutron reflectometer or coupled with Brewster angle microscopy. To demonstrate these modalities, experimental results are shown for two interfaces with very different rheological behaviors, inviscid and viscoelastic, undergoing well-defined shear and dilatational flows. These model interfaces help demonstrate rheological differences, as detected by surface pressure measurements, when undergoing pure shear and pure dilatational strains at the same mean molecular area (MMA), which is often associated with a particular phase behavior (e.g., liquid-expanded, liquid-condensed, solid-like). The existence of such phases is confirmed by in situ Brewster angle microscopy. Further, by implementing the new rheo-MAGIK configuration of combined Quadrotrough and neutron reflectometry, we demonstrate a technique capable of resolving the structure and rheology of an interface upon systematic processing, i.e., interfacial compression and shearing.
A. Apparent surface pressure isotherms at the air–water interface
To validate the dilatational/compressional mode of the Quadrotrough, the apparent surface pressure isotherms generated are compared to those observed in a traditional Langmuir trough, specifically the ribbon trough setup (KSV NIMA). The expected outcome is that an inviscid interface will generate comparable surface pressure isotherms between the two trough geometries, but a viscoelastic interface will be sensitive to the mixed flow field in a Langmuir trough. d-Stearic acid on a pure air–water interface displays negligible shear elasticity and, hence, is referred to as the inviscid interface under shear deformations. With negligible shear moduli, the measured surface pressure of d-stearic acid undergoing dilatational deformation only contains two contributions: dilatational viscoelasticity and compressibility, Παβ [Eq. (11)];31 hence, the apparent isotherm is unperturbed by the shear strain contributed in the mixed flow field and the interface displays comparable isotherms that are independent of Wilhelmy plate orientation [Fig. 7(a)]. The deviation of isotherm slope around 18 Å2 per molecule and below, where the stearic acid interface is in the solid-like phase, is speculated to derive from the different compression rates in each trough attributed to the measured dilatational viscoelasticity. In contrast, for a d-stearic acid monolayer over a 10−4M aluminum nitrate [Al(NO3)3] subphase, there is detectable shear elasticity characteristic of a viscoelastic interface. Because the viscoelastic interface exhibits both shear and dilatational elasticity, it has a convoluted rheological response to the complex interfacial strain field in the Langmuir trough, resulting in two distinct apparent surface pressure isotherms depending on plate orientation to barrier movements [Fig. 7(b)].
With the ability to deform the interface in a pure dilatational manner, the Quadrotrough produces surface pressure isotherms containing only contributions from the dilatational viscoelasticity and compressibility and avoids effects from the additional shear response produced in the Langmuir trough kinematics (Fig. 7). The isotherm measured under pure compression in the Quadrotrough exhibits an overall higher surface pressure as small surface areas compared to isotherm measurements in the Langmuir trough with both Wilhelmy plate orientations. Because the surface pressure measured in the Langmuir trough should contain contributions from both shear and dilatational rheological responses, the measured isotherms generated in the Langmuir trough must underestimate the interfacial dilatational modulus.31 Interestingly, this surface pressure difference at small areas arises from the onset of early mechanical buckling in Langmuir trough experiments due to the added shear component of the mixed flow field. This buckling effect is an instability that occurs where interfacial molecules no longer reside on the same interfacial plane and exhibits a plateaued or decrease in surface pressure upon further compression.50 Pepicelli et al. observed similar delay of buckling when compressing a glassy polymer interface of poly(tert-butyl methacrylate) (PtBMA) in the radial trough, where isotropic compression resulted in a “chaotic buckling” not detected in Langmuir trough measurements.15
The surface pressure isotherms observed in the Quadrotrough are as expected, Wilhelmy plate orientation independent, as shown in Fig. 8, for both an inviscid and viscoelastic interface. These apparent surface pressure isotherms further illustrate the value of the homogeneous compressional strain field in the Quadrotrough, which can elucidate the pure dilatational rheological response of the interface. To further separate the dilatational viscoelastic response from the compressibility contribution, one needs to obtain the compressibility term independently at various surface concentrations, as discussed in Sec. IV B.15,31
B. Compressibility surface pressure: A thermodynamic variable
To best determine the interfacial compressibility within an observable time frame, a series of step compressions to various mean molecular areas (MMA, i.e., area per molecule) are performed in the Quadrotrough and surface pressure relaxations are observed between compressions within a finite observation time. Interfaces with long surface pressure relaxations are fit to determine Παβ(Γ), as described elsewhere [Eq. (11)].15,40 Figure 9 is an illustration of the surface pressure isotherms created from continuous compression and step compression, where both the start of surface pressure relaxation (t = 0) and relaxed surface pressures (t = ∞) are plotted as points to observe meso-equilibrium surface pressures at quiescent phases.
d-Stearic acid at the air–water interface goes through phase transformations when compressed along the isotherm at 22 °C that are denoted by distinct changes in slope: liquid-condensed-gaseous phase (LC-G) at high MMA, liquid-condensed phase with alternating tilts (L2 and OV) at mid-MMA, and solid-like phase (S) at low MMA (Fig. 9).15,51 The surface pressure relaxation behavior varies systematically in the distinct phases, where negligible relaxation occurs at low MMA, mid-level relaxation is seen in the liquid-condensed regions, and high relaxation is seen based on dynamic nucleation occurring in the solid-like regions.52 On the other hand, the isotherm for d-stearic acid at an air–aqueous aluminum nitrate [Al(NO3)3] interface does not show distinct changes in slope; hence, no obvious phase transitions are apparent [Fig. 9(c)]. Rather, this interface is known to form an elastic, gel-like interface.53 Low level surface pressure relaxations occur at the first upturn in the isotherm at ∼23 Å2. At low MMA, the material becomes jammed and shows minimal surface pressure relaxation due to jamming effects [Figs. 9(c) and 9(d)]. These surface pressure relaxations enable deconvoluting the rheological and compressibility components of surface pressure that are not visible in the continuous compression isotherms. Using the Quadrotrough helps identify regions on the isotherm where the interface has a rheological response separate from the intrinsic Gibbs elasticity, Kgibbs, due to surface concentration differences [see Eq. (9)].41,54
C. Interfacial shear deformation
The Quadrotrough can also apply a pure shear strain at the interface at any desired surface concentration while maintaining a constant enclosed area at small strains within the moving boundary. This unique function allows one to investigate interfacial surface pressure differences arising from rheological and microstructural rearrangements. Furthermore, the trough can apply shear rates comparable to or below the minimum threshold in other interfacial rheometry methods, which is necessary for probing delicate interfaces.55
1. Experimental measurements
Surface pressure measurements taken during the shear stretch process help elucidate the shear strained behavior of interfacial materials at various surface concentrations as well as provides new insights into the rheological properties of the interface in different phases. To observe these effects, a step strain protocol was implemented using shear mode A in conjunction with surface pressure measurements as follows: compression, shear forward A (+γ), and shear backward A (−γ) [Fig. 1(a)]. Relaxation times were allowed between strain steps.
To help orient where measurements were taken on the isotherm, continuous compression, stepwise compression, and shear strain experiments are all overlaid in Figs. 10(a) and 10(c). For stepwise compression experiments, the starting surface pressure at each finite relaxation (t = 0) and near equilibrium surface pressure values (t = ∞) are plotted. For shear strain experiments, near equilibrium surface pressure values are plotted in addition to the stepwise compressions from the first step in the protocol. Transient shear strain measurements are plotted in Figs. 10(b) and 10(d) for d-stearic acid on an air–water interface and air–aqueous Al(NO3)3 interface, respectively.
For d-stearic acid at the air–water interface, the inviscid interface shows little shear elasticity. On the other hand, the data in Fig. 9(b) show that dilatational elasticity is present in LC and OV regions from mid-level surface pressure relaxations and is dominant in the solid-like region due to high level relaxations. With these observations in mind, it is expected that shear straining this particular interface would not perturb the residual surface pressure relaxations originating from compressional strain. An example of surface pressure measurements using the step strain protocol is shown in Fig. 10(b) in the LC region of 21.2 Å2, where the interface reaches equilibrium at long times. There is minimal deviation in the relaxed surface pressure upon holding the strained interface (shear forward) and upon returning the interface to its initial state (shear backward). On the other hand, when a finite elastic interface, such as d-stearic acid on an aqueous Al(NO3)3 interface, is subjected to step shear strain at 13.9 Å2 [Fig. 9(d)], the surface pressure drops significantly, which then does not recover to its original surface pressure upon returning to its initial state. This irreversible behavior indicates that shearing relaxes trapped stresses in the interface due to the initial compression. For more complex fluid interfaces, we demonstrate how isolating the shear and dilatational response helps elucidate meso-equilibrium phases that are created and dependent on a deformation history.
2. Linear velocity strain limitations
Currently, for the shear stretching mode, stepper motors apply linear velocities to displace the elastic barriers along the principal axes nearing constant area. Outside the small strain limit, this movement can cause deviations from constant area for finite strains of large deformation and appear during transient shear stretching as a surface pressure artifact.
For example, at an area of 5832 mm2, two interfacial nominal strains of 26.2% and 37.7% were applied using a constant linear velocity of 1 mm/min for the motors moving along the compressional axes. In Fig. 11, the deviation from constant area is evaluated as ΔA/A0 = (A − A0)/A0, where A and A0 are the deformed and reference state (t = 0) areas, respectively. Figure 11 shows that the higher the finite strain, the larger the deviation from constant area, which appears as a dilatational artifact in surface pressure measurements.
At large deformations, the dilatational artifact is coupled with the shear rheological response in transient surface pressure measurements upon shear stretching forward and backward. We evaluate the degree of this artifact using d-stearic acid at the air–water interface, which has a negligible shear response, undergoing two nominal strains of 26.2% and 37.7% by plotting the transient surface pressure change as
where is the reference starting surface pressure at t = 0 when straining the interface, either as a shear forward or shear backward (Fig. 12). Predictions of the range of evolving surface pressure deviation over time are also shown using the apparent dilatational modulus, Kapp, and Gibbs modulus, Kgibbs, of the interfacial sample such that
where ΔΠi(t) = Πi(t) − Π0 is the normalized surface pressure difference relative to a reference surface pressure, Π0, associated with a reference configuration, A0. Details of the transient surface pressure measurements and associated determination of the moduli are provided in the supplementary material (Fig. S6–S7). Measurements are performed on inviscid interface of d-stearic acid at the air–water interface. The results presented in Fig. 12 show that the surface pressure deviation in experimental measurements follows the same trajectory as the expected dilatational artifact for both shear strain cases. Additionally, the predicted change in surface pressure due to the artifact is higher for the larger nominal strain case [Fig. 12(b)] in comparison to the smaller strain case [Fig. 12(a)], as observed experimentally. Slight deviations seen in the predicted and calculated transient surface pressure changes are attributed to the small shear response of the interface, as there is still a detectable change in surface pressure at rest, Παβ, after shear straining (Fig. S6). Additionally, shear straining a clean air–water interface showed negligible changes in surface pressure measurements due to hydrodynamic effects (Fig. S8). Therefore, for this inviscid interface, the dilatational artifacts dominate surface pressure measurements at large deformations, which are coupled with transient shear responses. However, near equilibrium surface pressures at held strains can still give insight on the altered interfacial properties due to deformation history. Further improvements are being made to an improved setup with motors that can hold a fixed strain rate at constant area in order to isolate the shear modulus [Eqs. (15a) and (15b)].
3. Surface pressure dependence on plate orientation for pure shear mode
Equations (15a) and (15b) predict that surface pressure measurements upon shear straining are dependent on Wilhelmy plate orientation. In the limit of small deformations, the dilatational artifact seen at large strains is mitigated and the dominating shear response signal of the interface is detected. We observe these effects using both d-stearic acid at the air–aqueous Al(NO3)3 interface and d-stearic acid at the air–water interface. The first step strain protocol (protocol 1) implemented is as follows: compression, shear forward A (+γ), shear backward A (−γ), shear forward B (−γ), shear backward B (+γ) [refer to Fig. 1(a)]. Relaxation times were allowed between strain steps. The second step strain protocol (protocol 2) implemented is the reverse order of the first protocol and is as follows: compression, shear forward B (−γ), shear backward B (+γ), shear forward A (+γ), shear backward A (−γ). Between the two step strain protocols, the interface is expanded out to the maximum area and then compressed to the same area again to reset the interface as a pre-shear step. A nominal strain of is applied to the interface as a small deformation and surface pressure evolution is observed.
When the two step strain protocols () are implemented for d-stearic acid at the air–aqueous Al(NO3)3 interface, we immediately observe the opposite trends in surface pressure measurement between the two experiments due to the Wilhelmy plate orientation (Fig. 13). As seen in Fig. 13(a), when +γ is applied after compression, the surface pressure immediately drops as predicted in Eq. (15) where Π⊥ is detected. When a shearing backward A (−γ) is applied consecutively after due to shearing back, the opposite is seen where the surface pressure increases as Π‖ is detected [Eq. (16)]. Since shearing forward B is a continuing −γ, the surface pressure continues to increase, which then drops again when +γ is applied during shear backward B. The opposite effect is seen for the second protocol when applying −γ first, where we see the immediate increase in surface pressure [Fig. 13(b)]. These observations validate Eqs. (15a) and (15b) as to how surface pressure measurements depend on the Wilhelmy plate orientation relative to the principal axes if the interface has a detectable shear modulus.
When d-stearic acid at the air–water interface is subjected to the same two step strain protocol (), we observe negligible surface pressure dependence based on Wilhelmy plate orientation due to the negligible shear response of the inviscid interface (Fig. 14).This behavior is in stark contrast to that of the highly elastic interface of d-stearic acid at the air–aqueous Al(NO3)3 interface. These observations further validate the hypothesis that the anisotropic components of surface pressure extracted by orthogonal Wilhelmy plate orientations reflect the shear rheological properties of the interface. Furthermore, because the surface pressure measurements are equal and oppositely dependent on deviatoric shear stress based on plate orientation with respect to the major principal axis [Eqs. (15) and (16)], these results show that both the dilatational and shear modulus can be extracted from one Quadrotrough experiment, unlike the procedure for Langmuir trough experiments, which require comparing two measurements with orthogonal plate orientations.13,15
D. Quadrotrough coupled with Brewster angle microscopy
The Quadrotrough used in tandem with Brewster angle microscopy provides insight into mesoscale structural changes under both dilation and shear strain kinematics. We demonstrate how the complementary techniques work by showcasing the shear strained d-stearic acid at the air–water interface and the air–aqueous Al(NO3)3 interface.
Shear strain was applied for d-stearic acid at the air–water interface at the liquid-condensed phase region, also known as tilted condensed phase (Fig. 15). The initial microscopy images before shearing forward and shearing backward are shown in Figs. 15(a) and 15(c), respectively, for comparison. Upon straining the interface, the liquid-condensed domains align with the extensional axis [Fig. 15(b)], which then revert to the opposite direction upon shearing backward [Fig. 15(d)]. Because this interface is inviscid, shear straining the interface minimally alters the surface pressure in this region, as seen in Fig. 10(b), but the structure of the interface is easily altered upon straining by its fluid-like nature. These results agree with dodecanoic acid monolayers, another fatty acid, which also produce flow-induced anisotropy in the liquid-condensed phases.56
d-Stearic acid at the air–aqueous Al(NO3)3 interface exists as heterogeneous slab-like structures that fuse together upon compression (Fig. 16). When shear strain is applied, the interface also rotates toward the extensional axes, as visualized by the heterogeneous structures [Fig. 16(b), Multimedia view] in comparison to the initial phase [Fig. 16(a)]. A long-range fracture appears from defects in the slab network that runs along the height of the image and is irreversible [Figs. 16(c) and 16(d), Multimedia view]. The fractured effects of the interface upon strain can be seen more clearly in multimedia view. It should also be noted that the contrast between domains is lessened over time, which is attributed to relaxation effects at the interface. These Brewster angle microscopy images enable identifying the structural origin of surface pressure differences upon shear straining as due to mesoscale domain orientation realignment or network structure breakdown.
E. Rheo-MAGIK: Interfacial rheology coupled with neutron reflectometry
The Quadrotrough instrument can be coupled with a neutron reflectometer to resolve the out-of-plane interfacial structure with a well-defined strain history, known as rheo-MAGIK. To demonstrate this, the Quadrotrough was implemented on a horizontally modified reflectometer, known as MAGIK, at the NIST Center for Neutron Research (NCNR, Gaithersburg, MD). While the reflectometer modifications are still undergoing improvements independently, the first order measurements help verify the capabilities of this new technique when compared to a well-established method of coupling a Langmuir trough to another horizontal reflectometer, the NG7 Reflectometer, at NCNR (Fig. 17). Due to the lower signal-to-noise ratio for rheo-MAGIK measurements from a lower flux, spectra for these samples are plotted with Gaussian filtered spectra and confidence interval error bands; unfiltered data are provided in the supplementary material (Fig. S11). d- Stearic acid at the air–aqueous Al(NO3)3 interface is used as the sample for measurements, where the aqueous subphase is prepared as a 78:22 (vol. % H2O:D2O) mixture with ρ = 1 × 10−6 Å−2 and show agreement between the rheo-MAGIK and NG7 Reflectometer measurements [Fig. 17(a)]. Reflectivity measurements are made under increasing interfacial compression in both setups and show an increase in intensity due to an increase in deuterated stearic acid area density or decrease in mean molecular area [Figs. 17(b) and 17(c)]. It should be briefly noted that partial curves are plotted for some spectra due to beam footprint constraints and its construction is explained in the supplementary material. Additionally, model fits using a two-slab model of the same d-stearic acid interface measured on the NG7 Reflectometry are added as reference data. The favorable comparison showcases the ability to quantify the molecular area and length scale of the interface using the Quadrotrough on a neutron reflectometer. The details of the analysis of the spectra are presented in the supplementary material (Fig. S12). The molecular resolution of the interface will improve with ongoing developments of the MAGIK horizontal reflectometer.
Besides undergoing compression, the d-stearic acid interface was also subjected to the step shear strain A protocol [Fig. 1(a)] at ∼16 Å2/molecule and , where reflectivity measurements were taken under compression, shear forward A, and shear backward A (Fig. 18). Spectra measurements are shown by error bands from confidence intervals and indicate no significant differences in spectra due to shear strain. This indicates that the molecular structure in the surfactant-rich gel domains, which dominate the scattering intensity, remains constant along the isotherm. Rather, as shown by Brewster angle microscopy, the significant structural rearrangements occur on the mesoscale (Fig. 16).
Future improvements to the rheo-MAGIK setup will allow measurement across a larger Q-range with higher resolution. Improvements in progress for the horizontal reflectometer include constraining slit width, modifying slit placement for background reduction, and improving precision of slit motor positions to minimize offset. With these improvements, this technique can help determine the molecular structure and absolute surface concentration upon interfacial compression and strain. Nevertheless, these initial measurements show promising results that demonstrate the interfacial length scale impacted by shear strain and compression, which is potentially useful for complex fluid interfaces that need this multi-length scale structural information to form comprehensive conclusions about material behavior.
V. CONCLUSIONS AND FUTURE WORK
We describe the development of the Quadrotrough, a new interfacial trough instrument with the ability to monitor surface pressure changes under pure dilatational and pure shear strains. We validated the instrument using two samples with different interfacial mechanics, an inviscid and viscoelastic interface, to illustrate the behavior of surface pressure under different strains and strain histories. Coupling this instrument with structural measurement techniques, such as particle image velocimetry and neutron reflectometry, enables identifying the structural changes, spanning from the mesoscale to atomic length scales, responsible for surface pressure differences under well-defined strain histories. A near inviscid microgel interface at the pure air–water interface acts as a control sample validating the kinematics of the dilatational and shear strain deformation in the Quadrotrough by PIV. Experiments on a monolayer of d-stearic acid on pure water illustrate how isotropic compression leads to trapped stresses that can show long-relaxations depending on the specific surface phase. For more complex interfacial systems, such as the highly elastic gelled interfaces of d-stearic acid on an air–aqueous Al(NO3)3 interface, trapped stresses upon isotropic compression are shown to be relaxed by subsequent shear straining. The internal structures of the gel domains are largely independent of specific molecular area, while the dilatational interfacial properties depend on an evolving mesoscale microstructure in the plane of the surface. These results show that this newly developed instrument has the ability to resolve the structural origins of this strain-dependent behavior of complex fluid interface by the novel application of independently controlled strain and compression/dilation experiments. The instrument is available for use at NIST Center for Neutron Research (Gaithersburg, MD).
Ongoing developments are currently being made to improve the Quadrotrough and rheo-MAGIK techniques. Improvement in the mechanical motion controlling the flexible barrier will enable higher fidelity control of constant Hencky strain and shear rates. Improvements in rheo-MAGIK include modifications to the horizontal neutron reflectometer to narrow the beam footprint on the trough center and improve beam collimation control. The unique capabilities of the Quadrotrough open the possibility of investigating the true rheological properties of complex fluid interfaces by applying well-defined deformation histories. Coupled with accurate in situ structural metrology of both the in-plane mesoscale microstructure and out-of-plane molecular structure, we can create structure–property relationships for complex fluid interfaces for a more complete understanding of interfacial mechanical and thermodynamic behavior.
See the supplementary material for PIV derivation analysis and Quadrotrough mechanical deformation analysis, transient surface pressure measurements, Wilhelmy plate sensitivity to air–water interface, and rheo-MAGIK beam footprint and capabilities.
This article was prepared under cooperative Agreement No. 370NANB17H302 from NIST, U.S. Department of Commerce. We acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the neutron research facilities used in this work. This work was carried out with the help of facilities provided in part by the National Science Foundation under Agreement No. DMR-0944772. Certain commercial equipment, instruments, or materials (or suppliers, or software) are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose. We acknowledge support from the Swiss National Science Foundation, Grant No. 200021-165974. We would also like to thank Christian Furrer, Stephan Busato, Alexandra Alicke, and Cedric Gagnon for help during instrument and instrument parts development and Martina Pepicelli for assistance while learning interfacial rheology and useful discussions.
Conflict of Interest
The authors have no conflicts to disclose.
Y. Summer Tein: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Benjamin R. Thompson: Investigation (supporting); Writing – original draft (supporting). Chuck Majkrzak: Conceptualization (equal); Investigation (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Brian Maranville: Conceptualization (equal); Investigation (equal); Resources (equal); Software (equal); Writing – original draft (equal). Damian Renggli: Formal analysis (equal); Methodology (equal). Jan Vermant: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Norman J. Wagner: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.
deformed area state
reference area state
left Cauchy–Green deformation tensor
- , Dshear,
deviatoric part of rate-of-deformation tensor, shear strain rate tensor
rate-of-expansion tensor, dilatational rate-of-deformation
determinant of x tensor
interfacial deformation gradient tensor
- Fdil, θ
dilatational strain deformation tensor
- Fshear, γ
shear strain deformation tensor
surface shear moduli
surface dilatational moduli
apparent dilatational moduli
magnitude of x tensor
trace of x tensor
principal linear strain
principal stretch ratio
surface excess concentration
shear strain rate
flow type parameter
compressibility surface pressure between phases α and β
total surface stress tensor
interfacial/surface tension; compressibility stress
surface extra stress tensor; deviatoric stress
Hencky deviatoric stress tensor
dilatational strain rate, rate-of-expansion, rate of area change
dilatational strain, area change
velocity gradient tensor
While we use the conventional term “Langmuir” trough for the instrument and method of studying monolayers, we wish to acknowledge the seminal work of A. Pockels who developed the original method for studying monolayers that became the basis for Irving Langmuir’s work on this subject (see Ref. 21).