1. Introduction & Scope
Interfacial rheology investigates the mechanical response—deformation and flow—of the region (interphase or interface) between two immiscible phases, like oil and water or air and water. This field is crucial for understanding systems with extensive interfacial areas, such as emulsions and foams, where the properties of this boundary dictate overall behavior and stability.
Real systems often contain surface-active molecules (amphiphiles) such as surfactants, proteins, or asphaltenes, which adsorb at the interface, forming structures ranging from simple monolayers to complex viscoelastic films. The mechanical properties of these interfaces influence phenomena across diverse fields including food science, energy production (Enhanced Oil Recovery - EOR), pharmaceuticals, cosmetics, and biological systems [cite: 4-27].
2. Deformation Modes & Moduli
Two primary modes describe how interfaces deform:
- Shear: Tangential deformation occurring at a constant interfacial area. The resistance to this is quantified by the complex shear modulus, \(G^*\).
- Dilation (or Compression/Expansion): Deformation involving a change in interfacial area, often assumed at constant shape. The resistance is quantified by the complex dilational modulus, \(E^*\).
Interactive Deformation Modes
Shear
Constant Area
Dilation / Compression
Area Change
These moduli are complex numbers representing both elastic (storage) and viscous (loss) components:
Here, \(G'\) and \(E'\) are the elastic (storage) moduli, representing the energy stored and recovered per cycle of deformation (solid-like behavior). \(G''\) and \(E''\) are the viscous (loss) moduli, representing energy dissipated as heat per cycle (liquid-like behavior). The imaginary unit is \(i = \sqrt{-1}\). All these moduli have units of force per length, typically mN/m.
3. Historical Context
Early studies by Hadamard, Rybczynski, Boussinesq, Silvey, Frumkin, and Levich established foundational concepts [cite: 25, 37-40]. Boussinesq introduced the idea of a 2D "surface viscosity" to explain droplet sedimentation anomalies. Later, Frumkin and Levich correctly attributed these effects to the presence of surfactants, which create surface tension gradients when displaced during flow. This led to the understanding of the Gibbs-Marangoni effect.
4. The Gibbs-Marangoni Effect
This is a critical phenomenon in interfacial science. Gradients in interfacial tension (\(\nabla \gamma\)), typically caused by variations in surfactant concentration (\(\nabla \Gamma\)) along the interface, generate tangential stresses known as Marangoni stresses. These stresses drive fluid flow from regions of low tension to high tension (Marangoni flow).
In the context of dilational rheology, when an interface is expanded or compressed, the change in area alters the local surfactant concentration, creating tension gradients. The resulting Marangoni stress acts as a restoring force opposing the deformation. This effect is a primary contributor to the elastic dilational modulus (\(E'\)). It plays a major role in stabilizing thin liquid films (like those between bubbles in a foam or droplets in an emulsion) against drainage and rupture.
The relative importance of this effect can be characterized by the Marangoni number: \( M = E / \gamma \).
Note: It's important to distinguish the dynamic dilational modulus \(E^*\) from the equilibrium Gibbs elasticity \(E_G = \Gamma (\partial \Pi / \partial \Gamma)_T = -A (\partial \gamma / \partial A)_T\), which relates equilibrium changes in surface pressure (\(\Pi = \gamma_0 - \gamma\)) or tension (\(\gamma\)) to equilibrium changes in surface concentration (\(\Gamma\)) or area (\(A\)) [cite: 94-101]. \(E^*\) incorporates dynamic and kinetic effects.
5. Dynamic Aspects & Frequency Dependence
Interfacial rheological properties are often strongly dependent on the frequency (\(\omega\)) of the applied deformation. This dependence arises because the interface's response involves molecular relaxation processes with characteristic timescales (\(\tau\)), such as surfactant adsorption/desorption, diffusion along the interface or between the interface and bulk, and conformational changes of adsorbed molecules (e.g., proteins, polymers) [cite: 65, 86, 102-109].
Conceptually:
- At low frequencies (\(\omega \ll 1/\tau\)), molecules have time to relax or exchange with the bulk during deformation. Soluble surfactants can maintain near-equilibrium tension, reducing Marangoni stresses and leading to lower elasticity (\(E'\)) and predominantly viscous behavior (\(E'' > E'\)).
- At high frequencies (\(\omega \gg 1/\tau\)), molecules cannot relax or exchange significantly within a deformation cycle. The interface behaves more like an insoluble, elastic film, leading to higher elasticity (\(E'\)) and predominantly elastic behavior (\(E' > E''\)).
- Near the crossover frequency (\(\omega \approx 1/\tau\)), both elastic and viscous contributions are significant, and the loss modulus (\(E''\) or \(G''\)) often exhibits a maximum.
Viscoelastic Response vs. Frequency
Adjust the relaxation time (\(\tau\)) to see how the response curve shifts relative to frequency (\(\omega\)). The crossover (G' ≈ G'') occurs near \(\omega = 1/\tau\).
Note: This plot illustrates the concept using a single Maxwell element model. Real interfacial systems often exhibit more complex behavior.
Strain vs. Stress Response
Adjust the phase angle (\(\varphi\)) between strain (input) and stress (response). \(\varphi=0^\circ\) is purely elastic, \(\varphi=90^\circ\) is purely viscous.
Boussinesq Number (Bo) Estimator
Estimate the conceptual Boussinesq number (\(Bo = \eta_s / (\eta_b \cdot L)\)) comparing interfacial (\(\eta_s\)) to bulk (\(\eta_b\)) viscous effects over length \(L\).
Note: This definition of Bo applies primarily to steady flows. Frequency-dependent or oscillatory flow definitions also exist. Values \(Bo\) > 1 are generally considered favorable, indicating that interfacial viscosity effects are strong enough relative to bulk viscosity to be reliably measured and potentially dominate the response. Extremely high \(Bo\)≫1 suggests interfacial dominance, simplifying analysis but potentially posing challenges for the experimental technique itself if the interface becomes very rigid.