# Dilational Rheology of Fluid/Fluid Interfaces: Foundations and Tools

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## Abstract

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## 1. Introduction

## 2. Interfacial Rheology: Foundations

_{0}. However, the use of the interfacial tension only provides an equilibrium description of the true picture of the interface and, as previously mentioned, this is not enough in most practical applications of fluid/fluid interfaces. In fact, fluid/fluid interfaces with technological and scientific relevance are commonly subject to external mechanical perturbations that result in a modification of their size or shape [30]. The understanding of the response of fluid/fluid interfaces to mechanical processes is essential, because even the simplest deformation processes can yield to very complex responses, including multiple dynamic processes or deformation mixing different interfacial modes [31].

_{s}as a combination of two contributions: (i) the interfacial energy, which accounts for the energetic cost associated with the presence of a fluid interface of a fixed area, and provides information of any process changing the interfacial concentration and affecting the interfacial tension, and (ii) the Marangoni stresses emerging as a result of spatial interfacial tension gradients [9,10]. Thus, it is possible to define the interfacial stress tensor according to the following expression.

_{s}is the surface unit tensor, and $\gamma $ is the interfacial tension, which is a state variable depending on the interfacial concentration and temperature. The second contribution to the interfacial stress tensor is the anisotropic tensor or interfacial extra stress (Τ

_{ij}), and accounts for the energy required to deform the interface [1,37]. The surface stress tensor can be considered as a 2D second-order symmetric and tangential tensor embedded in 3D space [9].

## 3. Interfacial Dilational Rheology: General Aspects

_{0}account for the temporal evolution of the interfacial pressure and the initial interfacial pressure, respectively. Thus, it possible to define the time evolution of the viscoelastic dilational modulus in terms of the following expression:

_{s}) and imaginary components (E

_{v}), which correspond to the storage and loss moduli, respectively [43,44]. Assuming an oscillatory deformation of small amplitude and a fixed frequency ω, it is possible to define the complex dilational viscoelastic modulus as complex magnitude according to the following expression:

_{0}, defined as:

## 4. Rheological Tools for Evaluating the Response of Planar Fluid/Fluid Interfaces against Dilation

#### 4.1. Experimental Tools

#### 4.1.1. Experimental Techniques

#### Drop/Bubble Shape Tensiometers

^{−3}–0.2 Hz [55].

#### Capillary Pressure Tensiometers

#### Langmuir Troughs

#### Wave Damping

^{5}Hz) that can be probed by the evaluation of the damping of waves generated at fluid/fluid interfaces, and the contactless character of this type of technique, the interpretation of the data is not straightforward, which limits their applicability [72,76]. Recently, Slavchov et al. [68] reviewed the most fundamental aspects of capillary wave damping as a tool for evaluating the mechanical response of fluid/fluid interfaces and other of its potential applications. Moreover, they analyzed some recent theoretical developments on the use of capillary waves. Rajan [77] has recently solved the problems associated with the use of the damping of interfacial waves for the determination of the rheological properties of liquid/liquid interfaces, providing results of the interfacial elasticity and viscosity of water/oil interfaces.

#### 4.1.2. Experimental Methods

#### Stress Relaxation Experiments

#### Creep Experiments

_{0}is suddenly compressed, i.e., as fast as possible, until it reaches a desired interfacial pressure value; then, the surface pressure is constantly maintained by changing the interfacial area. Thus, the excess of interfacial pressure is adjusted by considering the area relaxation process, allowing one to define the creep compliance as [85]:

#### Oscillatory Area Experiments

#### Surface Waves Experiments

_{B}is the Boltzmann constant and η the subphase viscosity. q and m define the wave-vector and the diffraction order, respectively. E is the complex dilational modulus (compression+shear), and D(ω) the surface wave dispersion relation defined by the following expression [36]:

^{3}–10

^{6}Hz [93].

^{3}Hz) [30].

#### 4.2. Theoretical Models

## 5. Evaluation of the Mechanical Relaxation Spectrum from Dilational Rheology Experiments

^{−3}–10

^{3}Hz [104]. This required combining an oscillatory drop tensiometer, a capillary pressure tensiometer, and an electrocapillary wave instrument to access the whole frequency range. Moreover, there are several studies where the combination of different techniques has provided information about the mechanical relaxation spectrum, and information regarding the real and imaginary part of the viscoelastic modulus values from the whole frequency range is not available [106,107,108].

## 6. Non-Linear Dilational Interfacial Rheology

## 7. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Sketch of different surface relaxation modes: (

**a**) in-plane-modes (dilation and shear) and (

**b**) out-of-plane modes (bending and splaying). Reprinted from Maestro and Guzmán [34], with permission under Open access CC BY 4.0 license, https://creativecommons.org/licenses/by/4.0/ (accessed 22 August 2022).

**Figure 2.**Sketch of the typical material response for a viscoelastic layer undergoing a single relaxation process upon the application of a dilational stress. Adapted from Mendoza et al. [30], with permission from Elsevier, Copyright (2014).

**Figure 3.**Sketch of the typical dependences of the (

**a**) real and (

**b**) imaginary parts of the viscoelastic modulus for fluid/fluid interfaces presenting different relaxation processes. The dashed lines correspond to a system presenting a relaxation process characterized by a relaxation process defined in terms of the Lucassen-van der Tempel model, and the continuous lines represent the typical behavior of a fluid/fluid interfaces where a relaxation process affecting only the interface is coupled to a Lucassen-van der Tempel-like relaxation. Reprinted from Liggieri et al. [104], with permission from Royal Society of Chemistry, Copyright (2011).

**Figure 4.**Summary of some of the most extended methodologies for evaluating the dilational response of fluid/fluid interface, together with the accessible frequency ranges for such techniques.

**Figure 5.**Re-scaled frequency dependences of the elastic modulus and loss tangent for chitosan-anionic surfactant mixtures with different surfactant concentrations (symbols of different color represents the set of data for solutions with different concentrations) and a fixed chitosan concentration of 2 g/L (pH 4.5 and ionic strength 120 mM NaCl). The symbols are the experimental results, and the solid lines represent the best fit of the experimental curves to a model, including a Lucassen-Van der Tempel-like relaxation process and a relaxation process affecting the interface. Adapted from Akanno et al. [51], with permission from Royal Society of Chemistry, Copyright (2020).

**Figure 6.**Definition of minimum and large-strain moduli. (

**a**) Minimum and large deformation dilational moduli in extension. (

**b**) Minimum and large deformation dilational moduli in compression. Adapted from Sagis and Fischer [113], with permission from Elsevier, Copyright (2014).

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**MDPI and ACS Style**

Guzmán, E.; Maestro, A.; Carbone, C.; Ortega, F.; Rubio, R.G.
Dilational Rheology of Fluid/Fluid Interfaces: Foundations and Tools. *Fluids* **2022**, *7*, 335.
https://doi.org/10.3390/fluids7100335

**AMA Style**

Guzmán E, Maestro A, Carbone C, Ortega F, Rubio RG.
Dilational Rheology of Fluid/Fluid Interfaces: Foundations and Tools. *Fluids*. 2022; 7(10):335.
https://doi.org/10.3390/fluids7100335

**Chicago/Turabian Style**

Guzmán, Eduardo, Armando Maestro, Carlo Carbone, Francisco Ortega, and Ramón G. Rubio.
2022. "Dilational Rheology of Fluid/Fluid Interfaces: Foundations and Tools" *Fluids* 7, no. 10: 335.
https://doi.org/10.3390/fluids7100335