**Interfacial rheology, surface rheology**

*Interfacial or surface rheology is a branch of interfacial analysis which is concerned with the reaction of the interfacial tension (IFT) between two liquids respectively the surface tension (SFT) of a liquid to the deformation of an interface. It is relevant for studies on surfactant-stabilized systems such as emulsions and foams.*

**Viscoelastic behavior of interfaces**

The SFT/IFT *γ* is the amount of work involved in increasing the area of a given interface. For pure liquids, this amount of work is proportional to the change of area independent from its extent or speed, so that the SFT/IFT is a substance constant.

However, if surfactants are present, the SFT/IFT depends on the concentration of molecules adsorbed at the interface (surface excess concentration Γ). When dilating the interface, the concentration with respect to the area decreases spontaneously so that the SFT/IFT increases. This process is referred to as the *elastic behavior* of an interface. It is reversible, i.e. the initial value of the SFT/IFT is reached when the original area of the interface is restored. For simplification, the following figure illustrates this process using an insoluble surfactant. The molecules are only at the interface but not in the bulk phase so that no equilibration of the concentration occurs.

If surfactant molecules are in the bulk phase, diffusion to the interface and adsorption thereat occurs. This process leads to a time-dependent, irreversible decrease of the SFT/IFT that is referred to as *viscous behavior*. Both elastic and viscous processes take place at the interface of a surfactant solution so that its reaction to dilatation is called *viscoelastic*.

**Physical description**

The viscoelastic reaction of an interface when stretching it is expressed by the complex viscoelastic modulus E*. It consists of the elastic modulus E' and the viscous modulus E'':

The viscoelastic modulus E is the absolute value of the complex viscoelastic modulus E*:

As a whole, E' and E'' together characterize the effect of the dilatation of an interface on the SFT/IFT, whereby E' stands for the impact of the changed area on the concentration change, while E'' describes the time-dependent change due to gradual concentration equilibration.

**Significance**

Emulsions and foams are systems with a large inner surface, which are created and stabilized with the help of surfactants. The viscoelastic modulus describes the increase in SFT/IFT when dilating an interface, i.e. the resistance of the interface against such deformation. E' can be interpreted as the elastic counterforce of the system, while E'' describes how fast the initial value of the SFT/IFT is restored after deformation. Various correlations between these quantities and different aspects of foam and emulsion stability have been found.

**Measurement**

When displaying the moduli E*, E' and E'' in the complex number plane it becomes clear from the real part *Re* and the imaginary part *Im* that E' and E'' result from trigonometric functions.

During the measurement, the interface area of a pendant drop or a gas bubble, which is recorded with a camera, is changed periodically and exactly sinusidoidally by means of a dosing drive that operates with extreme precision. In doing this, the SFT/IFT is measured using drop shape analysis with a Young-Laplace fit to obtain its dependency from surface area *A* and time *t*. Due to the underlying sine function, the angle φ (see figure above) corresponds to the phase shift between the change in surface area and the SFT/IFT, which also displays a sine-curve progression.

Taking the relation between trigonometric functions and the exponential function into account, the interfacial rheological quantities E*, E', and E'' can be calculated (ω = angular frequency, A_{0} = mean value of the area, Δγ = amplitude of the SFT/IFT, ΔA = amplitude of the area).

**Bibliography**

- S.C. Russev, N. Alexandrov, K.G. Marinova, K.D. Danov, N. D. Denkov, L. Lyutov, V. Vulchev, C. Bilke-Krause, “Instrument and methods for surface dilatational rheology measurements”, Rev. Sci. Instr. 2008, 79, 104102.
- F. Thomsen, "Stretching exercises for drops". KRÜSS Application Report AR 246.

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