## What is surface and interfacial tension?

All liquids – water, organic solvents, oils, and so on – have strong intermolecular cohesive forces. Molecules in the bulk experience this attractive force in all directions. However, liquid molecules at the surface formed between the liquid and a gas, such as air, do not have neighbors of the same kind on all sides like in Fig. 1. Molecules at the surface experience a net attractive force towards the bulk of the liquid, resulting in a surface “film” that makes it more difficult to move an object through that surface than through the bulk. This “film” acts like an elastic membrane with a tension force acting in all directions along the surface, i.e. surface tension. When an interface forms between two immiscible liquids, a similar situation occurs. In this case it is called interfacial tension.

**Table of Contents**

Surface and interfacial tension are typically represented by the lowercase Greek letter gamma, γ (sigma, σ, is also sometimes used). Several units are traditionally used, the most common being mN/m or the equivalent dyn/cm, which have dimensions force/length. Sometimes the equivalent unit mJ/m^{2} is used, which has dimensions of energy/area. Thus, surface and interfacial tension can be interpreted in two ways: as a tension force acting along all directions of the surface per unit length, or as the energy required to increase the surface area. In both interpretations, the surface wants to minimize its surface area or potential energy. Therefore, a surface with no (or negligible) external forces will form a sphere which has the smallest surface area to volume ratio.

## How is surface and interfacial tension measured?

Techniques to measure surface and interfacial tension are separated here into two main categories: force tensiometry where the force the surface imparts on a probe is directly measured; and optical tensiometry where the profile of a drop or bubble extracted from an image is fit to a theoretical equation.

### Force tensiometry

In force tensiometry the force experienced by a probe at the liquid surface is directly measured. With the Attension Sigma tensiometers, three techniques are available: the Du Nouy ring method, the Wilhelmy plate method, and the platinum rod. In each technique, the respective probe is hung from a sensitive balance while being brought in contact with the liquid. The size and shape of the probe, contact angle between the probe and liquid, and in some cases liquid density, are incorporated into models to relate the measured force to surface or interfacial tension. To simplify the model, Attension probes are made of a platinum/iridium alloy which has a high surface energy and allows the contact angle to go to zero.

##### Du Nouy ring method

The Du Nouy ring method brings a platinum ring in contact with the surface or interface [1]. First, the ring is fully submerged and then slowly lifted up to form a meniscus-like in Fig. 2. During this process, the force of the meniscus pulling on the ring is measured with the balance as shown in the plot in Fig. 2. Prior to the meniscus tearing, the meniscus will exert a maximum force (point 7 in Fig. 2). This maximum force is used in determining the surface tension in the OneAttension software. While the original theory assumed a ring with infinite diameter and neglected excess liquid pulled up in the circular meniscus, current methods include correction factors in software calculations [2,3]. These correction factors require the liquid density to be known.

##### Wilhelmy plate

The Wilhelmy plate method brings a platinum plate into contact with the surface as in Fig. 3 [4]. In this case, the model assumes the plate is fully wetted and the plate is in contact with, but not submerged in, the liquid. Therefore, the position of the plate relative to the surface is critical. As the plate is lowered and brought into contact with the liquid, this “zero depth of immersion” height is recorded. The plate is then completely lowered into the liquid and raised back to the zero depth of immersion. At this point the force recorded by the balance is used to determine the surface tension according to the equation:

ΔF = 2γ(t + w) (1)

where ΔF is the net wetting force, 2(t + w) is the wetted perimeter of the probe, and γ is the surface tension. Note that the contact angle between the plate and the liquid is assumed to be zero. In contrast to the Du Nouy ring method, the density of the liquid is not necessary to measure surface tension.

### Optical tensiometry

Optical tensiometry uses drop or bubble profiles to indirectly measure surface and interfacial tension. In the literature, this technique is commonly referred to as axisymmetric drop shape analysis or ADSA [5]. ADSA can be subdivided into pendant drops/bubbles where a drop or bubble is suspended from a needle or pipette tip; sessile drops where a drop rests on a solid surface; and liquid bridges where a drop is attached to two independent surfaces such as a needle above and a solid surface below. Perhaps the most common optical tensiometry technique for measuring surface tension is the pendant drop/bubble method which can be performed with the Attension Theta and Theta Lite platforms.

The theoretical foundation of the pendant drop/bubble method was described by Bashforth and Adams in 1883 [6]. The technique revolves around extracting a drop or bubble shape from the Young-Laplace equation,

Δρgz = -γκ (2)

where Δρ is the difference in density between the drop/bubble phase and the continuous (surrounding) phase, g is gravitational acceleration, z is the vertical distance from the drop/bubble apex (Fig. 4), γ is the surface or interfacial tension, and κ is the curvature. Equation 2 describes the balance between gravity pulling the drop down in Fig. 4 (left-hand side of eq. 2) and surface tension resisting this pull (right-hand side). Discretizing with respect to arc length along the drop surface, s, leads to three coupled differential equations shown in Fig. 4 [7] that are solved numerically in the OneAttension software.

In practice the drop shape extracted from Eq. 2 is fit to an experimental drop shape extracted from an image. Thus, the quality of the measurement hinges heavily on the quality and proper calibration of the image. In the Young-Laplace equation, the parameter is adjusted to find the optimal drop shape is the drop shape factor β = ΔρgR_{0}^{2} / γ which contains two unknowns: the surface tension γ, and the radius at the drop apex R_{0}. This quantity is also sometimes referred to as the Bond number [8].

When β is large, gravity influences the drop shape much more than surface tension, leading to a “sagging” (pendant) drop shape. In this case, a unique combination of R0 and γ produces a suitable drop shape from Eq. 2 to fit the experimental image, and an accurate value for γ is returned. However, when β is very small, surface tension influences the drop shape more than gravity, leading to a spherical drop shape. A wide range of R_{0} and γ combinations can produce a spherical drop shape, and a unique solution is difficult to find, leading to measurement errors. Therefore, the experimentalist must be conscious of the density difference between the drop/bubble and surrounding fluid, drop size, and expected surface tension when designing the experiment.

The pendant drop method and ADSA, in general, has several advantages including the small sample size (on the order of tens of microliters), non-invasive measurements, and capability of measuring very low interfacial tensions. High quality surface and interfacial tension measurements with accuracy ± 0.01 mN/m can be easily made with the Attension Theta and Theta Lite platforms, and the Attension Theta High Pressure platform allows for measurements up to 200 ℃ and 400 bar.

## Effect of temperature and pressure on surface and interfacial tension

Surface and interfacial tension are sensitive to temperature and pressure. In the case of temperature, surface tension has been shown experimentally to decrease nearly linearly with temperature. The increase in temperature causes a corresponding decrease in cohesive intermolecular forces and this surface tension decreases. For example, the surface tension of water and air is 72.8 mN/m at 20°C, while the surface tension at 25°C is 72.0 mN/m. Figure 5 plots the surface tension of water and air for a range of temperatures from ref. [9]. As for pressure, surface tension is relatively unaffected by pressure changes until reaching high pressures as would be found in enhanced oil recovery and supercritical fluids. The increasing pressure causes dissolution of gas in the liquid, thereby decreasing the surface tension.

## Effect of surfactants on the surface and interfacial tension

Surfactants lower surface and interfacial tension by adsorbing to the surface or interface. In general, surfactants are amphiphilic molecules, meaning they have both hydrophobic (water-fearing) and hydrophilic (water-loving) components. In the presence of a water-air or water-oil interface, for example, the surfactant aligns itself at the interface such that the hydrophilic component is in the water and the hydrophobic component is in the other phase. The presence of the surfactant disrupts the cohesive forces between water molecules at the interface and reduces the surface or interfacial tension.

### Surface tension isotherms and the critical micelle concentration (CMC)

In general, as the surfactant concentration in the bulk phase is increased at constant temperature and pressure, the surface concentration of surfactant (Γ) that is adsorbed increases, resulting in lower surface or interfacial tensions. The relationship between bulk concentration and surface tension at a constant temperature is called an isotherm, and an equation describing this relationship can be derived from the Gibbs adsorption isotherm. One commonly used and relatively simple example is the Szyskowski equation [10],

γ = γ_{0} – nȒTΓ∞ ln(1+KC), (3)

where γ_{0} is the surface tension without surfactant, Ȓ is the universal gas constant, T is temperature, Γ∞ is the maximum possible surface concentration of surfactant, K is the surface-bulk equilibrium constant, and n is a coefficient that is 1 for non-ionic surfactants such as tetraethylene glycol monododecyl ether (C12E4) [11]. In Fig. 6, a plot of eq. 3 for a C_{12}E_{4} – air surface is shown using K and Γ∞ values from [11].

As the bulk concentration continues to increase, a point is reached where the surface concentration no longer increases. Instead surfactants in the bulk begin aggregating into structures called micelles. This point is called the critical micelle concentration or CMC and is driven by thermodynamics. The bottom of Fig. 6 demonstrates this transition schematically. After the CMC, surface tension tends to not vary significantly with bulk concentration.

### Surfactant adsorption and time dependent surface tension

When a new surface or interface is formed, such as when a pendant drop is generated, a dynamic process involving surfactant adsorption from the bulk to the surface and desorption from the surface back to the bulk begins. Additional bulk transport processes such as diffusion and convection impact the net rate of adsorption to the surface. The net adsorption rate, and similarly the change in surface or interfacial tension with time, is dependent on the surfactants involved and the fluid properties. The time it takes for the surface tension to reach an equilibrium can be as short as minutes and as long as days. When taking surface and interfacial tension measurements, particularly when performing a surface tension isotherm, it is important to consider this time dependence.

**References:**

[1] P.L. du Noüy. A new apparatus for measuring surface tension. J. Gen. Physiol 1 (1919) p. 521-524.

[2] C. Huh, S.G. Mason. A rigorous theory of ring tensiometry. Colloid Polymer Sci. 253 (1975) p. 566-580.

[3] H.H. Zuidema, G.W. Waters. Ring method for the determination of interfacial tension. Ind. Eng. Chem., Anal. Ed. 13 (1941) p. 312-313.

[4] A.W Neumann, R.J. Good, R.R. Stromberg. Surface and Colloid Science, Vol. 11, pp. 31-91. Plenum Press, New York, NY (1979).

[5] S.M.I. Saad, A.W. Neumann. Axisymmetric Drop Shape Analysis (ADSA): An Outline. Adv. Colloid Interface Sci. 238 (2016) pp. 62-87.

[6] F. Bashforth, J.C. Adams, An attempt to test the theories of capillary action. Cambridge University Press. London, England, 1883.

[7] Y. Rotenberg, L. Borkuva, A.W. Neumann. Determination of surface tension and contact angles from the shapes of axisymmetric fluid interfaces. J. Colloid Interface Sci. 93 (1983) p. 169-183.

[8] N.J. Alvarez, L.M. Walker, S.L. Anna. A non-gradient based algorithm for the determination of surface tension from a pendant drop: Application to low Bond number drops. J. Colloid Interface Sci. 333 (2009) pp. 557-562.

[9] N.B. Vargaftik, B.N. Volkov, L.D. Voljak. International tables of the surface tension of water. J. Phys. Chem. Ref. Data 12 (1983) p. 817-820.

[10] A.J. Prosser, E.I. Franses. Adsorption and surface tension of ionic surfactants at the air-water interface: review and evaluation of equilibrium models. Colloids Surf. A 178 (2001) p. 1-40.

[11] C.-T. Hsu, M.-J. Shao, S.-Y. Lin. Adsorption kinetics of C12E4 at the air-water interface: adsorption onto a fresh interface. Langmuir 16 (2000) p. 3187-3194.