Surface Tension and Interfacial Tension

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Surface tension and interfacial tension (Author: Prof. Dr. H.-J. Radusch)

Surface tension

Definition

The surface tension or specific surface energy corresponds to the force that acts tangentially to the interface in liquids at the boundary to a gas or vacuum [1, 8–11]. The surface tension is defined as the work dW that must be performed per area dA in order to increase the surface area of a liquid. The ratio of the work performed and the resulting increase in surface area is called surface tension or surface work:

$\sigma ={\frac {dW}{dA}}\ ({\frac {J}{m^{2}}})$.

(1)

The work to be performed dW = Fdx is therefore proportional to the force F parallel (tangential) to the surface, which must be applied to increase the surface area by the amount dA = l dx:

$\sigma ={\frac {dW}{dA}}={\frac {F\ dx}{l\ dx}}={\frac {F}{l}}$

$({\frac {J}{m^{2}}})=({\frac {N}{m}})$.

(2)

This means that the surface tension is also a force per length. The cause of the surface tension σ is that the cohesive forces between the molecules in the liquid cancel each other out (B), while the outwardly directed forces are absent on the surface (A), resulting in a force in the interior of the liquid (see Fig. 1).

Fig. 1:

Surface tension: Forces acting on a molecule on the surface (MO) and inside (Mi) of a substance

From a thermodynamic point of view, the surface tension σ is the derivative of the free enthalpy G with respect to the surface area A at constant temperature T and constant pressure p:

$\sigma ={({\frac {dG}{dA}})}_{T,p}$.

(3)

The free enthalpy G has the dimension of an energy. Thus, σ has the dimension of an energy per area (J/m² = N/m).

Measurement of surface tension

The surface tension can be measured using different methods based on the principle of measuring the existing surface force using a so-called tensiometer or the capillary effect [2, 14–16]. A distinction is made between the static surface tension (measured in equilibrium with a mechanically unchanged interface) and the dynamic surface tension (measured during the change of the surface). The most important methods are summarised in Table 1:

Table 1: Test methods and principles for determining surface tension

Method	Measuring principle
Bow method (Lenard)	Measurement of the tensile force when a wire bow is pulled out of a liquid (break-off point)
Ring method (Du Noüy)	Measurement of the force of a liquid lamella pulled up by a ring
Plate method (Wilhelmy)	Measurement of the force resulting from the wetting of a vertically suspended plate
Capillary effect method	Measurement of the height of rise of a liquid in a capillary
Contact angle measurement / Sessile drop method	Measurement of the contact angle between the drop of liquid and the substrate. Calculation of the surface tension using Young's equation from the cosine of the contact angle
Spinning drop method	Measurement of the diameter of a rotating drop in a heavier phase
Pendant drop method	Measurement of the size of the droplets dripping from a defined capillary. Optical detection of the drop geometry
Bubble pressure method	Passing a gas stream through a capillary and measuring the pressure curve to form bubbles..
Drop volume method	Measurement of the number of drops into which a given volume of liquid is divided. Suitable for measuring the dynamic surface tension
Test ink method	Application of a coloured liquid (‘ink’) with a defined surface tension to the surface to be tested (‘brush stroke’) and assessment of the ink flow
Expanding/Oscillating drop method (EDM/ODM)	Determination of the surface rheological properties of liquids. Describes the dependence of surface tension on the degree and speed of surface expansion of a drop, which is either rapidly expanded and then stands still (EDM) or undergoes sinusoidal oscillation (ODM)

Interfacial tension

Definition

Surface tension in the narrower sense can only be referred to when the liquid is adjacent to its own saturated vapour. In all other cases, the correct term is interfacial tension [1, 3]. Both cohesive forces into the liquid and adhesive forces in the direction of the neighbouring medium act on an interfacial molecule.

The interfacial tension refers to forces that occur at the boundary between two different phases that are in contact with each other. They form a common interface (see also: Phase boundary surface) that is under interfacial tension. Phases can be liquid, solid or gaseous. Different phases are phases that do not mix, such as water and oil or polyethylene ( abbreviation: PE) and polyvinyl chloride ( abbreviation: PVC).

A molecule inside a uniform phase is subjected to the same attractive intermolecular forces from all sides by neighbouring molecules, which compensate each other. For molecules in the interface between two immiscible phases (A + B), the interactions of the particle in the interface with molecules in the neighbouring phase are weaker and therefore associated with a lower energy gain than those of the particle in one phase. As a result, the molecules in the interface are in a higher energy state than the molecules inside the phases. This additional energy is referred to as interfacial energy. The result for these molecules is a force directed towards the inside of the phase.

In order to minimise the interfacial energy, the system tries to keep the interface to other phases and thus the number of molecules in the interface as small as possible. If the interface is enlarged, molecules must be moved from the interior of the volume to the interface with the higher energy. The work that must be applied to enlarge the interface A_G of a phase is referred to as the interfacial work W_AG. The interfacial work is proportional to the change in area:

$dW_{AG}=\gamma \cdot dA_{G}$

(4)

The proportionality coefficient γ is called interfacial tension. In the specific case of the interface between a condensed phase and a gas, this corresponds to the surface tension σ. Both have the dimension of energy per area.

$[\gamma ]=[\sigma ]={\frac {J}{m^{2}}}={\frac {N}{m}}$

(5)

Measurement of interfacial tension

As with the surface tension, a distinction is made between the static interfacial tension (measured at equilibrium with a mechanically unchanged interface) and the dynamic interfacial tension (measured during the change in the interface). In addition, the state of the phases (liquid/solid) must be taken into account when measuring. The most important methods are summarised in Table 2:

Table 2: Test methods and principles for determining interfacial tension

Method	Measuring principle
Liquid-liquid Interface
Ring method (Du Noüy)	Measurement of the force acting on an optimally wettable ring when the ring is moved from one phase to the other by the tension of the pulled-out liquid lamella
Plate method (Wilhelmy)	Measurement of the force acting on an optimally wettable plate immersed vertically in the lower phase
Rod method	Like the plate method, whereby a cylindrical rod with a smaller wetted length is used for the measurement with a smaller liquid volume
Drop volume method	Measurement of the volume of a drop of a liquid produced on a vertical capillary in another liquid at the moment it breaks off
Spinning drop method	A horizontal capillary filled with a surrounding phase and a drop phase is set in rotation. The diameter of the drop, which is elongated by centrifugal force, correlates with the interfacial tension.
Pendant drop method	The shape of a drop on a cannula in a surrounding liquid phase is determined, among other things, by the interfacial tension. The interfacial tension can be determined from the image of the drop using drop contour analysis.
Solid-liquid interface
Contact angle measurement	From the contact angle of several liquids with a solid, the interfacial tension with the respective liquids can be calculated in addition to its free surface energy.

Examples

Example 1: Determination of the surface tension or interfacial energy by measuring the contact angle on a sessile droplet

At an interface between a liquid and a solid, the angle between the liquid surface and the contact surface is referred to as the contact angle θ. The contact angle is a measure of the wettability of a solid by a liquid (Fig. 2).

σL – Surface tension of the liquid σS – Surface energy of the solid σLS – Interfacial energy between liquid and solid Θ – Contact angle

Fig. 2:

Sessile drop method: Stresses on a drop of liquid on a substrate and definition of the contact angle

Equation (6) (YOUNG's equation [4, 12, 13]) establishes a relationship between the specific surface free energy σ_S of a plane solid to the surrounding gas (index S for solid-gas), the specific interfacial energy σ_LS between a solid and a drop of liquid on it (index LS for liquid-solid), the surface tension σ_L of the liquid to the surrounding gas (index L for liquid-gas) and the contact angle θ between the two:

$cos\Theta ={\frac {\sigma _{S}-\sigma _{LS}}{\sigma _{L}}}\Rightarrow \sigma _{S}=\sigma _{LS}+\sigma _{L}cos\Theta$

(6)

This means that there is a quantitative relationship between the forces occurring at the phase boundary of a solid and a liquid substance.

Example 2: Calculation of the interfacial tension using the WU & FOWKES method

The interfacial tension γ (σ_SL) can be calculated using the two surface tensions σ_S and σ_L and the similar interactions between the phases. These interactions are interpreted as the harmonic mean value of a dispersive component σ^D and a polar component σ^P of the surface tension or surface free energy [5–7, 16].

$\gamma =\sigma _{SL}=\sigma _{L}+\sigma _{S}-4({\frac {\sigma _{L}^{D}\cdot \sigma _{S}^{D}}{\sigma _{L}^{D}+\sigma _{S}^{D}}}+{\frac {\sigma _{L}^{P}\cdot \sigma _{S}^{P}}{\sigma _{L}^{P}+\sigma _{S}^{P}}})$

(7)

To determine the surface free energy of the solid, at least two liquids with known dispersive and polar surface tension components are required, whereby at least one of the liquids must have a polar component > 0. For the calculation, an equation is set up for each possible combination of two liquids, i.e. n liquids result in (n²– n)/2 equations with a corresponding number of partial results. The resulting surface free energy is the arithmetic mean of the partial results. The empirical basis of the method is formed by interfacial tension measurements between polymer melts, i.e. materials with predominantly low surface tension of the individual phases. Accordingly, the WU & FOWKES method for surface energy calculation is mostly used for polymers with low surface energy (up to 40 mN/m).

Acknowledgements

The editors of the encyclopaedia would like to thank Prof. Dr. Hans-Joachim Radusch, Martin Luther University Halle-Wittenberg and Polymer Service GmbH Merseburg for this guest contribution.

See also

• Phase boundary surface
• Surface energy

References

[1]	Elias, H. G.: Makromoleküle, Struktur – Eigenschaften – Synthesen – Stoffe, Chapter 7.3 Grenzflächenphänomene, Hüthig & Wepf, Basel, Heidelberg (1972); (ISBN 978-3-7785-0677-6; see AMK-Library under N 41)
[2]	Stamm, M. (Ed.): Polymer Surfaces and Interfaces: Characterization, Modification and Applications, Chapter 1: Polymer Surface and Interface Characterization Techniques; Chapter 6: Characterization of Polymer Surfaces by Wetting and Electrokinetic Measurements – Contact Angle, Interfacial Tension, Zeta Potential, Springer, Berlin, Heidelberg (2008) (ISBN 978-3-540-73864-0)
[3]	van Krevelen, D. W.: Properties of Polymers, Chapter 8 – Interfacial Energy Properties, Elsevier B.V. (2009); (ISBN 978-0-08-091510-4)
[4]	Young, T.: An Essay on the Cohesion of Fluids. Philosophical Transactions of the Royal Society of London, The Royal Society, London (1805), Vol. 95, pp. 65–87
[5]	Wu, S.: Calculation of Interfacial Tensions in Polymer Systems. In: J. Polym. Sci. 43 (1971), pp.19–30; https://doi.org/10.1002/polc.5070340105
[6]	Wu, S.: Polar and Nonpolar Interaction in Adhesion. In: J. Adhesion 5 (1973), pp. 39–55; https://doi.org/10.1080/00218467308078437
[7]	Fowkes, F. M.: Attractive Forces at Interfaces. In: Industrial and Engineering Chemistry 56 (1964) 12, pp. 40–52; https://doi.org/10.1021/ie50660a008
[8]	http://www.chemie.de/lexikon/Oberfl%C3%A4chenspannung.html
[9]	https://www.spektrum.de/lexikon/physik/oberflaechenspannung/10567
[10]	https://tu-dresden.de/die_tu_dresden/fakultaeten/fakultaet_mathematik_und_naturwissenschaften/fachrichtung_physik/studium/lehrveranstaltungen/praktika/pdf/OS.pdf
[11]	https://uol.de/f/5/inst/physik/ag/physikpraktika/download/UWI/pdf/Basispraktikum/Oberflaechenspannung.pdf
[12]	https://de.wikipedia.org/wiki/Kontaktwinkel
[13]	https://de.wikipedia.org/wiki/Youngsche_Gleichung
[14]	http://www.surface-tension.de/
[15]	http://www.dataphysics.de
[16]	http://www.kruess.de/

Standards

• ISO 19403: Paints and Varnishes – Wettability –

• Part 1 (2022-06): Vocabulary and General Principles
• Part 2 (2024-09): Determination of the Surface Free Energy of Solid Surfaces by Measuring the Contact angle
• Part 3 (2024-09): Determination of the Surface Tension of Liquids Using the Pendant Drop Method
• Part 4 (2024-10): Determination of the Polar and Dispersive Fractions of the Surface Tension of Liquids from an Interfacial Tension
• Part 5 (2024-10): Determination of the Polar and Dispersive Fractions of the Surface Tension of Liquids from Contact Angles Measurements on a Solid with only a Disperse Contribution to its Surface Energy
• Part 6 (2024-10): Measurement of Dynamic Advancing and Receding angle by Changing the Volume of a Drop
• Part 7 (2024-10): Measurement of the Dynamic Contact Angles and the Roll-off Angle on a Tilt Stage

• ISO 1409 (2020-08): Plastics/Rubber – Polymer Dispersions and Rubber Latices (Natural and Synthetic) – Determination of Surface Tension
• ISO 304 (1985-12): Surface active agents – Determination of Surface Tension by Drawing up Liquid Films
• DIN EN 828 (2013-04): Adhesives – Wettability – Determination by Measurement of Contact Angle and Surface Free Energy of Solid Surface
• ISO 15989 (2004-12): Plastics – Film and Sheeting – Measurement of Water-contact angle of Corona-treated films
• DIN EN 14370 (2004-11): Surface Active Agents – Determination of Surface Tension
• ASTM D 3825 (2009): Standard Test Method for Dynamic Surface Tension by the Fast-Bubble Technique (withdrawn 2016)
• DIN 13310 (1982-08): Surface Tension in Fluids – Concepts, Quantities, Symbols, Units
• DIN EN 14210 (2004-03): Surface Active Agents – Determination of Interfacial Tension of Solutions of Surface Active Agents by the Stirrup or Ring Method
• ISO 6889 (1986-03): Surface Active Agents – Determination of Interfacial Tension by Drawing up Liquid Films

Additional References

• Ehrenstein, G. W.: Strukturverhalten. Struktur und Eigenschaften von Kunststoffen, Oberflächenspannung, Spannungsrisse. Carl Hanser Munich (2021) (ISBN 978-3-446-46703-3; E-Book ISBN 978-3-446-46710-1; see AMK-Library under F 25)