Contact angle
This text helps to familiarize yourself with the scientific background of the contact angle measurement technique, the calculation of surface free energy of solids and the surface tension of pendant drops. All methods described within this article are integrated in the KRÜSS Drop Shape Analysis programs DSA1 and DSA2.
1. Model considerations concerning interfacial tension
DUPRÉ defined the work of cohesion [Mimetex cannot convert this formula] as the work done in dividing a homogeneous liquid per parting surface produced. As during division two individual parting surfaces 2A are produced, [Mimetex cannot convert this formula] can be calculated from the surface tension [Mimetex cannot convert this formula] (which is defined as the work per surface difference) according to the following equation
[Mimetex cannot convert this formula] (Equation 1)
If a liquid column consists of 2 immiscible liquids then, when the column is separated, 2 new parting surfaces are formed at the interface and the boundary surface disappears. Therefore, according to DUPRÉ, the following relationship exists for the work of adhesion:
[Mimetex cannot convert this formula] (Equation 2),
where[Mimetex cannot convert this formula] represents the interfacial tension between the two phases.
ANTONOW has calculated the interfacial tension from the difference between the surface tensions of the individual phases:
[Mimetex cannot convert this formula] (Equation 3)
with the surface tensions[Mimetex cannot convert this formula], [Mimetex cannot convert this formula] of the individual components (this observation also forms the basis for the method according to ZISMAN described below (see Section 2.1)). However ANTONOW’s approach proved to be an approximation that was not sufficiently accurate.
GOOD and GIRIFALCO describe the work of cohesion as being dependent on the geometric mean of the interactive energies between the particles of the two individual phases:
[Mimetex cannot convert this formula] (Equation 4)
By combining Equations 2 and 4 and transposition for [Mimetex cannot convert this formula]the following relationship is obtained:
[Mimetex cannot convert this formula] (Equation 5)
The interaction parameter [Mimetex cannot convert this formula] introduced here is a complex function of molecular quantities and initially could only be determined empirically.
FOWKES was the first to prepare the way for the calculation of interfacial tensions from surface tension data. He specified the interactions represented by the parameter [Mimetex cannot convert this formula] by assuming that only the same types of interactions could occur between the phases. For example, according to this only a nonpolar substance, i.e. a purely disperse interactive substance, can interact with the disperse fractions of the surrounding second phase:
[Mimetex cannot convert this formula] (Equation 6)
The disperse character of the interactions is expressed by the index D.
While dispersion forces exist in all atoms and molecules, polar forces are only found in certain molecules. Polar forces have their source in the differing electronegativity of different atoms in the same molecule. For polar liquids OWENS, WENDT, RABEL and KAELBLE (1969) assumed that there was a polar fraction of the surface tension. According to their model, the surface tension was the sum of the disperse and polar fractions:
[Mimetex cannot convert this formula] (Equation 7)
For the interfacial tension between two phases with polar fractions the following Equation (8) is obtained as an extension of Equation 6:
[Mimetex cannot convert this formula] (Equation 8)
In the “Extended FOWKES” method a further interactive fraction is also differentiated; the interactions caused by hydrogen bondings:
[Mimetex cannot convert this formula] (Equation 9)
with the corresponding extension of Equation 8 for the calculation of the interfacial tension by a further square root term:
[Mimetex cannot convert this formula] (Equation 10)
Equations 6, 8 and 10 use the geometric mean of the particular surface tension components of the individual phases. They produce satisfactory results throughout a wide range of surface energies.
The model according to OSS and GOOD is also based on the geometric mean, but the polar fraction is described with the help of the Acid-Base-Model according to LEWIS. The polar fraction is divided into an acid part [Mimetex cannot convert this formula] and a base part [Mimetex cannot convert this formula]; this leads to the following equation:
[Mimetex cannot convert this formula] (Equation 11)
For low-energy systems (surface energies up to « 35mN/m) the method according to WU can be used as an alternative. WU uses the harmonic mean instead of the geometric mean and limits it to the disperse and polar fractions.
[Mimetex cannot convert this formula] (Equation 12)
In this way WU obtained more accurate results for low energy systems. However, the use of the harmonic mean is not suitable for high-energy materials (e.g. mercury, glass, metal oxides, graphite, polar polymers).
With the aid of the methods described here it is possible to calculate the interfacial tensions between liquids, provided that their surface tensions and disperse and polar fractions (and, if applicable, their hydrogen bridge fractions) are known. In addition the surface energies of solids can also be calculated. A requirement for this is the knowledge of the contact angles of the corresponding liquids during phase contact with the solid surface.
Contact angle and surface energy
2. Contact angle and surface energy
In 1805 YOUNG had already formulated a relationship between the interfacial tensions at a point on a 3-phase contact line.
- Fig. 1: Contact angle formation on a solid surface according to YOUNG
Indices s and l stand for “solid” and “liquid”; the symbols [Mimetex cannot convert this formula] and [Mimetex cannot convert this formula] describe the surface tension components of the two phases; symbol [Mimetex cannot convert this formula] represents the interfacial tension between the two phases, and [Mimetex cannot convert this formula] stands for the contact angle corresponding to the angle between vectors [Mimetex cannot convert this formula] and [Mimetex cannot convert this formula].
YOUNG formulated the following relationship between these quantities:
[Mimetex cannot convert this formula] (Equation 13)
The methods implemented in the DSA1 program allow the determination of the surface energy of solids from contact angle data. They are mainly based on combining various starting equations for [Mimetex cannot convert this formula] with the equation from YOUNG to obtain equations of state in which [Mimetex cannot convert this formula] represents a function of the phase surface tensions and, if applicable, the (polar and disperse) tension components [Mimetex cannot convert this formula], [Mimetex cannot convert this formula], [Mimetex cannot convert this formula], [Mimetex cannot convert this formula]und [Mimetex cannot convert this formula] .
As liquids with known surface tension data and known polar and disperse fractions are used it is possible to include [Mimetex cannot convert this formula] and [Mimetex cannot convert this formula] in the equations. All methods assume that the interactions between the solid and the gas phase (or the liquid vapour phase) are so small as to be negligible. The methods are described in the following sections.