KRÜSS | Theory Measuring Techniques - Contact Angle Measurement

Contact angle measurement – a theoretical approach

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 W _üas the work done in dividing a homogeneous liquid per parting surface produced. As during division two individual parting surfaces 2A are produced, W _ü can be calculated from the surface tension σ (which is defined as the work per surface difference) according to the following equation:

W _ü

2σ

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:

σ i + σ j - g

Equation 2

where

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:

γ₁₂

σ 1 - σ 2

Equation 3

with the surface tensions σ 1, σ 2 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:

Equation 4

By combining Equations 2 and 4 and transposition for γ ₁₂the following relationship is obtained:

Equation 5

The interaction parameter Φ 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 Φ 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:

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:

Equation 7

For the interfacial tension between two phases with polar fractions the following Equation (8) is obtained as an extension of Equation 6:

Equation 8

In the “Extended FOWKES” method a further interactive fraction is also differentiated; the interactions caused by hydrogen bondings:

Equation 9

with the corresponding extension of Equation 8 for the calculation of the interfacial tension by a further square root term:

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 σ⁺ and a base part σ^- ; this leads to the following equation:

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.

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.

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 σ_sand σ_ldescribe the surface tension components of the two phases; symbol γ_sl represents the interfacial tension between the two phases, and Θ stands for the contact angle corresponding to the angle between vectors σ_land γ_sl .
YOUNG formulated the following relationship between these quantities:

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

with the equation from YOUNG to obtain equations of state in which cos Θ represents a function of the phase surface tensions and, if applicable, the (polar and disperse) tension components

_l,D,

_l,P,

_S,D,

_S,P .
As liquids with known surface tension data and known polar and disperse fractions are used it is possible to include

_l,Dand

_l,Pin 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.

2.1 The ZISMAN method

In the ZISMAN method the surface energy of the solid is determined by using the critical surface tension (explained below) of the liquid. This method is based on a revised version of the ANTONOW method , it is implemented in the DSA1 program primarily for historical reasons and should not be used for routine measurements.

The method is based on the following consideration:
A liquid wets a solid completely when the work of cohesion for the formation of a liquid surface W _llis smaller than the work of cohesion for the formation of the interface boundaryW _sl.
The difference between these two quantities is known as the spreading pressure S _{l | s}:

S _{l | s}

W _sl- W _ll

Equation 14

The solid will be wetted completely when the spreading pressure is positive; at a negative spreading pressure the solid will not be wetted completely.
In addition, the following relationship exists between the work of cohesion W _sl, the contact angle Θ and the surface tension of the liquid (see also 2.3.2):

Equation 15

As the work of cohesion W _llis defined as2 · σ_laccording to DUPRÉ (see p. 2) then, for a contact angle of 0° ( cos Θ = 1) the work of cohesion will be the same as the work of adhesion; this results in a spreading pressure of 0. This means that the contact angle of 0° can be called the limiting angle for spreading (=complete wetting). Theoretically, a positive spreading pressure corresponds with negative contact angles which cannot be measured in practice.
The method according to ZISMAN uses this relationship by plotting cos Θ against the surface tension for various liquids and extrapolating the compensation curve to cos Θ = 1. The corresponding value for the surface tension is known as the critical surface tension σ_crit.

Fig 2: Determining the critical surface tension according to ZISMAN

ZISMAN equates this value with the surface energy of the solid σ_s.

Setting up a linear relationship between cos Θ and the surface tension σ_l is based on the now outdated assumption of ANTONOW that the interfacial tension is determined by the difference between the surface tensions. In fact this linear relationship only applies when the relationship between the disperse and polar interactions is the same between the solid and the liquid. This practically only occurs when a purely disperse interactive solid and liquid are involved; i.e. only under exceptional circumstances. This means that other methods should normally be used for determining the surface energy.

2.2 Equation of State

The equation of state was obtained during the search for a method of determining the surface energy of a solid from a single contact angle measurement by using a liquid with known surface tension.
Starting with the equation of Young

Y _{sl +}

· cos

Equation 16

it can be seen that a second equation is required which also describes the surface energy of the solid as a function of the interfacial tension solid/liquid and the surface tension of the liquid:

Equation 17

From thermodynamic considerations it was first demonstrated that such an equation valid for all systems must exist. By using an enormous volume of contact angle data the required equation of state was determined empirically:

Equation 18

The value 0.0001247 was determined for the constant β in the exponent. If the equation of state is inserted in Young’s equation then a new equation is obtained which allows the calculation of the surface tension of the solid σ_sfrom a single contact angle if the surface tensionσ_l is known:

Equation 19

In the calculation of the surface energy with the help of the equation of state the type of interactions which lead to the formation of the interfacial tensions (polar or disperse interactions) are not taken into account. However, the assumption that the knowledge of the surface tension of the liquid alone is sufficient has been disproved by experiments in which the contact angles of liquids with similar high surface tensions and differing fractions of polar interactions were measured. It appears that the disperse and polar fractions of the surface tensions must be taken into account; this means that the equation of state only provides useful results when only disperse interactions are present or when these are in the majority.

2.3 The method according to FOWKES

By using the FOWKES method the polar and disperse fractions of the surface free energy of a solid can be obtained. Strictly speaking this method is based on a combination of the knowledge of FOWKES on the one hand and that of OWENS, WENDT, RABEL and KAELBLE on the other, as FOWKES initially determined only the disperse fraction and the latter were the first to determine both the components of the surface energy. The difference between the FOWKES method used by KRÜSS and the OWENS, WENDT, RABEL and KAELBLE method is that in the FOWKES method the disperse and the polar fractions are determined in succession, i.e. in two steps, while in the OWENS, WENDT, RABEL and KAELBLE method both components are calculated by using a single linear regression.

The calculation steps described below are only intended to explain the methods. When calculating the surface energy according to FOWKES you do not have to proceed in several steps; when the calculation is carried out these steps are processed internally by the program. The same applies for the “Extended FOWKES” method described in Section 2.4.

2.3.1 Step 1: Determining the disperse fraction

In this first step the disperse fraction of the surface energy of the solid is calculated by making contact angle measurements with at least one purely disperse liquid.
By combination of the surface tension equation of FOWKES for the disperse fraction of the interactions

Equation 20

with the YOUNG equation (Equation 16) the following equation for the contact angle is obtained after transposition:

Equation 21

and, based upon the general equation for a straight line,

y = mx + b

Equation 22

cos

is then plotted against the term

and

can be determined from the slope m. The straight line must intercept the ordinate at the point defined as b=-1 (0/-1) As this point has been defined it is possible to determine the disperse fraction from a single contact angle: however, a linear regression with several purely disperse liquids is more accurate.

Fig. 3: Determining the surface energy according to FOWKES (1)

2.3.2 Step 2: Determining the polar fraction

For the 2nd step, the calculation of the polar fraction, Equation 20 is extended by the polar fraction:

Equation 23

It is also assumed that the work of adhesion is obtained by adding together the polar and disperse fractions:

Equation 24

and then as a third step YOUNG’s equation

Equation 25

is added to the equation of DUPRÉ

Equation 26

to obtain the following relationship for the work of adhesion:

Equation 27

Now all the components required for the calculation of the polar fraction of the surface energy have been assembled. A combination of Equations 23, 24 and 27 produces

Equation 28

Based upon this relationship the contact angles of liquids with known polar and disperse fractions are measured and W_sl^Pis calculated for each liquid. In this case a single liquid with polar and disperse fractions would be sufficient, although the results would again be less reliable.
As according to Equation 23 the polar fraction of the work of adhesion is defined by the geometric mean of the polar fractions of the particular surface tensions

Equation 29

then, by plotting

against

and following this with a linear regression, the polar fraction of the surface energy of the solid can be determined from the slope. As in this case the ordinate intercept b is 0, the regression curve must pass through the origin (0;0).

Fig. 4: Determining the polar surface energy according to FOWKES (2)

2.4 The Extended FOWKES method

In the Extended FOWKES method the work of adhesion is not split up into just two fractions but into three: the disperse and polar fractions as well as the fraction W_sl^H resulting from the hydrogen bridges:

Equation 30

The calculation of the surface energy accordingly is carried out in three steps instead of two.
As in the first step of the FOWKES method the disperse fraction of the surface energy of a solid is determined from the contact angle data of a purely disperse liquid (see 2.3.1).
In the second step liquids with known polar and disperse surface tension fractions are selected (σ_{l ^D}and σ_{l ^P}> O) , with a hydrogen bridge fraction σ_{l ^H}of O. In this way, as in the second step of the FOWKES method (see 2.3.2), the polar fraction of the surface free energy is first obtained with the aid of contact angle measurements (by subtracting the disperse fraction from the total work of adhesion).

Equation 31

The determination of the polar fraction of the surface free energy of the solid is carried out as in the FOWKES method.
In a third step work is again carried out in a similar manner for the calculation of the hydrogen bridge fraction. Contact angles of liquids with known polar, disperse and hydrogen bridge fractions (σ_{l ^H}< 0) of the surface tension are measured. By extending Equation 23 by the hydrogen bridge fraction we obtain

Equation 32

As in Equation 28 the required fraction of the work of adhesion, i.e. the fraction W_sl^Hresulting from the hydrogen bridges, can be calculated for each contact angle by subtracting the known fractions (by including the YOUNG Equation (25)):

Equation 33

Finally the hydrogen bridge fraction σ_{s ^H} of the surface energy of the solid can now be determined as described in Step 2 of the FOWKES method. According to Equation 32 the following relationship applies to W_sl^H :

Equation 34

If W_sl^H is plotted against

then

, i.e. the hydrogen bridge fraction of the surface energy of the solid, is obtained from the slope of the regression curve.

2.5 The Owens, Wendt, Rabel and Kaelble method

According to OWENS, WENDT, RABEL and KAELBLE the surface tension of each phase can be split up into a polar and a disperse fraction:

Equation 35

Equation 36

The FOWKES method for calculating the surface energy has already been developed from this relationship. In contrast to the FOWKES method, in the OWENS, WENDT, RABEL and KAELBLE method the calculation of the surface energy of the solid takes place in a single step.
OWENS and WENDT took the equation for the surface tension

Equation 37

as their basis and combined it with the YOUNG equation

Equation 38

The two authors solved the equation system by using the contact angles of two liquids with known disperse and polar fractions of the surface tension. KAELBLE solved the equation for combinations of two liquids and calculated the mean values of the resulting values for the surface energy. RABEL made it possible to calculate the polar and disperse fractions of the surface energy with the aid of a single linear regression from the contact angle data of various liquids. He combined Equations 37 and 38 and adapted the resulting equation by transposition to the general equation for a straight line

Y = mx + b

The transposed equation is shown below:

Equation 39

Equation 40

In a linear regression of the plot of y against x, σ_{s ^P} is obtained from the square of the slope of the curve m and σ_{s ^D}

from the square of the ordinate intercept b.

Fig. 5: Determination of the disperse and polar fractions of the surface tension of a solid according to RABEL

2.6 The WU method

In his observations on interfacial tension WU also started with the polar and disperse fractions of the surface energy of the participating phases. However, in contrast to FOWKES and OWENS, WENDT, RABEL and KAELBLE, who used the geometric mean of the surface tensions in their calculations, WU used the harmonic mean. In this way he achieved more accurate results, in particular for high-energy systems.
At least two test liquids with known polar and disperse fractions are required for this method; at least one of the liquids must have a polar fraction >0.

WU’s initial equation for the interfacial tension between a liquid and a solid phase is as follows:
:

Equation 41

If YOUNG’s equation is inserted in Equation 41

Equation 42

then the following relationship is obtained:

Equation 43

Y _{sl +}

· cos

In order to determine the two required quantities σ_{s ^D} and σ_{s ^P} , WU determined the contact angles for each of two liquids on the solid surface and then, based on Equation 43, he drew up an equation for each liquid. After a factor analysis the resulting equations were as follows:

(b₁ + c₁ - a₁ )+ c₁ (b₁ - a₁) + b₁ (c₁ - a₁) - a₁ b₁ c₁ = 0 Equation 44

(b₂ + c₂ - a₂ )+ c₂ (b₂ - a₂) + b₂ (c₂ - a₂) - a₂ b₂ c₂ = 0 Equation 45

The variables a₁, b₁, c₁ for the first liquid and a₂, b₂, c₂for the second liquid express the following terms:

a₁	1 σl,₁ ( cos Θ₁ + 1) 4
b₁	σ^D l,₁
c₁	σ^Pl,₁
a₂	1 σl,₂ ( cos &Thet₂ + 1) 4
b₂	σ^D l,₂
c₂	σ^Pl,₂

The solution of the equations produces the surface energy of the solid σ_sand its polar end disperse components σ_{s ^P}

and σ_{s ^D}. However, the following point must be taken into consideration: as quadratic equations are involved this means that two solutions are obtained for both σ_{s ^P}

and σ_{s ^D} only one of these solutions describes the actual surface energy.

2.6.1 Selecting the correct solution

The user now has the problem that, from the two solutions obtained above, the one which supplies the physically correct result for the system under investigation must be selected. This is very easy when one of the solutions has a negative sign. As negative values for the surface energy do not make sense from a physical point of view, in this case the second solution (with a positive sign) provides the result of the measurement.
For example:

	Solution 1	Solution 2
Surface energy of the solid	35.2 mN/m	15.7 mN/m
Disperse fraction	37.2 mN/m	12.2 mN/m
Polar fraction	-2.0 mN/m	3.5 mN/m

In this example Solution 1 can be rejected as it supplies a negative polar fraction of the surface energy. Solution 2 is the correct result. In such a case the DSA1 program automatically ignores the negative solution and presents the positive solution as the result.
However, it is often the case that both solutions make sense from a physical point of view. In such cases the decision can be simplified by including further information

Which of the two solutions has the order of magnitude which is to be expected from a knowledge of the properties of the substance?
Which of the two solutions agrees best with the results obtained with other pairs of liquids?
Which of the two solutions is closest to results obtained by calculations according to FOWKES, or OWENS, WENDT, RABEL and KAELBLE?

2.6.2 Measurements with more than two liquids

Although the equation system drawn up by WU can be solved with the contact angle data obtained with two liquids, as in other methods the selection of a larger number of test liquids increases the reliability of the measurements. As WU uses two equations for two liquids to calculate the surface energy, a part-result is obtained for each of the possible pairings of the test liquids.
For example: The surface energy of a solid is to be determined by using the contact angles of 4 test liquids: water, diiodomethane, ethylene glycol and benzyl alcohol. The calculation is carried out for each of the six possible pairings:

	Liquid 1	Liquid 2
1st pair	water	diiodomethane
2nd pair	water	ethylene glycol
3rd pair	water	benzyl alcohol
4th pair	diiodomethane	ethylene glycol
5th pair	diiodomethane	benzyl alcohol
6th pair	ethylene glycol	benzyl alcohol

This means that the 4 test liquids supply 6 part-results; as described above, each of these results has two solutions. This means that the choice of the right solution must be made for each individual pair of liquids. The pairing of two purely disperse liquids (σ^Pl = 0 ) produces no solution for the equation system; they are not included in the calculation.
The final result of the surface energy determination is the arithmetic mean of the selected part-results.

Continuation of the theoretical approach
on the following page

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