Contact angle

3. Measuring the contact angle

The previous section explained the various methods of calculating the surface energy from contact angle data. In this section the theory of contact angle measurement is explained. All calculation methods (except for SCHULTZ 2) are based on the sessile drop method, i.e. drops of liquid are deposited on a solid surface (as smooth and horizontal as possible).

A differentiation is made between the various ways of measuring the drop:

• A contact angle can be measured on static drops. The drop is produced before the measurement and has a constant volume during the measurement.
• A contact angle can be measured on dynamic drops. The contact angle is measured while the drop is being enlarged or reduced; the boundary surface is being constantly newly formed during the measurement. Contact angles measured on increasing drops are known as “advancing angles”; those measured on reducing drops as “retreating angles”.

3.1 Static contact angles

In a static contact angle measurement the size of the drop does not alter during the measurement. However, this does not mean that the contact angle always remains constant; on the contrary, interactions at the boundary surface can cause the contact angle to change considerably with time. Depending on the type of time effect the contact angle can increase or decrease with time.

Fig.10: Alteration of the static contact angle as a function of time

For example, these interactions could be:

• Evaporation of the liquid
• Migration of surfactants from the solid surface to the liquid surface
• Substances dissolved in the drop migrating to the surface (or in the opposite direction),
• Chemical reactions between the solid and liquid,
• The solid being dissolved or swollen by the liquid.

It may be a good idea to choose to measure the static contact angle when its variation as a function of time is to be studied. A further advantage of static contact angle measurement is that the needle does not remain in the drop during the measurement. This prevents the drop from being distorted (particularly important for small drops). In addition, when determining the contact angle from the image of the drop it is possible to use methods which evaluate the whole drop shape and not just the contact area.

Certain materials which don’t show a fully rigid surface (e.g. rubber) are better being tested with static measurements. In such cases, dynamic contact angles are poorly reproducable.
However, changes with time often interfere with the measurement. There is also a further source of error: as the static contact angle is always measured at the same spot on the sample any local irregularities (dirt, inhomogeneous surface) will have a negative effect on the accuracy of the measurement. This error can be averaged out in dynamic contact angle measurements.

3.2 Dynamic contact angle

Dynamic contact angles describe the processes at the liquid/solid boundary during the increase in volume (advancing angle) or decrease in volume (retreating angle) of the drop, i.e. during the wetting and dewetting processes.

A boundary is not formed instantaneously but requires some time before a dynamic equilibrium is established. This is why a flow rate which is too high should not be selected for measuring advancing and retreating angles, as otherwise the contact angle will be measured at a boundary which has not been completely formed. However, it should also not be too slow as the time effects mentioned above will then again play a role. In practice flow rates between 5 and 15 ml/min can be recommended; higher flow rates should only be used for the simulation of dynamic processes.
For high-viscosity liquids (e.g. glycerol) the rate will tend to approach the lower limit.

During the measurement of the advancing angle the syringe needle remains in the drop throughout the whole measurement. In practice a drop with a diameter of about 3-5 μl (with the needle of 0.5 mm diameter which is used in KRÜSS measurement systems) is formed on the solid surface and then slowly increased in volume. At the beginning, the contact angle measured is not independent from the drop size because of the adhesion to the needle. At a certain drop size the contact angle stays constant; in this area the advancing angle can be measured properly. (Fig.11).

As a result of the wetting process, advancing angles always simulate a fresh surface for the contact angle; this is formed immediately after the creation of the contact between the liquid and the surface. This type of measurement is therefore the most reproducible way of measuring contact angles. As a result, advancing angles are normally measured in order to determine the surface free energy of a solid.

3.2.2 Receding angle

During the measurement of the receding angle the contact angle is measured as the size of the drop is being reduced, i.e. as the surface is being de-wetted. By using the difference between the advancing and the receding angles it is possible to make statements about the roughness of the solid or chemical inhomogeneties; however, the receding angle is not suitable for calculating surface energies.

In practice a relatively large drop with a diameter of approx. 6 mm is deposited on the solid and then slowly reduced in size with a constant flow rate.

Fig.12: Measuring receding angles

The same guiding limits and conditions apply here as for the measurement of the advancing angle (see Section 3.2.1).

3.3 Methods of evaluating the drop shape

The basis for the determination of the contact angle is the image of the drop on the drop surface. In the DSA1 program the actual drop shape and the contact line (baseline) with the solid are first determined by the analysis of the grey level values of the image pixels. To describe this more accurate, the software calculates the root of the secondary derivative of the brightness levels to receive the point of greatest changes of brightness. The found drop shape is adapted to fit a mathematical model which is then used to calculate the contact angle. The various methods of calculating the contact angle therefore differ in the mathematical model used for analyzing the drop shape. Either the complete drop shape, part of the drop shape or only the area of phase contact are evaluated. All methods calculate the contact angle as tanq at the intersection of the drop contour line with the solid surface line (base line).

In the following sections the different drop shape analysis methods are briefly described.

3.3.1 Tangent method 1

The complete profile of a sessile drop is adapted to fit a general conic section equation. The derivative of this equation at the intersection point of the contour line with the baseline gives the slope at the 3-phase contact point and therefore the contact angle. If dynamic contact angles are to be measured, this method should only be use when the drop shape is not distorted too much me the needle.

3.3.2 Tangent method 2

That part of the profile of a sessile drop which lies near the baseline is adapted to fit a polynomial function of the type () The slope at the 3-phase contact point at the baseline and from it the contact angle are determined using the iteratively adapted parameters.

This function is the result of numerous theoretical simulations. The method is mathematically accurate, but is sensitive to distortions in the phase contact area caused by contaminants or surface irregularities at the sample surface.

As only the contact area is evaluated, this method is also suitable for dynamic contact angles. Nevertheless, this method requires an excellent image quality, especially in the region of the phase contact point.

3.3.3 Height-width method

In this method the height and width of the drop shape are determined. If the contour line enclosed by a rectangle is regarded as being a segment of a circle, then the contact angle can be calculated from the height-width relationship of the enclosing rectangle. The smaller drop volume, the more accurate the approximation for smaller drops are more similar to the theoretically assumed spherical cap form.

As the drop height cannot be determined accurately when the needle is still in the drop, the height-width method is not suitable for dynamic drops. This method also has the disadvantage that the drops are regarded as being symmetrical, so that the same contact angle is obtained for both sides, even when differences between the two sides can be seen in the actual drop image.

3.3.4 Circle fitting method

As in the height-width method, in this method the drop contour is also fitted to a segment of a circle. However, the contact angle is not calculated by using the enclosing rectangle, but by fitting the contour to a circular segment function. The same conditions apply to the use of this method as to the height-width method with the difference that a needle remaining in the drop disturbs the result far less.

3.3.5 Young-Laplace (sessile drop fitting)

The most complicated, but also the theoretically most exact method for calculating the contact angle is the YOUNG-LAPLACE fitting. In this method the complete drop contour is evaluated; the contour fitting includes a correction which takes into account the fact that it is not just interfacial effects which produce the drop shape, but that the drop is also distorted by the weight of the liquid it contains. After the successful fitting of the YOUNG-LAPLACE Equation the contact angle is determined as the slope of the contour line at the 3-phase contact point.

If the magnification scale of the drop image is known (determined by using the syringe needle in the image) then the interfacial tension can also be determined; however, the calculation is only reliable for contact angles above 30°. Moreover, this model assumes a symmetric drop shape. Therefor it cannot be used for dynamic contact angles where the needle remains in the drop.

4. Measuring the surface tension of pendant drops

If a drop of liquid is hanging from a syringe needle then it will assume a characteristic shape and size from which the surface tension can be determined. A requirement is that the drop is in hydromechanical equilibrium.
When in hydromechanical equilibrium the force of gravity acting on the drop and depending on its particular height corresponds to the LAPLACE pressure, which is given by the curvature of the drop contour at this point. The LAPLACE pressure results from the radii of curvature standing vertically upon one another in the following way:

(Equation 59)

This equation describes the difference between the pressure below and above a curved section of the surface of a drop with the principal radii of curvature and .

The pressure difference is the difference in pressure between the outside of the drop and its inside.

4.1 The basic drop contour equation

For a pendant drop which is rotationally symmetrical in the z-direction then, based on Equation 59, it is possible to give an analytically accurate geometric description of the principal radii of curvature. The tangent at the intersection of the z-axis with the apex of the drop forms the x-axis. The drop profile is given by pairs of values (x,z) in the x-z-plane.

Fig.13: Geometry of the pendant drop

In hydromechanical equilibrium the following relationship applies

(Equation 60)

( = pressure difference at apex; \Delta\ \rho_P = pressure difference at Point P (x,z); = difference in density between the drop liquid and its surroundings; = acceleration due to gravity).

With the principal curvatures k (reciprocal value of principal curvature radius r) and the YOUNG-LAPLACE Equation (Equation 59) we obtain:

(Equation 61)

(Equation 62)

= principal curvatures at apex
= principal curvatures at Point P (x,z)

Because of the axial symmetry of the drop, the principal curvatures at the apex are the same in all directions ( ). From differential geometry the analytical expressions for the curvatures of the principal normal sections at Point P (x,z) are known:

(Equation 63)

= } (Equation 64)

From Equations 60 to 64 we obtain:

(Equation 65)

( = length of arc along the drop profile; = angle between the tangents at Point P (x,z) and the x-axis)

Equation 65 describes the profile of a pendant drop in hydromechanical equilibrium. The Equation is converted into a dimensionless form to solve it. The following definitions are used:

B = dimensionless form parameter of the pendant drop
a = capillary constant

With these definitions Equation 65 can also be expressed in the following way:

(Equation 66)

At the apex the limiting conditions apply. This results in:

(Equation 67)

B is the only parameter to determine the shape of the drop profile. It is therefore known as the form parameter. In addition, it can be seen that the surface tension tex> \sigma\ can be calculated for a known difference in density if the relative size ratio a of a measured drop can be determined for the corresponding theoretical drop profile.

Equation 67 is, together with the limiting conditions from Equation 66, known as the fundamental equation for a pendant drop.

By varying the form parameter B it is possible to calculate theoretical drop profiles after carrying out a numerical integration method. If the theoretical drop profile corresponds to the measured drop profile then the surface tension can be calculated. The problem in measuring the interfacial tension therefore consists in determining the correct theoretical drop profile for the measured drop exactly and rapidly.

4.2 The robust shape comparison method

Various groups of methods exist for solving the problem mentioned above. In the DSA1 program the robust shape comparison method is used. This method is a statistical method which is characterized by its stability against “outliers”. In this way even low-quality drop images can still be evaluated.
A series of drop profile co-ordinates is used for the evaluation. The measured profile is compared with the theoretical profile. The comparison is not made directly via the profile points, but via their vectors. An advantage of this method is that it is possible to optimize the individual parameters used independently.

The error function E used in the optimization is a function of the form parameter B, the capillary constant a (which includes the surface tension), the position of the apex () (co-ordinate origin) and the angular variation of the drop from the plane of symmetry.

(Equation 68)

The angular variation of the drop from the plane of symmetry describes the variation of the vertical drop axis from the normal axis (z-axis). For small variations () the correction does not cause any problems.

Fig.14: Angular variation Θ of the drop from the plane of symmetry

In the evaluation of the image a further quantity appears which also needs to be taken into consideration: the height-width ratio AR (Aspect Ratio) of the image pixels in the drop image. By adaptation of the described parameters B,  and AR the error function E rsc (robust shape comparison) from Equation 68 can be minimized by the robust shape comparison:

(Equation 69)