- Continuation
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2.7 The SCHULTZ method
The method for calculating the surface energy according to SCHULTZ is only intended for use with high-energy solid surfaces. In the DSA1 program there are two methods implemented which are based on SCHULTZ: “SCHULTZ 1” and “SCHULTZ 2”. The theoretical requirements are the same for both methods; the difference lies in the test arrangement. As a result this section first described the theoretical principles and only then explains the differences between the two SCHULTZ methods.
2.7.1 Theoretical principles for the two SCHULTZ methods
High-energy solids are normally completely wetted by all liquids, so that their surface energy cannot be determined by using conventional contact angle measurements. In order to be able to investigate such systems at all the test arrangement must be altered: instead of being measured in air, the contact angle of a liquid drop (“drop phase”) on a solid is measured in a surrounding liquid phase (“bulk phase”).
The calculation of the surface energy assumes that the YOUNG equation also applies to a liquid/liquid/solid system:
With the equations of FOWKES and OWENS, WENDT, RABEL and KAELBLE (Equation 37, p.10) adapted for a liquid/liquid/solid system, the following equations are obtained for the drop phase and the surrounding phase:
If Equations 46, 47 und 48 are combined then the following relationship is obtained:
In the “SCHULTZ 1” method only a single drop liquid is used and the surrounding phase is changed instead. The drop liquid used is normally water; the bulk phase is a liquid which is immiscible with water and with a lower density than water.
As in the FOWKES method (see Section 2.3) the calculation of the polar and disperse fractions of the surface energy is carried out in two steps.
.
At first the contact angle of water on the solid is measured in a range of purely disperse interacting liquids. Owing to the nonpolar character of the surrounding phase the term W P sl bulk can be deleted from Equation 49. Equation 49 can then be adapted to conform with the general equation for a straight line:
to give:
is can be calculated directly from the slope and W P sl drop from the y-axis intercept
According to FOWKES, this adhesion energy can be calculated from the geometric mean between the polar fractions of the surface tensions of the participating phases:
2.7.3 SCHULTZ 2
In the SCHULTZ 2 method it is not the heavier liquid which is used as the drop liquid but the lighter one; the heavier liquid forms the surrounding phase. In order for this to be possible the test arrangement must be inverted: the drop is not present as a sessile drop on the solid but is suspended from it as a pendant drop:
As in SCHULTZ 1 the disperse fraction of the surface energy of the solid σDs is measured first by using purely disperse interacting liquids. The difference from the calculation for SCHULTZ 1 is that the term W P sl bulk is obtained from the intercept of the regression curve with the y-axis (see Fig.7) on the plot, whereas the term W P sl drop is deleted from Equation 49.
In the second step the term W P sl drop in Equation 49 is calculated from the contact angles of drop liquids with polar fractions for each test liquid. The polar of the surface energy of the solid σPs is obtained in a similar way to SCHULTZ 1 by using the equation
OSS and GOOD also differentiate between a polar and a disperse fraction of the surface energy. However, in contrast to the previously described authors, they describe the polar fraction with the help of the acid-base model according to Lewis. According to this model, the polar fraction of the surface energy of the solid and the surrounding drop liquid is split into an electron acceptor fraction corresponding to a Lewis acid (=”electron receiving” fraction σ + and an electron donor corresponding to a Lewis base (=“electron donor” fraction) σ-.
Owing to the attraction of opposite charges there are interactions between the particular counter poles of the polar components of the solid and the liquid. The Equation for the surface tension of FOWKES and OWENS, WENDT, RABEL, KAELBLE (Equation 37) is adapted accordingly:
Moreover, at least one of the liquids must have equal basic and polar parts. Usually water is chosen for this purpose because it serves as neutral point in the LEWIS scale.
2.9 Predicting the wetting behavior: the “wetting envelope”
The “wetting envelope” is not an independent calculation method for the polar and disperse fractions of the surface energy of a solid, but only a special type of presentation. It can be used for all surface energy calculation methods which provide a polar fraction and a disperse fraction in the result.
With the help of the wetting envelope and a knowledge of the polar and disperse fractions of the surface energy of a solid it is possible to predict whether a particular liquid, whose surface tension components are also known, will wet the solid completely. The following relationships make this possible:
A liquid will wet a solid surface completely when the work of adhesion W sl between the solid surface and the liquid is greater than work of cohesion W ll within the liquid. The difference between these two quantities is known as the spreading pressure S l /s :
The work of cohesion can also be described with the help of the contact angle between the liquid and the solid and surface tension of the liquid:
The following figure shows the connection between contact angle and wettability:
The procedure is demonstrated below using two liquids as an example:
| Liquid |
Disperse fraction
|
Polar fraction
|
Wetting behavior |
| Ethanol |
17.5
|
4.6
|
wetted completely |
| Cyclopentanol |
27.2
|
5.5
|
not wetted completely |
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.
- 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.
3.2.1 Advancing angle
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).
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.
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 (y=a + bx + cx0,5 + d/lnx + e/x2) 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:
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.
Δρ apex - Δρ P = z · Δρ · g
Δρ = difference in density between the drop liquid and its surroundings; g = 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:
Δρ P = σ · (k p,1 + k p,2 )
Equation 62
k p,1 (2) = principal curvatures at Point P (x,z)
k apex). From differential geometry the analytical expressions for the curvatures of the principal normal sections at Point P (x,z) are known:
Equation 64
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:
a = capillary constant
With these definitions Equation 65 can also be expressed in the following way:
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 (x0, z0) (co-ordinate origin) and the angular variation Θ of the drop from the plane of symmetry.
from the plane of symmetry
