Page 26 - Handbook of Adhesion Promoters

2.6 Surface free energy and wetting 19

the surface free energy of liquid (surface tension of a liquid)
γ l
θ wetting angle (popularly known as contact angle) (angle of solid-liquid interphase)
Contact angle can be measured by various types of instruments (goniometers, high-
resolution cameras) and calculated using the following equation:
h
θ = 2arctan --- [2.7]
r
where:
h height of a droplet
r radius of a droplet

The equation [2.6] applies only to one-dimensional spreading and, therefore, it has
mostly historical importance (it is not used for calculations or modeling).
The work of adhesion is measured using Dupré’s equation, as follows
(
W = γ 1 + cos θ) [2.8]
l
a
but the surface tension of the liquid is taken from one the equations given below.
Considering that the surface tension is a combination of dispersion forces (van der
Waals forces) and polar forces (hydrogen bonding), Fowkes proposed the following equa-
tion for description of the total surface tension:

d
γ tot = γ + γ p i [2.9]
i
i
where: d
γ i dispersion component
γ p i polar component
The dispersive component contains all the London forces such as dispersion (Lon-
don-van der Waals), orientation (Keesom-van der Waals), induction (Debye–van der
Waals) and Lifshitz-van der Waals (LW) forces), and the polar component consists of
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hydrogen bonding (acid/base) component. The equation only considers the dispersive
interactions of the system and, therefore, it is not reliable in the case of complex sys-
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tems. It can provide useful approximations for simple systems. 30
Geometric means theory was developed by Owens and Wendt resulting in the fol-
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lowing equation:
p p
d d
γ sl = γ + γ – 2 γ γ – 2 γ γ [2.10]
l
s l
s l
s
Due to the presence of a polar term, at least two liquids of known surface tension have to
be used in the test to calculate solid surface components. 31,32 Several other theories are
popular among researchers. Their broader account can be found in the review papers. 8,30
Currently the surface tension of adhesive components is determined using several (5-10)
liquids of different surface tensions and proportions between dispersion and polar compo-
nents.
The effect of the surface treatment on the surface properties of carbon fiber in the
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epoxy laminates has been analyzed. The shear strength and mode I fracture energy of
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laminate was correlated to the polar component of the surface free energy.