Wetting behavior analysis (WBA/wetting envelopes) and intrinsic equilibrium contact angle (IECA)

2019-2-15 15:01:08

**4.1 Wetting behavior analysis and wetting envelopes**

It may be imagined that up until now we have a *rule *which
tells us if wetting will occur when placing a liquid on a solid. This
comes from knowing the surface tension of the liquid and the surface
energy of the solid. Thus the rule which has been formulated states that
‘a liquid, having a lower surface tension than the solid surface
energy, will wet that solid’. In practice it is found that this is not
always the case and it is not, therefore, an immutable rule

A2D map of wetting can be constructed by using the components of surface tension and a plot produced which is designed to show where wetting will occur. To illustrate this Owens–Wendt model described previously has been used to construct a plot.

It is suggested above that there are other ways of understanding how contact angles, hence degrees of wetting, arise from an understanding of the forces existing in the materials and between the materials. Rather than considering surface tension (or surface energy) as a single component it can be seen that surface tension is the sum of individual components, dispersive and polar components for instance, and these can be summed to yield the surface tension. It is possible to take these components which have been generated by the empirical expressions and draw (2D) maps of wetting.

The experimental programme required to produce the necessary information is relatively simple and follows the rationale described in Section 2. The contact angles for a solution on two standard substrates, for instance glass and poly (vinyl chloride) (PVC), are determined. One of the surfaces is polar and the other is non-polar. Following this the contact angles of two standard liquids on the substrates are determined (iodomethane and water, for instance). The contact angle on the substrate of the liquid of particular interest is subsequently determined. From this information the polar and dispersive components are derived by using the Owens.

Wendt equation. For the unknown sample these components are plotted against each other. An example plot is shown in Figure 1.22

**Fig. 1.22** Wettability envelopes for a substrate with a series of liquids

The
dispersive component is plotted along the y-axis against the polar
component along the x-axis. Four points labeled A, B, C and D are shown
on the plot. An envelope, the ‘wettability
envelope’, is also plotted. The envelope is created when the
Owens–Wendt model is solved for the case of a contact angle of 90°; So the area bounded by the axes and the curve is less than 90° and that outside this boundary is greater than 90°.
For each of the four liquids it can be seen that A and B will wet the
substrate. From this it is clear that the two materials A and B, which
have different overall surface tensions, can be plotted on a map, and
both can be seen to be wetting. The values of polar and dispersive
contributions have been calculated previously. This starts to give an
understanding of wettability. It is also clear that D is de-wetted,
since it sits outside that envelope. In the case of liquid C it has a
contact angle of 90° and so is on the border
of wetting and de-wetting. It is possible to make a comparison here
between this type of approach and the calculation of solubility
parameters. The
Hildebrandt solubility parameter [103] gives a single value of
solubility, whereas it is also possible to break the solubility
parameter into components, as is done in the Hansen solubility
parameters [103,104], or partial solubility parameters. In the latter a
three-component coordinate set of parameters can be identified, which
map out the solubility in a more detailed manner and give a greater
insight into the solubility of materials, polymers for instance, in
different solvents. Thus the solubility of a material is put onto a 3D
map which has axes of polar, dispersive and hydrogen bonding. The same
principle is adopted here, but using surface tensions, to see how
wettability is influenced by the components of surface tension. Other
models such as **Wu1, and Wu2,** can also be used to draw these
envelopes. This gives another view and method of predicting the wetting
behavior of substrates with liquids.

**4.2 Intrinsic equilibrium contact angle (IECA)** [Radmor approach]

In
addition to experimental procedures to determine the thermodynamic
equilibrium contact angle, it can also be evaluated from the measured
advancing and receding contact angles as derived by Radmor [105, 106] by
the relation of the as-placed contact angle to drop size.

The combination of the Young equation
and the Wenzel equation gives the relation between the surface tensions
and the global energy minimum equilibrium contact angle θ_{0} for the liquid drop:

(1.78)

where are the interfacial tensions (or interfacial energies [107–109])
between phases i and j , and the indexes S, L and V stand for solid,
liquid and vapor, respectively (though vapor may sometimes refer to
another liquid medium surrounding the drop). The reason a drop can have a
contact angle that is different from θ_{0} is related to the pinning of the three phase contact line to its
position which induces a force resisting drop motion. One can describe
the force per length associated with this pinning, k/r, by Eq. (1.79)
(see [110-115]) (where r is the radius of the circle the drop makes with
the surface). In this description k has opposite values, k_{A} and kR corresponding to advancing and receding contact angles:

(1.79a)

(1.79b)

where r_{A} and r_{R} are drop radii that correspond to the advancing and receding curvatures and γ ≡ γ_{LV}.

From this, the relation between the θ_{A}, θ_{R} and θ_{0} is given by [111]

(1.80)

where

and

Equation
1.69 has been derived from a combination of Young and Wenzel equations
and recognizing that the equilibrium contact angle results from the
global energy minimum in the system [105,106]. The advancing and
receding contact angles result from pinning of the three-phase contact
line, resisting motion of the drop. He assumed that the resistance to
the motion *out* for advancing drop was just equal to the resistance to the motion *in* of the receding drop. This is because both of these resistance are due
to the three-phase contact line pinning to the similar protrusions.[105]
in other words, the irregularities on the surface are isotropic with
respect to their nature and distribution [105,106]. However, this is
rather a weak assumption in the derivations.

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