Page 31 - Nanotechnology an introduction

According to the Derjaguin approximation, a sphere of radius r (material 3) interacting with an infinite planar surface (material 1) has the following
free energies of interaction as a function of z, the perpendicular distance between the plane and the nearest point of the sphere:

(3.14)
where ℓ is the equilibrium contact distance (about 0.15 nm);
0
(3.15)
where χ is the decay length for the ab interactions; and where electrostatic charges are present,

(3.16)
where ψ are the electrostatic surface potentials of materials 1 and 3 and 1/κ is the Debye length (inversely proportional to the square root of the
ionic strength—this is why electrostatic interactions tend not to be very important in salty aqueous systems).

3.2.2. Critique of the Surface Tension Formalism

As Bikerman has pointed out [20], there are some difficulties with applying the concept of surface tension to solids. It is generally recognized that
most real, machined solid surfaces are undulating as shown in Figure 3.3. Capillary pressure P :
c
(3.17)
where r is the radius of the undulation, would tend to flatten such undulations since the pressure where r is small would engender a strain easily
exceeding the yield strength of most metals, yet such undulations persist.

Figure 3.3 Sketch of the undulations of a somewhat idealized real machined surface. The arrows indicate the direction of capillary pressure.
The equation for the three-phase line between a vapor (phase 1) and two liquids (phases 2 and 3) is well known to be

(3.18)

where the angles , etc. enclose phase 1, etc. The assumption that solids have a surface tension analogous to that of liquids ignores the
phenomenon of epitaxy (in which a crystal tends to impose its structure on material deposited on it). As an illustration of the absurdities to which the
uncritical acceptance of this assumption may lead, consider a small cube of substance 2 grown epitaxially in air (substance 3) on a much larger
cube of substance 1 (Figure 3.4). The angles A and B are both 90° and correspond to and respectively in equation (3.18). It immediately
follows that the surface tension of substance 1 equals the interfacial tension between substances 1 and 2, a highly improbable coincidence.
Moreover, since equals 180°, the surface tension of substance 2 must be zero, which also seems highly unlikely.

Figure 3.4 Illustration of a small cube of substance 2 grown epitaxially on a large cube of substance 1. After Bikerman [20].
Given these difficulties in the application of the surface tension concept to rigid solids, it is appropriate to exercise caution when using the
approach described in Section 3.2.1. Many of the applications in nanotechnology (especially those concerned with the nano/bio interface) involve,
however, soft matter, to which the application of the formalism may be reasonable.

3.2.3. Experimental Determination of Single-Substance Surface Tensions

The general strategy is to measure the advancing contact angles θ on the material 3 whose surface tension is unknown using three appropriate
liquids with different surface tension components (determined, e.g., from hanging drop measurements; the shape of the drop, in particular its two
radii of curvature r and r , are measured for a pressure difference ΔP and the surface tension calculated from the Laplace law, equation (2.2), i.e.,
2
1
) [127]. With these values, three Young–Dupré equations:

(3.19)
can be solved to yield the unknowns , and . Values of the surface tension parameters for some common materials have been given in
Table 3.1 and Table 3.2. Methods for computing ψ for ionized solids (e.g., polyions, protonated silica surfaces) are based on the Healy–White
ionizable surface group model [72].

A major practical difficulty in experimentally determining surface tensions of solutes is that under nearly all real conditions, surfaces are