The Derjaguin Approximation. The force, F
c
, between two gently curved surfaces is proportional to the
interaction energy per unit area, E
f
, between two flat ones at the same separation. This relationship, known as
the Derjaguin approximation, allows straightforward interpretations to be made of surface force measurements
between crossed cylinders (or between one sphere and another or between a sphere and a flat plate). It is also
helpful in certain adhesion measurements, as described below.
The Derjaguin approximation is derived (see Ref 7, for example) by considering the force between each
element of one curved surface and each element of the other, and then integrating over the two surfaces to
obtain the total force. As long as the radius of curvature is much larger than the range of the surface force, this
is approximately equivalent to integrating the force per unit area, F
f
, between flat surfaces, from the minimum
separation of the curved surfaces, D, to an effectively infinite upper limit, with some geometrical factors to
account for the shape of the surfaces. The integral simply gives the interaction energy between flats, E
f
(D),
which is the work done against the surface forces in moving the flat surfaces from infinity to D. For two
spheres of radius R
1
and R
2
, the geometrical factor is a constant, giving:
F
c
(D) = 2π R E
f
(D)
where 1/R = 1/R
1
+ 1/R
2
. It can be shown that the geometry of crossed cylinders of equal radii, R
c
, is equivalent
to a sphere of radius R
c
approaching a flat plate, or to two spheres of radius 2 R
c
approaching each other.
Substrate Materials. The original and still most common solid material used in surface force measurements is
mica, chosen because it satisfies the requirements (thin and transparent) of the optical interference technique
used in the SFA and because it is easy to prepare large areas of molecularly smooth surface by cleavage.
Experiments have been conducted on mica surfaces immersed in many different liquid and vapor environments
(Ref 8).
Recently, there has been some success in extending these measurements to a wider range of surfaces. One
approach is to coat mica surfaces by various techniques, including Langmuir-Blodgett deposition, surfactant or
polymer adsorption from solution, plasma modification, and evaporative coating of thin metal, carbon, and
metal-oxide films. An alternative approach is to find a means of preparing other transparent materials as
micron-thick foils with very smooth surfaces. This has been done for sapphire, silica, pyrex glass, and certain
polymers. It is reasonable to expect that the range of materials studied will continue to increase in the near
future.
Currently, the best way to prepare metal surfaces for SFA appears to be thin-film evaporation onto mica or
another smooth substrate. Because the optical technique requires some light to pass through the two films, their
thicknesses cannot be more than a few tens of nanometers. It is possible to use the metal films themselves as
one or both optical interferometer mirrors, but the fringes of equal chromatic order would disappear from the
visible spectrum if the two metal surfaces were brought closer together than about 1 μm (40 μin.). In that case,
an alternative method of measuring separation, such as capacitance, would be required.
Environments. Tests with the surface force apparatus can be conducted in many different liquids or vapors, as
long as they are compatible with the materials of the SFA system (namely, stainless steel, silica, Kel-F, and
Teflon). There is a provision to heat the chamber to around 100 °C (212 °F). Use of an appropriate heating
jacket could extend the temperature range from, perhaps, -50 to 150 °C (-60 to 300 °F). At about 150 °C (300
°F), the silver layers used for interferometer mirrors degrade. This limit might be raised by using other optical
coatings. The next limitation of the current design would be the maximum operating temperature of the Teflon
seals, which is 250 °C (480 °F). In principle, the same or comparable techniques could be extended to operate
at several hundred degrees, but in practice this would require a major redesign of the apparatus.
At present, the SFA is intended only to operate at or near ambient pressure. With some modifications to the
seals, it could be made to hold moderate vacuum, say 10
-4
Pa (10
-6
torr). A total redesign would be required to
build a device for making comparable measurements in ultrahigh vacuum (UHV) conditions.
Preparation of Surfaces and Fluids. Any solid to be investigated by the SFA method should be smooth,
compared to the range of forces under examination. Because the adhesion between surfaces is often dominated
by very short-range forces, atomically smooth surfaces would be required to make fundamental and
reproducible measurements of these. However, rough surfaces still adhere, and so measurements can be made
without insisting on atomic smoothness. The drawback, in that case, is that it would be more difficult to obtain
a straightforward interpretation of the results.
Because SFA measurements involve extremely small surface separations, there are stringent requirements for
cleanliness. One speck of dust in the wrong place can spoil the entire test. The relative importance of surface
cleanliness is again related to the range of force under investigation. For very short-range forces, even a