# Calculations and measurement of forces

## Experimental techniques

• Forces are easier to measure than energies.
• The Derjaguin approximation is used to convert to energies.
• Most of these techniques only give the force (or energy) at a single distance (or concentration).
• A surface force apparatus can be used for distance dependence.

## Surface force apparatus

The surface force apparatus is pioneered by D. Tabor, R.H.S. Winterton, J.N. Israelachvili in the early 1970s at Cambridge University. This device can resolve distances to within 0.1 nanometer and forces at the $10^{-8}$ N level. Compared to an atomic force microscope, the surface force apparatus is more ideally suited to measuring surface-surface interactions, and can measure much longer-range forces more accurately.

The surface force apparatus (SFA) is a scientific instrument and technique pioneered by D. Tabor, R.H.S. Winterton, J.N. Israelachvili in the early 1970s at Cambridge University. By the mid-70's Israelachvili had adapted the original design to operate in liquids, notably aqueous solutions, while at the Australian National University.

Measurement of adhesion forces and interfacial energy can be analyzed by JKR (Johnson, Kendal, Roberts) theory for large soft objects, or DMT(Derjaguin, Muller, Toporov) for small hard objects

Johnson, Kendal, Roberts Formula:

$F_{adhesion}=3\pi \gamma R$

Derjaguin, Muller, Toporov Formula:

$F_{adhesion}=4\pi \gamma R$

## How strong are vdW Forces?

 Measurable? Yes, even non-ideal gases show the effects. Thermally? Hamaker constants are about 1 to 5 kT. Significant if separation is less than size. Mechanically? Gravitational force is $F_{\downarrow }=\rho L^{3}g$ vdW force is: $F_{vdW}=\frac{A}{6\pi l^{3}}L^{2}=F_{\uparrow }$ Pick A = 1 kT, 10 nm separation, 1 gm/cm^3 Supports a 8 cm^3 bug.

Parsegian, p. 21

## Lifshitz theory

 The first clear evidence for forces between neutral molecules came from van der Waals (in a PhD thesis, 1873). Hamaker extended the idea of molecule pair-wise interaction, using London's equation, to that between large bodies.  The modern view (Lifshitz et al) is that the attraction is really due to random fluctuations of electrons (producing fluctuating electric fields).  http://www.erpt.org/051Q/leed-01.pdf The first three pages of this paper explains very well how the Van der Waals theory has developed to Lifshitz theory.