The Van Der Waals Force
The van der Waals force
2 pBolyeV=NkT (pvdW+a/V )(V‐b)=NkT
1660 Boyle’s law 1873 van der Waal’s gas equation Coefficient a≥0, p ≤p N, number of particles, vdW Bolye because of attractive forces; k, Boltzmann constant Volume of particles, b≥0 The van der Waals force
Field around a dipole in 3D 1/r3
1/r6 dipole‐dipole interaction: Permanent dipoles (Keesom force) Dipole‐induced dipoles (Debye force) Instantaneous dipoles (London dispersion force)
Neutral atoms attract.
In matter in general, attraction is the norm! Atom‐atom interaction potential can be generalized to condensed media, macroscopic bodies
Extension to condensed media, two half spaces, pairwise summation of dipole interactions (Derjaguin 1934, Hamaker 1937) Geckos adhere to surfaces by the van der Waals force!
Autumn et al., PNAS (2002) breakdown of pairwise force‐laws Saw already that for van der Waals interactions atom‐ atom interaction energy, 1/r6 When generalized to parallel plates, this gives interaction energy/area, E/A 1/r2
When trying to use this 1/r6 atom‐atom interaction potential to describe the behaviour of colloidal particles, Overbeek found that at large distances a power law dependence of 1/r7 rather than 1/r6 was required.
Explained on the basis of retardation:
1/r6 1/r7 because at large distances the reflected field is out of phase with the source dipole. i.e., quasi‐static picture fails for separations ~ , where is the frequency of dipolar fluctuation.
15 ‐1 e.g., e=3.310 s for the Bohr atom retardation sets in at d 100 nm.
.... but Hendrik Casimir suggested a different picture [Casimir & Polder, Phys Rev. (1948)] Casimir Effect
Theoretical Physicist Studied with Ehrenfest, Pauli and Bohr
‟Casimir understood what Pauli did not, that fundamentally new knowledge and novel technology are inter‐dependent. In his later years, Casimir would develop his experience into a model for research which he called the Science‐Technology spiral. He liked to say that by going from fundamental research into industrial research management, he exemplified Hendrik Casimir 1909‐2000 as a person what happens to ideas all the time.ˮ The Casimir picture
, d , d 2 720
Casimir force considered as ‟proofˮ of the reality of the zero point energy The Casimir force Extension to condensed media, two half spaces, pairwise summation of dipole interactions (Derjaguin 1934, Hamaker 1937)
Modern macroscopic point of view: focus on EM waves
Casimir’s ideally Generalized to any two conducting flat plates flat surfaces of any in vacuum: 1940s material Lifshitz, Dzyaloshinksii& Pitaevski 1950s Measurement of the Casimir Force
1 ∝
l
Consider a slope of 10‐5: 100 nm rise over 1 cm (0.01 mm over 1 m)
for l=100 nm, force/Area is only 30% of expected value Proximity Force Approximation (Derjaguin approximation)
R2 z z2 x
z1
R1 z
x dx Proximity Force Approximation (Derjaguin approximation) f(Z) is force per unit area of 2 d two flat plates R 2 2 2 z2 1 1 x 2 2 dZ z1
W(D) is energy per unit area of R1 = 2 two flat plates at distance, D
For R2pR1 2 Force between two spheres (expt.) = Energy per unit area of two flat plates (theory)
Valid as long as sphere radii, R p range of interaction and separation, D 10 cm force
1 nN is the force exerted by 10‐10 kg: 2.5 cm a fine grain of sand! Lamoreaux, PRL (1997) Casimir Force