© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

9.2.1 Van der Waals Adhesion Forces

In dry or vacuum environments, adhesion forces between <100 micron diameter particles and surfaces (nanorobot and other) separated by 100 nm or less are usually dominated by van der Waals interactions.¹¹⁴⁹ Consider a spherical particle of radius r lying a small distance z_sep <~ 100 nm from a flat plate of surface area A >> p r². As a crude approximation, ignoring retardation effects that may become important in solution for z_sep > 5 nm,¹⁰ the van der Waals adhesive force is approximated by:¹¹⁴⁶

{Eqn. 9.1}

for z_sep << r, where H is the Hamaker constant for the interaction. (Eqn. 9.1 also applies to two crossed cylinders of equal radius.¹¹⁵²) The Hamaker constants for the materials comprising the sphere, the plate, and the surrounding medium are H_s, H_p, and H_m, respectively (Table 9.1), giving:¹¹⁴⁹

{Eqn. 9.2}

The adhesion separation distance z_sep is often taken as ~0.4 nm for particles in intimate (but chemically unbonded) contact with a surface.^1146,1148 Hence the force required to overcome the van der Waals adhesive attraction of a perfectly rigid r = 0.5 micron particle adhered to a diamond plate in vacuo (with z_sep = 0.4 nm, H = 340 zJ) is F_vdW ~ 180 nN.

Real particles are not perfectly rigid but deform under the influence of this adhesive force. If the adhesion surface area is increased from point contact to a circle of radius r_adhesion, then the van der Waals adhesion force increases to:¹¹⁴⁶

{Eqn. 9.3}

According to the JKR theory of adhesion mechanics¹¹⁵⁵ for a sphere on a flat surface of the same material with a work of adhesion W_adhesion (joined surface energy; Section 9.2.3) and elastic modulus K_e:¹¹⁴⁹

{Eqn. 9.4}

Assuming W_adhesion ~10 J/m² and K_e ~ 10¹² N/m² for smooth flawless diamond and r = 0.5 micron gives r_adhesion ~ 36 nm. Taking z_sep = 0.4 nm contact with a flat diamond plate over a ~4100 nm² circular adhesion spot in vacuo, then F_vdW ~ 1300 nN, a mean surface pressure of p_mean = F_vdW / p r_adhesion² ~ 3200 atm over the contact circle and a peak pressure p_peak = 1.5 p_mean = 4800 atm at the center. As the sphere is pulled away, separation occurs abruptly when the contact radius falls to r_adhesion / 4^1/3 = (~63%) r_adhesion = 23 nm.

Van der Waals forces also depend upon the roughness of the surface. If a smooth sphere approaches a surface having uniform rugosity or roughness b_rug with roughness feature size << r, then the van der Waals force is approximately:¹¹⁴⁸

{Eqn. 9.5}

For z_sep = 0.4 nm and b_rug = 10 nm, F_rug = 2% F_vdW ~ 4 nN for the rigid sphere (in vacuo) in the examples given above. Surface dimpling is a common practice in MEMS fabrication, for example, to reduce adhesion between polysilicon layers and the substrate.¹³⁸²

A few other geometries of rigid bodies approaching contact are useful as well. For example, the van der Waals adhesive force between two flat plates of equal interfacial area A and uniform separation z_sep is:¹¹⁵²

{Eqn. 9.6}

Two (1 nm)² diamond plates separated by z_sep = 0.4 nm in vacuo require F_vdW ~ 0.07 nN (~7 atm) to overcome the van der Waals attractive force. Such plates are the same size as the contacting end surfaces of the logic rods employed in Drexler's mechanical computer design (Section 10.2.1); the design assumes a rod realignment force of 1 nN in the exemplar calculations.¹⁰ This result suggests that forces on the order of 10-100 pN must be applied to mobile nanoscale components inside mechanical nanodevices in order to overcome static van der Waals adhesion that arises due to the contact proximity of parts within the structure.

For another example, two rigid spheres of radii r₁ and r₂, separated by a distance z_sep << r₁, r₂, have a van der Waals adhesive force of:¹¹⁵²

{Eqn. 9.7}

where the reduced radius r_red = (r₁ r₂) / (r₁ + r₂). A pair of 1-micron diameter rigid diamond spheres separated by z_sep = 0.4 nm in vacuo have a van der Waals mutual adhesive force F_vdW ~ 90 nN.

As one final example, the van der Waals force between two parallel cylinders of length L, radii r₁ and r₂, and separation z_sep is (differentiating V_vdW in Drexler¹⁰):

{Eqn. 9.8}

A pair of 1-nm diameter, 10-nm long rigid diamondoid cylinders lying exactly z_sep = 0.4 nm apart along their entire length in vacuo feel a van der Waals force of F_vdW ~ 1.5 nN. In all three examples, as before, the force rises slightly if the objects deform and declines significantly if surface rugosity is increased. (See also entropic packing, Section 9.4.2.3.)

Last updated on 20 February 2003