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

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

9.2.2 Electrostatic Adhesion Forces

Two types of electrostatic forces may act to hold particles to surfaces. The first type of force is due to bulk excess charges present on the particle or surface which produce a classical Coulombic attraction known as the electrostatic image force. Consider two spheres of radii r₁ and r₂ located a distance z_sep apart, where z_sep << r₁, r₂. Then following Eqn. 4.40 the image force is given by:¹¹⁴⁸

{Eqn. 9.9}

where q₁ and q₂ are charge on the two spheres (coul), s₁ and s₂ are the surface charge densities (coul/m²), e₀ = 8.85 x 10^-12 farad/m (permittivity constant), k_e is the dielectric constant of the medium, and d_sep ~ r₁ + r₂ is the distance between charge centers. Taking s₁ ~ s₂ = 10^-5 coul/m² (~63 charges/micron²; cf. ~1000 charges/micron² for "highly charged surfaces"¹¹⁵¹), r₁ ~ r₂ = 0.5 micron and k_e = 1 in vacuo, then F_image ~ 0.009 nN.

The electrostatic image force between two flat plates of equal interfacial area A, equal charge density s, and uniform separation z_sep with z_sep << A^1/2, is:⁷²⁷

{Eqn. 9.10}

For s ~ 10^-5 coul/m² and A = 1 micron² in vacuo (k_e = 1), F_image ~ 0.006 nN (~0.00006 atm). At larger z_sep and a voltage differential V_plate between the plates with capacitance C_plate, then:

{Eqn. 9.11}

Thus if A = 1 micron², V_plate = 1 volt, and z_sep = 0.2 micron, then F_image ~ 0.1 nN.

The second and more important type of electrostatic force for very small particles is the electrostatic contact potentials, also known as induced electrical double layer forces.¹¹⁴⁶ When two different initially uncharged materials with different local energy states and work functions are brought into contact, charge flows between them until an equilibrium is reached. The resulting potential difference is called a contact potential, V_contact, which typically ranges from ~0-1 volt. In the case of two metals, only the surface layer which is x_contact ~ 1 nm thick (the gap for tunneling) carries contact charges; for semiconductors and insulators, these regions may extend into the bulk up to x_contact ~ 1 micron or deeper.¹¹⁴⁶ For a spherical particle of radius r at a distance z_sep from a surface, the induced charge density¹¹⁴⁷ is s = e₀ V_contact / z_sep and the contact force is:¹¹⁴⁶

{Eqn. 9.12}

Taking r = 0.5 microns, V_contact = 0.5 volts, and z_sep = 5 nm, then F_contact ~ 1 nN (~0.01 atm). Induced charge densities vary experimentally from 2 x 10^-7 coul/m² (~2 x 10^-11 atm contact pressure) for polystyrene on polystyrene¹¹⁵³ up to 2 x 10^-2 coul/m² (~200 atm) for SiO₂ on mica.¹¹⁵⁴ In this latter study, a pair of contact-adhered silica/silica sheets required a force of ~0.08 nN/nm to pull them apart and a mica/mica pair required ~ 0.11 nN/nm; however, due to contact electrification a silica/mica pair required 6.6-8.8 nN/nm to separate.¹¹⁵⁴ Proper nanomechanical design can avoid or enhance these effects, as required.¹⁰

Double layer electrostatic contact forces usually predominate over electrostatic image forces for small particles; image forces are important only for materials that can carry high surface charges, such as polymers of poor conductivity or other extreme insulators. Because of their r/z_sep dependency, contact electrical forces are usually more important than van der Waals forces only for large particles and at separations wider than ~100 nm, although electrical forces can dominate in smaller particles if the surface asperities of the particles are significant enough to remove the bulk of the particle from the contact point, thus greatly reducing the van der Waals interaction.¹¹⁴⁶ In either case, the total forces of adhesion far outweigh the force of gravity, given by:

{Eqn. 9.13}

For density r = 3510 kg/m³ for diamond, acceleration of gravity g = 9.81 m/sec², and particle radius r = 0.5 micron, F_gravity ~ 10^-5 nN; for r = 1 nm, F_gravity = 10^-13 nN.

Entropic and electrostatic effects can sometimes interact to produce an apparent repulsion between opposite charges -- as, for example, between positively charged bilayer surface and negatively charged colloidal particles.³⁶⁷⁴

Last updated on 20 February 2003