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

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

9.2.3 Immersive Adhesion Forces

For hydrophilic particles exposed to high humidity, or immersed and then withdrawn from a fluid, a liquid film can form on the surface of the particle by capillary condensation or by capillary action between particle and surface. For a spherical particle of radius r linked by a liquid bridge to a planar surface of the same material with contact area A, separation z_sep, and liquid surface tension g, then the capillary force between sphere and plane is F_capillary ~ 2 g A / z_sep for small contact angles between the liquid and the surfaces.¹¹⁴⁷ Assuming hydrophilic (wettable) surfaces with wetting coefficient cos(q) as defined in Section 9.2.4 and z_sep << r, the capillary force is given approximately by:^{1146,1147,1149}

{Eqn. 9.14}

which has been confirmed experimentally by direct measurement.¹¹⁴⁹ For the water-air interface, cos(q) ~ 1 and surface tension g = 75.6 x 10^-3 N/m at the freezing point (273 K), 72.75 x 10^-3 N/m at room temperature (293 K), 70.05 x 10^-3 N/m at human body temperature (~310 K), 58.90 x 10^-3 N/m at the boiling point at 1 atm (373 K), and 110 x 10^-3 N/m for ice at 273 K.^763,1149 Thus at room temperature, the capillary force on an r = 0.5 micron particle is F_capillary ~ 460 nN; an r = 10 nm particle feels F_capillary ~ 9 nN, a rather substantial force. Liquid bridges containing dissolved substances may evaporate and create solid crystalline bridges. Hydrophobic surfaces are generally unaffected by capillary forces.¹¹⁴⁶

Eqn. 9.14 also applies to nanorobots that are caught in an air-liquid interface such as the surface of a puddle of water, alveolar fluids in the lung, or an epidermal pool of sweat. However, adding capillary-active solute to a solvent may lower surface tension considerably. For dilute surface concentrations of a solute of molecular area A_mol in solvent of surface tension g₀ at temperature T, the resulting surface tension of the solution drops to:³⁹⁰

{Eqn. 9.15}

For palmitic or stearic acid (e.g., soap), A_mol ~ 0.21 nm²;²¹⁷⁸ added to water (g₀ = 70.05 x 10^-3 N/m) at 310 K, surface tension falls to g = 49.70 x 10^-3 N/m. More concentrated solutions further reduce surface tension, nonlinearly with concentration. For example, g ~ 25 x 10^-3 N/m for a 0.1 M aqueous fatty-acid solution of hexanoic acid (CH₃(CH₂)₄COOH; MW = 116 daltons), a total of ~6 x 10⁷ molecules, ~0.01 micron³, or ~0.01 picograms of hexanoic acid per micron³ of solution.³⁹⁰ For the aqueous solution-air interface, inorganic electrolytes (e.g., NaCl), organic acid salts, low-MW bases, and certain nonvolatile nonelectrolytes such as glucose and glycerin are capillary inactive.³⁹⁰

Surface tension is reduced even more dramatically by biological surfactants. Lung surfactant, whose activity is largely attributed to the phospholipid palmitoylphosphatidylcholine,⁹⁹⁶ produces a nonlinear change in surface tension with area. As the lungs fill with air, more-inflated alveoli have higher g's than less-inflated alveoli, serving to stabilize alveoli of different sizes.⁵²⁶ Aqueous solutions of surfactant from mammalian lung extracts show a surface tension reduction to as low as g ~ 0.7 x 10^-3 N/m at 20% of maximum film distension.²¹⁷⁹ Survanta, an intratracheal pulmonary surfactant commonly used in the treatment of respiratory distress syndrome (aka. hyaline membrane disease) in premature infants, lowers minimum aqueous surface tension to g <~ 8 x 10^-3 N/m.²¹¹⁹

Adhesion forces are greatly reduced for particles and surfaces that remain completely immersed in liquid. Surface tension effects are largely eliminated.^1146,1147 Electrostatic image forces (Eqns. 9.9 and 9.10), already quite small, are further reduced by a factor of k_e^-1 ~ 1-2% (k_e = 74.3 for pure water at 310 K, usually reduced to ~40 in a hydrophobic environment). Electrostatic contact forces may be greatly reduced because of sorption phenomena which tend to shield the charges (Sections 3.5.1 and 4.7.1). For particles and plates made of the same material, charges generally form the same double layers on both surfaces so that the double layer interactions are repulsive, thus reducing the net adhesive force, by an amount:¹¹⁴⁶

{Eqn. 9.16}

where k = 0.01381 zJ/K (Boltzmann constant), T is temperature, r is particle radius, c_ion is the volumetric charge density of ions in the liquid, and x_contact is the double layer thickness. Taking T = 310 K, r = 0.5 micron, c_ion ~10²⁶ ions/m³ (~1% or 0.15M NaCl aqueous solution, ~human blood), and x_contact ~ 0.8 nm (~K_dh^-1; Section 3.5.1), then F_shield ~ 34 nN. (Effective double layer thickness x_contact ~ 1 nm for a 0.1M solution of univalent electrolyte, ~10 nm for a 0.001M solution.¹¹⁶²) While different ions may collect in various alignments at unlike surfaces so that attractive (i.e.,contributing to adhesion) or repulsive shielding forces are possible, in most cases involving nanodevice surfaces that are fully immersed in human body fluids the electrostatic contact forces should be quite small.

The van der Waals force is also moderately reduced in fluids. For example, van der Waals adhesion between two diamondoid nanodevice surfaces fully immersed in water is only half of its vacuum value because the Hamaker constant (Eqn. 9.2) is reduced from 340 zJ to 153 zJ. Blood cells are subject to similar forces. It has been proposed that the ~15 nm gap frequently observed between the surfaces of aggregated red cells represents the position of the potential energy minimum where the forces of electrostatic repulsion (between negatively charged red cells) and the van der Waals attractive forces are equal.¹¹⁶² Taking H = 5 zJ, r_red = 3 microns and z_sep = 15 nm in Eqn. 9.7, the net attractive force between red cells is F_vdW ~ 0.01 nN.

Surface force experiments have demonstrated molecular control of adhesion at a fine level.¹¹⁵² According to the commonly-used JKR theory,¹¹⁵⁵ the mechanical force needed to separate two elastic spheres each of radius r already adhered by molecular contact is:

{Eqn. 9.17}

where W_adhesion is the Dupre reversible work of adhesion (energy per unit area). For example, hydrocarbon rubber spheres display W_adhesion = 71 x 10^-3 J/m² in air but only 6.8 x 10^-3 J/m² in water. When sodium dodecyl sulfate¹¹⁵⁵ or dodecylammonium chloride³⁵⁴⁶ is added to the water so the liquid can wet the surface, W_adhesion actually becomes negative -- the surfaces are pushed apart by the intervening fluid. Nonwetting liquids increase adhesion. For instance, polydimethyl siloxane rubber has W_adhesion = 43.6 x 10^-3 J/m² to itself, in air. In water, a nonwetting liquid, adhesion nearly doubles to 74 x 10^-3 J/m²; in methanol, a wetting liquid, adhesion is almost eliminated, falling to 6 x 10^-3 J/m².¹¹⁶⁰ In nanomedical engineering applications where minimization of adhesive forces is an important design objective, adhesion of specific classes of materials may be virtually eliminated by surface modifications that reduce W_adhesion, as is commonplace in biological surfaces and in engineered capillary coatings designed to reduce electroosmotic drag force during capillary zone electrophoresis.¹²²⁹

Last updated on 20 February 2003