Israelachvil, J. N. et al.“Surface Forces and Microrheology of .” Handbook of Micro/Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press LLC, 1999 © 1999 by CRC Press LLC © 1999 by CRC Press LLC 9 Surface Forces and Microrheology of Molecularly Thin Liquid Films Jacob N. Israelachvili and Alan D. Berman 9.1 Introduction 9.2 Methods for Measuring Static and Dynamic Surface Forces Adhesion Forces • Force Laws • The Surface Force Apparatus and the Atomic Force Microscope 9.3 van der Waals and Electrostatic Forces between Surfaces in Liquids van der Waals Forces • Electrostatic Forces 9.4 Solvation and Structural Forces: Forces Due to Liquid and Surface Structure Effects of Surface Structure • Effect of Surface Curvature and Geometry 9.5 Thermal Fluctuation Forces: Forces between Soft, Fluidlike Surfaces 9.6 Hydration Forces: Special Forces in Water and Aqueous Solutions Repulsive Hydration Forces • Attractive Hydrophobic Forces • Origin of Hydration Forces 9.7 Adhesion and Capillary Forces Adhesion Mechanics 9.8 Nonequilibrium Interactions: Adhesion Hysteresis 9.9 Rheology of Molecularly Thin Films: Nanorheology Different Modes of Friction: Limits of Continuum Models • Viscous Forces and Friction of Thick Films: Continuum Regime • Friction of Intermediate Thickness Films © 1999 by CRC Press LLC 9.10 Interfacial and Boundary Friction: Molecular Tribology General Interfacial Friction • Boundary Friction of Surfactant Monolayer-Coated Surfaces • Boundary Lubrication of Molecularly Thin Liquid Films • Transition from Interfacial to Normal Friction (with Wear) 9.11 Theories of Interfacial Friction Theoretical Modeling of Interfacial Friction: Molecular Tribology • Adhesion Force Contribution to Interfacial Friction • Relation between Boundary Friction and Adhesion Energy Hysteresis • External Load Contribution to Interfacial Friction • Simple Molecular Model of Energy Dissipation ε 9.12 Friction and Lubrication of Thin Liquid Films Smooth and Stick-Slip Sliding • Role of Molecular Shape and Liquid Structure 9.13 Stick-Slip Friction Rough Surfaces Model • Distance-Dependent Model • Velocity-Dependent Friction Model • Phase-Transition Model • Critical Velocity for Stick-Slip • Dynamic Phase Diagram Representation of Tribological Parameters Acknowledgment References 9.1 Introduction In this chapter the most important types of surface forces are described and the relevant equations for the force laws given. A number of attractive and repulsive forces operate between surfaces and particles. Some of these occur only in vacuum, for example, attractive van der Waals and repulsive hard-core interactions. Others can arise only when the interacting surfaces are separated by another condensed phase, which is usually a liquid medium. The most common types of surface forces and their main characteristics are list in Table 9.1. In vacuum , the two main long-ranged interactions are the attractive van der Waals and electrostatic (coulombic) forces, while at smaller surface separations — corresponding to molecular contacts at surface separations of D ≈ 0.2 nm — additional attractive forces can come into play, such as covalent or metallic bonding forces. These attractive forces are stabilized by the hard-core repulsion, and together they determine the surface and interfacial energies of planar surfaces as well as the strengths of materials and adhesive junctions. Adhesion forces are often strong enough to deform the shapes of two bodies or particles elastically or plastically when they come into contact for the first time. When exposed to vapors (e.g., atmospheric air containing water and organic molecules), two solid surfaces in or close to contact will generally have a surface layer of chemisorbed or physisorbed molecules, or a capillary condensed liquid bridge between them. These effects can drastically modify their adhesion. The adhesion usually falls, but in the case of capillary condensation the additional Laplace pressure or attractive “capillary” force between the surfaces may make the adhesion stronger than in inert gas or vacuum. When totally immersed in a liquid , the force between two surfaces is once again completely modified from that in vacuum or air (vapor). The van der Waals attraction is generally reduced, but other forces can now arise which can qualitatively change both the range and even the sign of the interaction. The overall attraction can be either stronger or weaker than in the absence of the intervening liquid medium, for example, stronger in the case of two hydrophobic surfaces in water, but weaker for two hydrophilic surfaces. Since a number of different forces may be operating simultaneously in solution, the overall force law is not generally monotonically attractive, even at long range: it can be repulsive, oscillatory, or the force can change sign at some finite surface separation. In such cases, the potential energy minimum, © 1999 by CRC Press LLC which determines the adhesion force or energy, occurs not at true molecular contact but at some small distance farther out. The forces between two surfaces in a liquid medium can be particularly complex at short range , i.e., at surface separations below a few nanometers or 5 to 10 molecular diameters. This is partly because, with increasing confinement, a liquid ceases to behave as a structureless continuum with properties TABLE 9.1 Types of Surface Forces Type of Force Subclasses and Alternative Names Main Features Attractive van der Waals Dispersion force (v & s) Induced dipole force (v & s) Casimir force (v & s) Ubiquitous force, occurs both in vacuum and in liquids Electrostatic Coulombic force (v & s) Ionic bond (v) Hydrogen bond (v) Charge–transfer interaction (v & s) “Harpooning” interaction (v) Strong, long-ranged force arising in polar solvents; requires surface charging or charge-separation mechanism Quantum mechanical Covalent bond (v) Metallic bond (v) Exchange interaction (v) Strong short-ranged forces responsible for contact binding of crystalline surfaces Hydrophobic Attractive hydration force (s) Strong, apparently long-ranged force; origin not yet understood Ion–correlation van der Waals force of polarizable ions (s) Requires mobile charges on surfaces in a polar solvent Solvation Oscillatory force (s) Depletion force (s) The oscillatory force generally alternates between attraction and repulsion; mainly entropic in origin Specific binding “Lock and key” binding (v & s) Receptor–ligand interaction (s) Antibody–antigen interaction (s) Subtle combination of different noncovalent forces giving rise to highly specific binding; main “recognition” mechanism of biological systems. Repulsive Quantum mechanical Hard-core (v) Steric repulsion (v) Born repulsion (v) Forces stabilizing attractive covalent and ionic binding forces, effectively determine molecular size and shape van der Waals van der Waals disjoining pressure (s) Arises only between dissimilar bodies interacting in a medium Electrostatic Arises only for certain constrained surface charge distributions. Solvation Oscillatory solvation force (s) Structural force (s) Hydration force (s) Monotonically repulsive forces, believed to arise when solvent molecules bind strongly to surfaces. Entropic Osmotic repulsion (s) Double-layer force (s) Thermal fluctuation force (s) Steric polymer repulsion (s) Undulation force (s) Protrusion force (s) Forces due to confinement of molecular or ionic species between two approaching surfaces. Requires a mechanism which keeps trapped species between the surfaces. Dynamic Interactions Nonequilibrium Hydrodynamic forces (s) Viscous forces (s) Friction forces (v & s) Lubrication forces (s) Energy-dissipating forces occurring during relative motion of surfaces or bodies. Note: v, Applies only to interactions in vacuum ; s, applies only to interactions in solution , or to surfaces separated by a liquid; v & s, applies to interactions occurring both in vacuum and in solution. © 1999 by CRC Press LLC determined solely by its bulk properties; the size and shape of its molecules begin to play an important role in determining the overall interaction. In addition, the surfaces themselves can no longer be treated as inert and structureless walls (i.e., mathematically flat) — their physical and chemical properties at the atomic scale must also now be taken into account. Thus, the force laws will now depend on whether the surface lattices are crystallographically matched or not, whether the surfaces are amorphous or crystalline, rough or smooth, rigid or soft (fluidlike), hydrophobic or hydrophilic. In practice, it is also important to distinguish between static (i.e., equilibrium) forces and dynamic (i.e., nonequilibrium) forces such as viscous and friction forces. For example, certain liquid films confined between two contacting surfaces may take a surprisingly long time to equilibrate, as may the surfaces themselves, so that the short-range and adhesion forces appear to be time dependent, resulting in “aging” effects. 9.2 Methods for Measuring Static and Dynamic Surface Forces 9.2.1 Adhesion Forces The simplest and most direct way to measure the adhesion of two solid surfaces, such as two spheres or a sphere on a flat surface, is to suspend one on a spring and measure — from the deflection of that spring — the adhesion or “pull-off” force needed to separate the two bodies. Figure 9.1 illustrates the principle of this method when applied to the interaction of two magnets. However, the method is applicable even at the microscopic or molecular level, and it forms the basis of all direct force-measuring FIGURE 9.1 Schematic attractive force law between two macroscopic objects, such as two magnets, or between two microscopic objects such as the van der Waals force between a metal tip and a surface. On lowering the base supporting the spring, the latter will expand or contract such that at any equilibrium separation D the attractive force balances the elastic spring restoring force. However, once the gradient of the attractive force between the surfaces dF / dD exceeds the gradient of the spring restoring force, defined by the spring constant K S , the upper surface will jump from A into contact at A ′ ( A for advancing). On separating the surfaces by raising the base, the two surfaces will jump apart from R to R ′ ( R for receding). The distance R – R ′ multiplied by K S gives the adhesion force, i.e., the value of F at R . © 1999 by CRC Press LLC apparatuses such as the surface forces apparatus (SFA) (Israelachvili, 1989, 1991) or the atomic force microscope (AFM) (Ducker et al., 1991). If K S is the stiffness of the force-measuring spring and ∆ D the distance the two surfaces jump apart when they separate, then the adhesion force F S is given by (9.1) where we note that in liquids the maximum or minimum in the force may occur at some nonzero surface separation (see Figures 9.3 and 9.4 below). From F S one may also calculate the surface or interfacial energy γ . However, this depends on the geometry of the two bodies. For a sphere of radius R on a flat surface, or for two crossed cylinders of radius R , we have (Israelachvili, 1991) (9.2) while for two spheres of radii R 1 and R 2 (9.3) where γ is in units of J/m 2 . 9.2.2 Force Law The full force law F ( D ) between two surfaces, that is, the force F as a function of surface separation D , can be measured in a number of ways. The simplest is to move the base of the spring (see Figure 9.1) by a known amount, say, ∆ D 0 . If there is a detectable force between the two surfaces, this will cause the force-measuring spring to deflect by, say, ∆ D S , while the surface separation changes by ∆ D . These three displacements are related by (9. 4) The force difference ∆ F between the initial and final separations is given by (9.5) The above equations provide the basis for measuring the force difference between any two surface separations. For example, if a particular force-measuring apparatus can measure ∆ D 0 , ∆ D S , and K S , then by starting at some large initial separation where the force is zero ( F = 0) and measuring the force difference ∆ F between this initial or reference separation D and ( D – ∆ D ), then working one’s way in increasing increments of ∆ D = ( ∆ D 0 – ∆ D S ), the full force law F ( D ) can be constructed over any desired distance regime. Whenever an equilibrium force law is required, it is essential to establish that the two surfaces have stopped moving before the “equilibrium” displacements are measured. When displacements are measured while two surfaces are still in relative motion, one also measures a viscous or frictional contribution to the total force. Such dynamic force measurements have enabled the viscosities of liquids near surfaces and in thin films to be accurately measured (Israelachvili, 1989). FF K D SS ==⋅ max ,∆ γ= πFR s 3, γ= π +       F RR s 3 11 12 , ∆∆∆DDD S == 0 . ∆∆FK D SS = . © 1999 by CRC Press LLC In practice, it is difficult to measure the forces between two perfectly flat surfaces because of the stringent requirement of perfect alignment for making reliable measurements at the angstrom level. It is far easier to measure the forces between curved surfaces, for example, two spheres, a sphere and a flat, or two crossed cylinders. As an added convenience, the force F ( D ) measured between two curved surfaces can be directly related to the energy per unit area E ( D ) between two flat surfaces at the same separation, D . This is given by the so-called “Derjaguin” approximation: (9.6) where R is the radius of the sphere (for a sphere and a flat) or the radii of the cylinders (for two crossed cylinders). 9.2.3 The Surface Force Apparatus and the Atomic Force Microscope In a typical force-measuring experiment, two or more of the above displacement parameters: ∆ D 0 , ∆ D S , ∆ D , and K S , are directly or indirectly measured, from which the third displacement and resulting force law F ( D ) are deduced using Equations 9.4 and 9.5. For example, in SFA experiments, ∆ D 0 is changed by expanding a piezoelectric crystal by a known amount and the resulting change in surface separation ∆ D is measured optically, from which the spring deflection ∆D S is obtained. In contrast, in AFM experiments, ∆D 0 and ∆D S are measured using a combination of piezoelectric, optical, capacitance, or magnetic techniques, from which the surface separation ∆D is deduced. Once a force law is established, the geometry of the two surfaces must also be known (e.g., the radii R of the surfaces) before one can use Equation 9.6 or some other equation that enables the results to be compared with theory or with other experiments. Israelachvili (1989, 1991), Horn (1990), and Ducker et al. (1991) have described various types of SFAs suitable for making adhesion and force law measurements between two curved molecularly smooth surfaces immersed in liquids or controlled vapors. The optical technique used in these measurements employs multiple beam interference fringes which allows for surface separations D to be measured to ±1 Å. From the shapes of the interference fringes, one also obtains the radii of the surfaces, R, and any surface deformation that arises during an interaction (Israelachvili and Adams, 1978; Chen et al., 1992). The distance between the two surfaces can also be independently controlled to within 1 Å, and the force sensitivity is about 10 –8 N (10 –6 g). For the typical surface radii of R ≈ 1 cm used in these experiments, γ values can be measured to an accuracy of about ±10 –3 mJ/m 2 (±10 –3 erg/m 2 ). Various surface materials have been successfully used in SFA force measurements including mica (Pashley, 1981, 1982, 1985), silica (Horn et al., 1989b), and sapphire (Horn et al., 1988). It is also possible to measure the forces between adsorbed polymer layers (Klein, 1983, 1986; Patel and Tirrell, 1989; Ploehn and Russel, 1990), surfactant monolayers and bilayers (Israelachvili, 1987, 1991; Christenson, 1988a; Israelachvili and McGuiggan, 1988), and metal and metal oxide layers deposited on mica (Coakley and Tabor, 1978; Parker and Christenson, 1988; Smith et al., 1988; Homola et al., 1993; Steinberg et al., 1993). The range of liquids and vapors that can be used is almost endless, and so far these have included aqueous solutions, organic liquids and solvents, polymer melts, various petroleum oils and lubricant liquids, and liquid crystals. Recently, new friction attachments were developed suitable for use with the SFA (Homola et al., 1989; Van Alsten and Granick, 1988, 1990b; Klein et al., 1994; Luengo et al., 1997). These attachments allow for the two surfaces to be sheared past each other at varying sliding speeds or oscillating frequencies while simultaneously measuring both the transverse (frictional or shear) force and the normal force or load between them. The externally applied load, L, can be varied continuously, and both positive and negative loads can be applied. Finally, the distance between the surfaces D, their true molecular contact area A, their elastic (or viscoelastic or elastohydrodynamic) deformation, and their lateral motion can all be monitored simultaneously by recording the moving interference fringe pattern using a video camera–recorder system. ED FD R () = () π2 , © 1999 by CRC Press LLC 9.3 van der Waals and Electrostatic Forces between Surfaces in Liquids 9.3.1 van der Waals Forces Table 9.2 lists the van der Waals force laws for some common geometries. The van der Waals interaction between macroscopic bodies is usually given in terms of the Hamaker constant, A, which can either be measured or calculated in terms of the dielectric properties of the materials (Israelachvili, 1991). The Lifshitz theory of van der Waals forces provides an accurate and simple approximate expression for the Hamaker constant for two bodies 1 interacting across a medium 2: (9.7) where ε 1 , ε 2 , and n 1 , n 2 are the static dielectric constants and refractive indexes of the two phases and where I is their ionization potential which is close to 10 eV or 2 × 10 –18 J for most materials. For nonconducting liquids and solids interacting in vacuum or air (ε 2 = n 2 = 1), their Hamaker constants are typically in the range (5 to 10) × 10 –20 J, rising to about 4 × 10 –19 J for metals, while for interactions in a liquid medium, the Hamaker constants are usually about an order of magnitude smaller. For inert nonpolar surfaces, e.g., of hydrocarbons or van der Waals solids and liquids, the Lifshitz theory has been found to apply even at molecular contact, where it can predict the surface energies (or tensions) of solids and liquids. Thus, for hydrocarbon surfaces the Hamaker constant is typically A = 5 × 10 –20 J. Inserting this value into the appropriate equation for two flat surfaces (Table 9.2) and using a “cut-off” distance of D = D 0 ≈ 0.15 nm when the two surfaces are in contact, we obtain for the surface energy γ (which is conventionally defined as half the interaction energy): (9.8) a value that is typical for hydrocarbon solids and liquids (for liquids, γ is sometimes referred to as the surface tension and is expressed in units of mN/m). TABLE 9.2 van der Waals Interaction Energy and Force Between Macroscopic Bodies of Different Geometries Geometry of Bodies With Surfaces D Apart van der Waals Interaction (D Ӷ R) Energy Force Two flat surfaces (per unit area) E = A/12pD 2 F = A/6pD 3 Sphere of radius R near flat surface E = AR/6DF = AR/6D 2 Two identical spheres of radius RE = AR/12DF = AR/12D 2 Cylinder of radius R near flat surface (per unit length) E = F = Two identical parallel cylinders of radius R (per unit length) E = F = Two identical cylinders of radius R crossed at 90° E = AR/6DF = AR/6D 2 AR D12 2 32/ AR D82 52/ AR D24 32/ AR D16 52/ AkT I nn nn = − +       + − () + () 3 4 16 2 12 12 2 1 2 2 2 2 1 2 2 2 32 εε εε , γ= = π ≈ 1 2 0 2 24 30E A D mJ m 2 , © 1999 by CRC Press LLC If the adhesion force is measured between a spherical surface of radius R = 1 cm and a flat surface using an SFA, we expect the following value for the adhesion force (see Table 9.2): (9.9) Using the SFA with a spring constant of K S = 100 N/m, such an adhesive force will cause the two surfaces to jump apart by ∆D = F/K S = 3.7 × 10 –5 m = 37 µm, which can be accurately measured (actually, for elastic bodies that deform on coming into adhesive contact, their radius R changes during the interaction and the measured adhesion force is 25% lower — see Equation 9.21). The above example shows how the surface energies of solids can be directly measured with the SFA and, in principle, with the AFM (if the geometry of the tip and surface at the contact zone can be quantified). The measured values are generally in good agreement with calculated values based on the known surface energies γ of the materials and, for nonpolar low-energy solids, are well accounted for by the Lifshitz theory (Israelach- vili, 1991). For adhesion measurements in vacuum or inert atmosphere to be meaningful, the surfaces must be both atomically smooth and clean. This is not always easy to achieve, and for this reason only inert, low- energy surfaces, such as hydrocarbon and certain polymeric surfaces, have had their true adhesion forces and surface energies directly measured so far. Other smooth surfaces have also been studied, such as bare mica, metal, metal oxide, and silica surfaces but these are high-energy surfaces, so that it is difficult to prevent them from physisorbing a monolayer of organic matter or water from the atmosphere or from getting an oxide monolayer chemisorbed on them, all of which affects their adhesion. Many contaminants that physisorb onto solid surfaces from the ambient atmosphere usually dissolve away once the surfaces are immersed in a liquid, so that the short-range forces between such surfaces can usually be measured with great reliability. Figure 9.2 shows results of measurements of the van der Waals forces between two crossed cylindrical mica surfaces in water and various salt solutions, showing the good agreement obtained between experiment and theory (compare the solid curve, corresponding to F = AR/6D 2 , where A = 2.2 × 10 –20 J is the fitted value, which is within about 15% of the theoretical FIGURE 9.2 Attractive van der Waals force F between two curved mica surfaces of radius R ≈ 1 cm measured in water and various aqueous electrolyte solutions. The measured nonretarded Hamaker constant is A = 2.2 × 10 –20 J. Retardation effects are apparent at distances above 5 nm, as expected theoretically. Agreement with the continuum Lifshitz theory of van der Waals forces is generally good at all separations down to five to ten solvent molecular diameters (e.g., D ≈ 2 nm in water) or down to molecular contact (D = D 0 ) in the absence of a solvent (in vacuum). F AR D R==π ≈× () − 6 4 37 10 0 2 3 γ N about 0.4 grams © 1999 by CRC Press LLC nonretarded Hamaker constant for the mica–water–mica system). Note how at larger surface separations, above about 5 nm, the measured forces fall off faster than given by the inverse-square law. This, too, is predicted by Lifshitz theory and is known as the “retardation effect.” From Figure 9.2 we may conclude that at separations above about 2 nm, or 8 molecular diameters of water, the continuum Lifshitz theory is valid. This can be expected to mean that water films as thin as 2 nm may be expected to have bulklike properties, at least as far as their interaction forces are concerned. Similar results have been obtained with other liquids, where in general for films thicker than 5 to 10 molecular diameters their continuum properties, both as regards their interactions and other prop- erties such as viscosity, are already manifest. 9.3.2 Electrostatic Forces Most surfaces in contact with a highly polar liquid such as water acquire a surface charge, either by the dissociation of ions from the surfaces into the solution or the preferential adsorption of certain ions from the solution. The surface charge is balanced by an equal but opposite layer of oppositely charged ions (counterions) in the solution at some small distance away from the surface. This distance is known as the Debye length which is purely a property of the electrolyte solution. The Debye length falls with increasing ionic strength and valency of the ions in the solution, and for aqueous electrolyte (salt) solutions at 25°C the Debye length is (9.10) where the salt concentration M is in moles. The Debye length also relates the surface charge density σ of a surface to the electrostatic surface potentials ψ 0 via the Grahame equation: (9.11) where the concentrations [M 1:1 ] and [M 2:2 ] are again in M, ψ 0 in mV, and σ in C m –2 (1 C m –2 corresponds to one electronic charge per 0.16 nm 2 or 16 Å 2 ). For example, for NaCl solutions, 1/κ ≈ 10 nm at 1 mM, and 0.3 nm at 1 M. In totally pure water at pH 7, where [M 1:1 ] = 10 –7 M, the Debye length is 960 nm, or about 1 µm. The Debye length, being a measure of the thickness of the diffuse atmosphere of counterions near a charged surface, also determines the range of the electrostatic “double-layer” interaction between two charged surfaces. The repulsive energy E per unit area between two similarly charged planar surfaces is given by the following approximate expressions, known as the “weak overlap approximations”: (9.12) where the concentration [M 1:1 ] and [M 2:2 ] are again in moles. Using the Derjaguin approximation, Equation 9.6, we may immediately write the expression for the force F between two spheres of radius R as F = πRE, from which the interaction free energy is obtained by a further integration as κ − = = = 1 11 12 22 0 304 0 174 0 152 . . . , : : : M M M for 1:1 electrolytes such as NaCl for 1:2 or 2:1 electrolytes such as CaCl for 2:2 electrolytes such as MgSO 2 4 σψ ψ = () ++ () − 0 117 51 4 2 01122 25 7 12 0 . sinh . , :: . MM e E mV J m for monovalent salts = 0.0211 M mV J m for divalent salts -2 2:2 -2 = [] () [] [] () [] − − 0 0482 103 2 103 11 12 2 0 12 2 0 . tanh tanh , : Me e D D ψ ψ κ κ [...]... 9.11C) where energy is liberated and therefore “lost” via heat or phonon release (Israelachvili and Berman, 1995) This is an area of much current interest and activity, especially regarding the fundamental molecular origins of adhesion and friction, and the relationships between them 9.9 Rheology of Molecularly Thin Films: Nanorheology 9.9.1 Different Modes of Friction: Limits of Continuum Models Most frictional... decay length of 1 to 2 nm in the range 0 to 10 nm, and then more © 1999 by CRC Press LLC FIGURE 9.8 Examples of attractive hydrophobic interactions in aqueous solutions (a) Low solubility/immiscibility of water and oil molecules; (b) micellization; (c) dimerization and association of hydrocarbon chains in water; (d) protein folding; (e) strong adhesion of hydrophobic surfaces; (f) nonwetting of water on... experimentally (Johnson et al., 1971; Israelachvili, 1991) Equations 9.20 and 9.21 are the basic equations of the JKR theory and provide the framework for analyzing the results of adhesion measurements of contacting solids, known as contact mechanics (Pollock et al., 1978; Barquins and Maugis, 1982), and for studying the effects of surface conditions and time on adhesion energy hysteresis (see next section) 9.8... induces roughness and chemical heterogeneity even though initially (and after separation and reequilibration) both surfaces are perfectly smooth and chemically homogeneous Energy dissipating processes such as adhesion and contact angle hysteresis arise because of practical constraints of the finite time of measurements and the finite elasticity of materials which prevent many loading–unloading or approach–separation... diameters (Snook and van Megan, 1979, 1980, 1981; van Megan and Snook, 1979, 1981; Kjellander and Marcelja, 1985a,b; Tarazona and Vincente, 1985; Henderson and Lozada-Cassou, 1986; Evans and Parry, 1990) In a first approximation the oscillatory force laws may be described by an exponentially decaying cosine function of the form ( ) E ≈ E0 cos 2πD σ e − D σ , (9.15) where both theory and experiments show... forces and interaction potentials between two mica surfaces as a function of the orientation (twist angle) of their surface lattices The forces were measured in air, in water, and in an aqueous salt solution where oscillatory structural forces were present In air, the adhesion was found to be relatively independent of the twist angle θ due to the adsorption of a 0.4-nm-thick amorphous layer of organics and. .. between perfectly smooth and chemically homogeneous surfaces supported by perfectly elastic materials, and can be responsible for such phenomena as “rolling” friction and elastoplastic adhesive contacts (Bowden and Tabor, 1967; Greenwood and Johnson, 1981; Maugis, 1985; Michel and Shanahan, 1990) during loading–unloading and adhesion–decohesion cycles Adhesion hysteresis may be thought of as being due to... observed in a number of systems, including the interactions of fibers Most interesting, the depth of the potential energy well at contact (–E0 at D = 0) is generally deeper but of similar magnitude to the value expected from the continuum Lifshitz theory of van der Waals forces (at a cutoff separation of D0 ≈ 0.15 – 0.20 nm), even though the continuum theory fails to describe the shape of the force law... monotonic components, and both theory and experiments show that the oscillations decay with distance with the maxima and minima, respectively, above and below the baseline of the van der Waals force (middle dashed curve) or superimposed on the continuum DLVO interaction 9.6.3 Origin of Hydration Forces From the previous discussions we can infer that hydration forces are not of a simple nature, and it may be... viscoelastic constants, and tribological yield points such as the friction coefficient and critical shear stress of fracture in solids) and their molecular diffusion and relaxation times can be ten orders of magnitude longer than in the bulk liquid or even in films that are just slightly thicker The three friction regimes are summarized in Table 9.3 9.9.2 Viscous Forces and Friction of Thick Films: Continuum . Israelachvil, J. N. et al.“Surface Forces and Microrheology of .” Handbook of Micro/ Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press. Films: Nanorheology Different Modes of Friction: Limits of Continuum Models • Viscous Forces and Friction of Thick Films: Continuum Regime • Friction of Intermediate