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- Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A general introduction to surface and interfacial forces, perfectly combining theoretical concepts, experimental techniques and practical applications.
In this completely updated edition all the chapters have been thoroughly revised and extended to cover new developments and approaches with around 15% new content. A large part of the book is devoted to surface forces between solid surfaces in liquid media, and while a basic knowledge of colloid and interface science is helpful, it is not essential since all important concepts are explained and the theoretical concepts can be understood with an intermediate knowledge of mathematics. A number of exercises with solutions and the end-of-chapter summaries of the most important equations, facts and phenomena serve as additional tools to strengthen the acquired knowledge and allow for self-study.
The result is a readily accessible text that helps to foster an understanding of the intricacies of this highly relevant topic.
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Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Title Page
Copyright
Preface
List of Symbols
Chapter 1: Introduction
Chapter 2: Experimental Methods
2.1 Surface Forces Apparatus
2.2 Atomic Force Microscope
2.3 Optical Tweezers
2.4 Total Internal Reflection Microscopy
2.5 Magnetic Tweezers
2.6 Summary
2.7 Exercises
Chapter 3: van der Waals Forces
3.1 van der Waals Forces between Molecules
3.2 The van der Waals Force between Macroscopic Solids
3.3 The Derjaguin Approximation
3.4 Retarded van der Waals Forces
3.5 Measurement of van der Waals Forces
3.6 The Casimir Force
3.7 Summary
3.8 Exercises
Chapter 4: Electrostatic Double-Layer Forces
4.1 The Electric Double Layer
4.2 Poisson–Boltzmann Theory of the Diffuse Double Layer
4.3 Beyond Poisson–Boltzmann Theory
4.4 The Gibbs Energy of the Electric Double Layer
4.5 The Electrostatic Double-Layer Force
4.6 The DLVO Theory
4.7 Electrostatic Forces in Nonpolar Media
4.8 Summary
4.9 Exercises
Chapter 5: Capillary Forces
5.1 Equation of Young and Laplace
5.2 Kelvin Equation
5.3 Young's Equation
5.4 Capillary Forces Calculated with the Circular Approximation
5.5 Influence of Roughness
5.6 Kinetics of Capillary Bridge Formation and Rupture
5.7 Capillary Forces in Immiscible Liquid Mixtures and Other Systems
5.8 Normal Forces Acting on a Particle at a Liquid Interface
5.9 Lateral Forces between Particles at a Fluid Interface
5.10 Summary
5.11 Exercises
Chapter 6: Hydrodynamic Forces
6.1 Fundamentals of Hydrodynamics
6.2 Hydrodynamic Force between a Solid Sphere and a Plate
6.3 Hydrodynamic Boundary Condition
6.4 Gibbs Adsorption Isotherm
6.5 Hydrodynamic Forces between Fluid Boundaries
6.6 Summary
6.7 Exercises
Chapter 7: Interfacial Forces between Fluid Interfaces and across Thin Films
7.1 Overview
7.2 The Disjoining Pressure
7.3 Drainage
7.4 Thin Film Balance
7.5 Interfacial Forces across Foam and Emulsion Films
7.6 Thin Wetting Films
7.7 Summary
7.8 Exercises
Chapter 8: Contact Mechanics and Adhesion
8.1 Surface Energy of Solids
8.2 Contact Mechanics
8.3 Influence of Surface Roughness
8.4 Elastocapillarity
8.5 Adhesion Force Measurements
8.6 Summary
8.7 Exercises
Chapter 9: Friction
9.1 Macroscopic Friction
9.2 Lubrication
9.3 Microscopic Friction: Nanotribology
9.4 Summary
9.5 Exercises
Chapter 10: Solvation Forces and Non-DLVO Forces in Water
10.1 Solvation Forces
10.2 Non-DLVO Forces in an Aqueous Medium
10.3 The Interaction between Lipid Bilayers
10.4 Force between Surfaces with Adsorbed Molecules
10.5 Summary
10.6 Exercises
Chapter 11: Surface Forces in Polymer Solutions and Melts
11.1 Properties of Polymers
11.2 Polymer Solutions
11.3 Steric Repulsion
11.4 Polymer-Induced Forces in Solutions
11.5 Bridging Attraction
11.6 Depletion and Structural Forces
11.7 Interfacial Forces in Polymer Melts
11.8 Summary
11.9 Exercises
Chapter 12: Solutions to Exercises
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Bibliography
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 Schematic of a dispersion.
Figure 1.2 Scanning electron microscopic image of a piece of a porous aluminum oxide membrane lying on top of a continuous membrane.
Chapter 2: Experimental Methods
Figure 2.1 Schematic of a surface forces apparatus. Two mica sheets are glued to silica half cylinders to form a crossed cylinder geometry (inset on upper left). Their surfaces are brought in close proximity by micrometer screws and a double-cantilever spring mechanism and can then be moved in and out of contact by a piezoactuator. The distance between the mica sheets is measured using optical interferometry and the force is deduced from the observed deflection of the second, much softer double-cantilever spring.
Figure 2.2 Atomic structure of muscovite mica () in side view. A unit cell is indicated by the dashed line. The number of atoms of a certain species in a layer for two unit cells is indicated at the top right. In the third molecular layer from top, every fourth Si atom is replaced by an Al atom.
Figure 2.4 Examples of the FECO patterns as observed in a real SFA experiment. (a) With a gap of 10nm between the mica surfaces. The curved shape resembles a cross section of the sphere–plate geometry equivalent to the crossed cylinders. (b) With surfaces in contact. The flat portion of the fringes resembles the width of the contact zone. Schematics of the layer structure of the silvered mica sheets for each situation are shown on the left.
Figure 2.3 Schematic of fringes of equal chromatic order for a parallel plate interferometer (a) and as obtained for the crossed cylinder configuration in the SFA (b).
Figure 2.5 Schematic of an AFM setup. The sample is in contact with a sharp tip that is located at the end of the microfabricated cantilever and is raster-scanned by a piezoactuator. The deflection of the cantilever due to the surface topography is measured by a laser beam that is reflected from the backside of the cantilever onto a photodetector.
Figure 2.6 Schematic of an AFM force measurement. (1) Zero force region, where probe and surface are far from each other; (2) “snap-in” to surface due to attractive force; (3) approach part of the constant compliance re gion, where probe and surface move up in parallel; (4) retract part of the constant compliance region, where the probe and the surface move down in parallel and adhesion leads to negative bending; and (5) “jump-out” occurs when the restoring force of the cantilever exceeds the adhesion force. (a) Detector signal versus piezoactuator position. (b) Force versus distance curve derived from (a).
Figure 2.7 (a) Schematic top view of a rectangular and triangular AFM cantilever with length , width , and opening angle ; (b) SEM image of an AFM cantilever with sharp tip; and (c) colloid probe prepared by sintering a polystyrene particle to a tipless AFM cantilever.
Figure 2.8 Schematic of the friction force microscopy operation of an AFM. When using a scan direction perpendicular to the long axis of the cantilever, friction between tip and surfaces leads to a lateral deflection of the laser beam, which is recorded by a quadrant photodiode.
Figure 2.9 Example of the friction force obtained in an AFM friction force measurement [147]. The friction force between probe and surface induces a tilt of the cantilever that is recorded as a lateral movement of the laser spot on the photodetector. The upper and lower parts of the so-called friction loop correspond to the trace and retrace scans, respectively. The width of the friction loop (i.e., the difference between trace and retrace) is proportional to the friction force.
Figure 2.10 Optical gradient force on a particle with refractive index higher than the surrounding. Conservation of momentum for the diffracted rays leads to a net force that pulls the particle into the focus.
Figure 2.11 Schematic of an optical tweezer setup. It consists of an optical video microscope (components within dotted line) and the components needed for the laser trap. The laser beam is widened by a beam expander and coupled into the microscope by a dichroic mirror. The beam is focused on a diffraction-limited spot by the high numerical aperture objective to form the laser trap for the bead. The light scattered by the bead is collected by the condenser and reflected onto the detector by a second dichroic mirror to monitor the bead position. Positioning of the laser trap can be done by either a displacement of lens (, slow) or an acoustooptical deflector (AOD, fast).
Figure 2.12 Schematic of a TIRM setup. A laser is reflected from the bottom of the measurement cell at an angle above the critical angle to achieve total internal reflection of the light. Light of the evanescent wave that is scattered by a colloidal particle is collected by a microscope objective and detected by a photomultiplier. From observed intensity fluctuations, the Brownian motion of the particle within the combined surface and gravitational potential can be determined and thus calculation of the surface potential is possible.
Figure 2.13 Schematic of a basic magnetic tweezer setup. A paramagnetic bead is attached to the bottom of the measurement cell via a linker molecule. The field gradient of the two permanent magnets induces a pulling force on the molecule. The bead can be moved vertically or rotated by moving the magnets. Displacement of the bead is measured by a combination of digital video microscopy and image processing.
Chapter 3: van der Waals Forces
Figure 3.1 Calculating the van der Waals force between a macroscopic body and a molecule.
Figure 3.2 Picture of a Gecko foot taken through a glass plate (a) and scanning electron microscope images of the setae and spatulae (b, c).
Figure 3.3 (a) Schematic of the function of a dielectric material with three separated peaks in the absorption spectrum. (b) Idealized schematics of the functions (dotted line) and (solid line). While diverges at the absorption peaks, is a monotonous decaying function. (c) Typical shape of for a real material.
Figure 3.4 Matsubara frequencies (drawn in units of Hz, while has the unit rad Hz), at which the terms of the sum in Eq. (3.41) have to be evaluated. All values are for room temperature. The infrared and optical range contribute only with a few values. Most contributions come from the ultraviolet range.
Figure 3.5 Cleaving a molecular crystal to calculate the surface energy of a solid.
Figure 3.6 Schematic of Derjaguin's approximation for a rotational symmetric body interacting with a planar surface.
Figure 3.7 Interaction between a cone and a planar surface.
Figure 3.8 Calculating the interaction between two spheres with Derjaguin's approximation.
Figure 3.9 Different configurations for which the van der Waals force was calculated.
Figure 3.10 Dependence of effective Hamaker constant on the separation distance between two half-spaces coated with a thin layer. Case (a): for , . Case (b): for , .
Figure 3.11 Measurement of the Casimir force between a gold-coated sphere and a gold-coated plate.
Chapter 4: Electrostatic Double-Layer Forces
Figure 4.1 Helmholtz and Gouy–Chapman model of the electric double layer.
Figure 4.2 (a) Potential versus distance for a surface potential of 50 mV and different concentrations of a monovalent salt in water. (b) Local co- and counterion concentrations are shown for a monovalent salt at a bulk concentration of 0.1 M and a surface potential of 50 mV. In addition, the total concentration of ions, that is, the sum of the co- and counterion concentrations, is plotted.
Figure 4.3 Potential versus distance for surface potentials of 50, 100, 150, and 200 mV (from bottom to top) with 2 mM monovalent salt. (a) Planar surface. Results calculated with the full solution equation (4.24) and the solution of the linearized Poisson–Boltzmann equation (4.9) are shown. (b) Potential around a sphere of 15 nm radius. The radial coordinate originating in the center of the particle is plotted rather than the distance from the surface of the particle. Solutions for the linearized (Eq. (4.27)) and the full (Eq. (4.28)) Poisson–Boltzmann are plotted.
Figure 4.4 Surface potential versus surface charge calculated with the full Grahame equation (4.32), (continuous line) and with the linearized version (4.33), (dotted).
Figure 4.5 Simple version of the Stern layer.
Figure 4.6 Stern layer at a metal surface. Owing to the high electrical conductivity, the potential in the metal is constant up to the surface. The inner (IHP) and outer (OHP) Helmholtz planes are indicated. In the first layer of primary bound water, the permittivity is typically 6. In the secondary layer of water, it is of the order of 30.
Figure 4.7 Change in the potential distribution when two parallel planar surfaces approach each other. The gap is filled with electrolyte solution.
Figure 4.8 Electrostatic double-layer force between a sphere of m radius and a flat surface in water containing 1 mM monovalent salt. Results were calculated for constant potentials ( 80 mV, 50 mV) with Eq. (4.68). For constant surface charge density ( Cme nm, Cme nm), we applied Eq. (4.69).
Figure 4.9 Gibbs interaction energy (in units of T) versus distance for two identical spherical particles of 100 nm radius in water, containing different concentrations of monovalent salt. The calculation is based on DLVO theory using Eqs. (4.71) and (4.23). The Hamaker constant was J, and the surface potential was set to 30 mV. Both Figure show the same curves, only at different scales.
Figure 4.10 Force versus distance between two curved mica cylinders measured with the surface forces apparatus [5]. The force is divided by the radius of curvature of the cylinders. The experiment was carried out in aqueous electrolyte containing different concentrations of KNO. Continuous lines are to guide the eye.
Figure 4.11 Specific electric conductivity (a) and zeta potential versus concentration of NaAOT in hexadecane () [491] and cyclohexane () [494]. Zeta potentials were measured on titania [494] and surface potentials on spherical poly(methyl methacrylate) (PMMA) particles [491].
Figure 4.12 Electrostatic force between charged PMMA particles of 1.2 m diameter in hexadecane for different concentrations of NaAOT [491]. Forces for successive concentrations have been offset by 50 fN. Solid lines are fits with the screened Coulomb potential (Eq. (4.74)).
Chapter 5: Capillary Forces
Figure 5.1 Illustration of the curvature of a sphere, a cylinder, and a drop placed between the flat ends of two cylinders.
Figure 5.2 Capillary condensation into a narrow conical pore (a) and a slit pore (b) at a given vapor pressure . For the conical pore, the perfectly wetting case () and the case of finite contact angle () are shown.
Figure 5.3 Rim of a liquid drop with a contact angle on a solid surface.
Figure 5.4 Spherical particle of radius a distance away from a planar surface. The angle describes the position of the three-phase contact line on the particle surface. The height of the liquid meniscus is .
Figure 5.5 Schematic of the contributions to calculate the volume between a sphere and a plane.
Figure 5.6 Capillary force versus distance for two similar spheres (m, contact angle with respect to water). Curves were calculated with two different boundary conditions: for constant relative humidity () and for constant volume (). Exact solutions are plotted in continuous lines. Approximations are dotted.
Figure 5.7 Capillary force versus distance for a sphere of 1.5 cm radius interacting with another sphere of similar radius (open circles) and with a plane (filled circles). The spheres are in aqueous medium and the menisci are formed by a mixture of di-
n
-butyl phthalate and a liquid paraffin. The organic liquid perfectly wets the surfaces (). Its volume was fixed to either 0.02 or 0.1 ml. The dotted lines are calculated with Eqs. (5.28) and (5.32) using an interfacial tension of 0.032 N . Results are redrawn from Ref. [562].
Figure 5.8 Capillary force versus distance and adhesion force versus humidity for a sphere and a plane and for a cylinder with a conical end and a plane calculated with the respective equations in Table 5.2.
Figure 5.9 Adhesion force versus relative humidity curves. (a) A hydrophilic glass sphere of 20 m radius interacting with a naturally oxidized silicon wafer as measured by AFM [520]. (b) Force between a microfabricated silicon nitride AFM tip and a silicon wafer [587].
Figure 5.10 Schematic of two particles in contact at different vapor pressures of a condensing liquid. Macroscopically, both particles are assumed to be spherical and described by the apparent radius . On the nanometer scale, they are rough.
Figure 5.11 Capillary adhesion force caused by condensing water for two rough spheres of 10 m apparent radius and a contact angle of . Roughness is characterized by the maximal asperity height . The force required to separate the two spheres versus humidity is plotted. See Ref. [592] for details.
Figure 5.12 Schematic of the approach and retraction of two particles with an adsorbed liquid layer.
Figure 5.13 Radius of a meniscus
l
versus time (in milliseconds) calculated with Eq. (5.36). Water vapor in air at normal pressure is condensing into the gap between a sphere of m and a flat surface, both being perfectly wetted by water. The arrows indicate the respective characteristic time constants calculated with Eq. (5.39).
Figure 5.14 Menisci of gas and vapor between two lyophobic surfaces (a) and a liquid phase B in a liquid phase A (b).
Figure 5.15 Phase diagram of a binary mixture of methanol and
n
-hexane with an upper critical solution temperature of C. Methanol (M) has a higher density than hexane (H) and will therefore form the bottom phase.
Figure 5.16 Schematic of a small particle at a liquid–fluid interface. At the left the spherical particle is in its equilibrium position and the interface is planar. When pulling on the particle the interface is deformed, leading to a capillary force. On the right, the particle is completely removed from the interface.
Figure 5.17 Capillary force versus extension for a silica particle (m, ) pulled out of a water surface. Experimental results (circles) can be well fitted with a complex analytical theory (continuous line) [624]. The force predicted with the simple approximations Eqs. (5.45) and (5.46) is plotted as a dashed line. As common in the literature we plotted the force with a negative sign since it is directed downward.
Figure 5.18 A spherical particle with contact angle at a liquid–fluid interface without (a) and including gravitation (b). Two particles at constant contact angle in the liquid–fluid interface (c). The density of the particles is assumed to be higher than that of liquid A.
Figure 5.19 Capillary flotation forces between two particles at a liquid surface can be attractive or repulsive depending on the meniscus slopes and .
Figure 5.20 Capillary immersion forces between particles in a liquid film on a solid support can be attractive (a) or repulsive (b) depending on and . In free-standing films (c), immersion forces are always attractive.
Figure 5.21 Polystyrene (PS) particles of m radius dried from aqueous suspension on a solid surface and imaged by scanning electron microscopy. Owing to immersion forces, the particles tend to aggregate rather than being isolated as in solution. Bare PS particles are hydrophobic and would aggregate in aqueous solution. Therefore, these PS have covalently bound carboxyl groups at their surface. At some places, the particles even formed an ordered hexagonal array.
Figure 5.22 Lateral immersion force versus center-to-center distance for two glass spheres of mm radius measured with a torsion balance [643]. The two spheres are partially immersed in aqueous surfactant solution with and mm. Each particle is kept at the liquid surface by a support from below, which plays the role of the solid substrate. With , we have contact line radii of and slope angles of . The continuous line was calculated with Eq. (5.48). The dotted line indicates the position of closest contact between the two spheres.
Chapter 6: Hydrodynamic Forces
Figure 6.1 The viscous force required to slide a plane of area
A
over another parallel plane at a distance z across a fluid of viscosity is
F
=
z
.
Figure 6.2 Fluid volume element in a flow field. In this case, the flow is supposed to be in the
x
-direction, with a gradient in the
z
-direction. As a result, a shear force is acting between the upper and the lower surface elements. The pressure acting on the right surface, , and on the left surface, , can be different.
Figure 6.3 Symbols and variables used to calculate the hydrodynamic force between a sphere and a plane.
Figure 6.4 Hydrodynamic force on a sphere approaching a planar surface versus the distance calculated with approximation (6.37) and accurately with Eq. (6.39). The force is scaled by division with so that for Eq. (6.37) is plotted while for (6.39) the function is shown. For comparison, we also plotted (Eq. 6.41). The distance is scaled by division with the radius of the particle.
Figure 6.5 Hydrodynamic force versus distance curves between a microsphere of = 9m and a silicon wafer in aqueous electrolyte measured with an AFM. The sphere is attached to a cantilever that was driven at a speed of 40m s. Curves simulated with Eq. (6.37) are plotted in gray.
Figure 6.6 Two circular, parallel disks (a) and two identical spheres (b) approaching each other in a fluid in axial direction.
Figure 6.7 Spherical particle in contact with a planar surface with a linearly increasing fluid flow.
Figure 6.8 Spherical particle at a distance
D
moving parallel to a planar wall with velocity and rotating with an angular velocity in a stationary fluid. We consider a force and a torque directed parallel to the surface.
Figure 6.9 Velocity and angular rotation frequency versus distance for a silica sphere of 3m radius falling in water at 25 C parallel to a planar wall. With a density of the silica of 2650 kg m, we have kg m and N. In the absence of a wall, the sphere would sink with a velocity of 36.4 m s. Results were calculated with Eq. (6.58). The continuous line was calculated with the full numerical solution for , , , and from Refs [[184, 687]]. For large distances, Eqs. (6.55) and (6.56) provide a good approximation (dashed line). For close distances, Eqs. (6.49) and (6.50) can be used (dotted).
Figure 6.10 Schematic of different hydrodynamic boundary conditions at a solid–fluid interface. (a) The layer adjacent to the solid surface is bound. (b) No-slip boundary condition. (c) Slip characterized by the slip length
b
.
Figure 6.11 Plots of surface tension versus concentration for
n
-pentanol [723], LiCl (based on Ref. [724]), and SDS in an aqueous medium at room temperature.
Figure 6.12 Bubble at the end of a capillary approaching a planar rigid surface. A dimple is formed at the intermediate stage.
Figure 6.13 Experimental results of aqueous film thickness plotted as a function of radial position
r
at discrete times, showing dimple formation [738]. Note the difference in the vertical and horizontal scales. From bottom to top, the curves correspond to times , 0.10, 0.18, 0.34, 0.62, 1.22, 2.62, 5.62, and 13.62 s. Results were provided by R. Horn.
Chapter 7: Interfacial Forces between Fluid Interfaces and across Thin Films
Figure 7.1 Different systems in which fluid interfaces interact. (a) Two drops of liquid A interacting in a continuous, immiscible liquid B. Different stages of the interaction are the approach (1), drainage of the intervening film (2), leading to a thin film (3), and rupture of the film with coalescence (4). (b) The interaction between a rising bubble and a fixed bubble in a liquid. (c) A particle interacting with a bubble (or a drop). Interacting fluid interfaces are also important for the structure and stability of foams (d) and for liquid films on solid surfaces (e) that can either form stable continuous films or rupture and disintegrate into small drops.
Figure 7.2 Schematic of the disjoining pressure between two parallel plates.
Figure 7.3 Schematic of how a liquid film can be formed by carefully lifting a frame from a surfactant solution.
Figure 7.4 Schematic cross section of a draining liquid.
Figure 7.5 Drainage of a slowly draining water film stabilized by surfactant. The film is initially slightly higher than 10 cm. After 15 min, the film has drained to a thickness of m at its bottom. The experimental points () show the successive interference fringes obtained with red light [759]. With increasing height
z
, the film thickness decreases. The top 1–2 mm of the film has thinned to a black film. It is plotted as a vertical line. In addition to the experimental results, the film thickness calculated with Eq. (7.9) is plotted. After 60 min, the film has further drained and the Newton black film has extended to cm. Experimental results () show a slightly thinner film than the one calculated with Eq. (7.9). This small difference can be due to various factors such as residual evaporation or imperfections in the film.
Figure 7.6 Schematic of thin film balance cells.
Figure 7.7 Schematic of a thin film balance. The gas pressure inside the sealed chamber can be adjusted via the syringe relative to the external reference pressure to enforce thinning of the film and balance the disjoining pressure. Film thickness is monitored via microinterferometry and film drainage can be observed by video microscopy.
Figure 7.8 Schematic of a foam film (a) and a film on a solid surface (b). The region of the meniscus (C), the transition regions (T), and the planar films (F) are indicated.
Figure 7.9 Thickness of two aqueous foam films versus the concentration of added salt. In one classical experiment (), carried out by Scheludko and Exerowa [471], a thin film balance was used to study an aqueous solution containing saponin plus different concentrations of KCl. In the second experiment (), a vertical film of SDS solution at various concentrations of LiCl was analyzed [472]. The curves were calculated with Eq. (7.20) with 65 mV and Pa (continuous line) and mV and Pa (dashed).
Figure 7.10 Photo of an aqueous foam film in a thin film balance with a hole diameter of 1 mm. The film was stabilized with a mixture of commercial alkylpolyglycosides (APG) with alkyl chain lengths of 12 and 14 carbon atoms at a concentration four times lower than the CMC. In addition, a positively charged polymer poly(diallyl dimethyl ammonium chloride) (DADMAC) at a concentration of 8 mM (with respect to the monomer concentration) was added. The polymer forms a layered structure in the film, which leads to stratification when the film is thin. Thinning is induced by increasing the external pressure. (The picture was provided by R. von Klitzing [835].)
Figure 7.11 Disjoining pressure versus film thickness of thin aqueous foam films stabilized by -CG and CE. The solutions contain 0.1 mM NaCl. The data are fitted with the DLVO theory from which the surface charge density is calculated to be = 1.1 mC m. The continuous line was calculated using Eqs. (7.17, 7.19) with = 28 nm and J. The vertical dashed line indicates the Newton black film of 5 nm thickness.
Figure 7.12 (a)Schematic of liquid helium climbing up the wall of a container due to the repulsive van der Waals force between the solid–liquid and liquid–vapor interfaces. (b) Film thickness of liquid helium at 1.35 K on single-crystal SrF and BaF surfaces (redrawn from Ref. [840]). For heights of up to 10 cm, the thickness was determined at saturated vapor pressure at different heights. For thinner films, the vapor pressure was varied and converted to a corresponding height (see Exercise 7.1). In (b), the height scale is converted to a disjoining pressure by gz (Eq. (7.21)) with = 125 kg m. The dashed line is a fit for .
Figure 7.13 Adsorption isotherms for He adsorbing to SrF and BaF at 1.35 K (a), H adsorbing to gold at 14 K (b), and HO adsorbing to SiO at 21C (c). The experimental results were redrawn from Refs [840, 846, 847]. Continuous lines were calculated with Eq. (7.25) with J, kg m, kg m for (a), J, kg m, kg m for (b), and J, kg m, kg m for (c). In (a), the same results are plotted as in Figure 7.12.
Figure 7.14 Schematic of a thin film balance. A capillary of radius is moved close to the planar surface. Gas is pressed through the capillary with a pressure P. Between the end of the capillary and the plane a bubble is formed with a radius P. At the end, the bubble is flattened due to repulsive surface forces.
Figure 7.15 Morphologies of different polymer films after dewetting. (a) A poly(dimethyl siloxane) film under water on a silicon wafer with a layer of grafted identical molecules. (b) Early stage of a dewetting polystyrene film on a silicon wafer. (c) Late stage of the same dewetting polystyrene film. Images are m.
Figure 7.16 Schematic of a drop, which is so small that gravitation can be neglected (I), a drop, which is already slightly deformed by gravitation (II), and a continuous pancake-like film of height
h
.
Chapter 8: Contact Mechanics and Adhesion
Figure 8.1 Cleavage of a crystal: thought experiment for the calculation of the surface energy.
Figure 8.2 Typical reconstructions of face-centered cubic (110) surfaces.
Figure 8.3 Indentation of an elastic half-space by a flat cylindrical punch with contact radius
a
.
Figure 8.4 Vertical stress in units of the applied pressure for flat cylindrical punch with contact radius
a
.
Figure 8.5 Radial displacements– for different values of the Poisson's ratio for indentation of an elastic half-space by a flat cylindrical punch with contact radius
a
.
Figure 8.6 Adherence of a flat cylindrical punch with contact radius
a
to an elastic half-space.
Figure 8.7 Contact geometry between a rigid sphere and an elastic half-space with contact radius
a
as derived from the Hertz theory.
Figure 8.8 Vertical stress (contact pressure) between a sphere and an elastic half-space forming a contact with radius
a
.
Figure 8.9 Loading (0–A) and partial unloading (A–B) process for deriving indentation and elastic energy in the JKR model.
Figure 8.10 Vertical stress (contact pressure) between a sphere and an elastic half-space forming a contact with radius
a
. The stress distribution according to the JKR theory (solid line) is the sum of a Hertzian contact pressure (dashed line, compressive stress) and a flat punch contact stress (dotted line, tensile stress).
Figure 8.11 Profiles of the contact of a rigid sphere with an elastic half-space for the Hertz (dotted line) and the JKR model (solid line). The insets show the rim of the contact zones in detail.
Figure 8.12 Stress distribution for the DMT model. In an annular zone outside the contact area, the so-called cohesive zone, surface forces lead to a tensile stress. Within the contact zone, the stress distribution is that of a Hertzian contact.
Figure 8.13 Dugdale potential used in the Maugis theory.
Figure 8.14 Relation between normalized contact radius and normalized load calculated from the Maugis theory for different values of . In the limit of the DMT theory, pull-off occurs at a value of . For the other curves, pull-off occurs at the point where the tangent to the curves becomes vertical.
Figure 8.16 Relation between normalized applied load and normalized indentation calculated from the Maugis theory for different values of . For the Hertz case, detachment occurs at zero load and indentation. In the DMT case, detachment occurs at a normalized load of and zero indentation. For all other cases, detachment under a constant applied force occurs at negative load between and and at negative indentation in the point where the tangent to the curve becomes horizontal.
Figure 8.17 Schematic of the centrifuge method to determine adhesion forces of particles on surfaces. Friction forces can also be analyzed when the particles are placed on a horizontal surface.
Figure 8.18 Adhesion force between silica spheres plotted versus the reduced radius (Eq. (8.40) [942]). A linear relation is observed as expected from the JKR or DMT theories.
Chapter 9: Friction
Figure 9.1 Amontons' law of friction: the frictional force does not depend on the contact area and is proportional to the load.
Figure 9.2 Optical micrograph of the contact between an acrylic plastic and a soda lime glass hand lapped with # 240 abrasive. In this inverted image, dark areas correspond to contact between the surfaces.
Figure 9.3 Example of a system that exhibits stick–slip friction. Stick–slip is also illustrated as a schematic plot of spring elongation versus time at constant pulling speed .
Figure 9.4 Dependence of friction on load for a single microcontact. The friction force between a silica sphere of 5 diameter and an oxidized silicon wafer is shown (filled symbols). Different symbols correspond to different silica particles. The solid line is a fitted friction force using a constant shear strength of the interface and the JKR model to calculate the true contact area (based on Eq. (8.67)). Results obtained with five different silanized particles (using hexamethylsilazane) on silanized silica are shown as open symbols.
Figure 9.5 (a) Schematic of a rubber block sliding over an asperity with a characteristic contact dimension at speed . (b) Schematic of storage modulus and loss modulus versus frequency for a rubber material. (c) Ratio of loss modulus over absolute value of complex modulus versus frequency.
Figure 9.6 Plot of friction coefficient versus logarithm of sliding velocity for a acrylonitrilebutadiene rubber sliding on silicon carbide emery paper at C. All data were recorded at low sliding speeds between cm and 3 cm . High-speed data were obtained by time-temperature superposition of data obtained at elevated temperatures. Shift factors necessary to overlay the experimental friction data into this single master curve were found to be the same as obtained from bulk rheology of this rubber.
Figure 9.7 Models for adhesive friction of rubber on rough or smooth surfaces.
Figure 9.8 Sphere or cylinder rolling over a planar surface.
Figure 9.9 Schematic of a pin-on-disk tribometer.
Figure 9.10 Working principle of the quartz crystal microbalance. The quartz crystal is excited to shear oscillate at its resonance frequency. Changes in adsorbed mass or viscous coupling of adsorbed layers lead to changes in resonance frequency and width of the resonance peak.
Figure 9.11 Simple example of a configuration in hydrodynamic lubrication.
Figure 9.12 Plots of the parameters and depending on .
Figure 9.13 Plot of the coefficient of friction versus expressed in units of . Friction increases as the square root of viscosity and velocity. For very thin lubrication layers, the experimentally observed values (dotted line) will be higher than expected due to the onset of boundary lubrication.
Figure 9.14 Different lubrication situations in a journal bearing. (a) At low velocities and high loads, boundary lubrication with a high coefficient of friction dominates. The shaft climbs the journal on the right side. (b) At high speeds and low loads, hydrodynamic lubrication leads to much lower friction. The build-up of the hydrodynamic wedge moves the shaft to the upper left (lubrication layer thickness and eccentricity are strongly exaggerated in the drawing to visualize the effect).
Figure 9.15 Schematic cross-section of a lubricated line contact between two rotating parallel cylinders. The solid line indicates the pressure distribution in the contact region with the characteristic pressure peak in the outlet region. The dotted line corresponds to the pressure distribution of the classical Hertzian contact at the same loading force.
Figure 9.16 Stribeck diagram for a lubricated friction contact. Coefficient of friction versus is plotted, where is the density of the lubricant, the velocity, and
P
the contact pressure. From left to right, there are three distinct friction regimes: boundary lubrication with high friction and wear, mixed lubrication with intermediate friction and wear, and hydrodynamic lubrication with low friction and (almost) no wear.
Figure 9.17 Schematic of the Prandtl–Tomlinson model.
Figure 9.18 Energy for a sliding AFM tip in the one-dimensional Prandtl–Tomlinson model versus position. The position is given in units of the periodic surface potential (). Parameters were and . The energy is plotted for a support moving to the right with a speed (i.e., is assumed) leading to . The tip is assumed to reside in a minimum at at time . The three different lines indicate the shape of the potential for three different times, where the position of the support is at (continuous line), (dashed line), and (dotted line), respectively.
Figure 9.19 Friction forces measured between a NaCl(001) surface and an AFM tip sliding along the (100) direction in UHV. At an applied load of 4.7 nN (a) and 3.3 nN (b), stick–slip motion of the tip is observed with a hysteresis between forward and backward directions. For an applied load of (at 0.7 nN adhesion force), motion changes to a continuous sliding without detectable hysteresis, indicating dissipationless sliding. This effect is denoted as static superlubricity. Parts (d)–(f) show corresponding numerical results using a Tomlinson model with values of , , and .
Figure 9.20 Friction force microscope pictures (a, b) of a graphite(0001) surface as obtained experimentally with FFM and results of simulations (c, d) of the stick–slip friction using a two-dimensional equivalent of the Prandtl–Tomlinson model. The friction force parallel to the scan direction (a, c) and the lateral force perpendicular to the scan direction (b, d) are shown. The scan size is .
Chapter 10: Solvation Forces and Non-DLVO Forces in Water
Figure 10.1 Schematic Figure of the structure of a simple liquid confined between two parallel walls. The order changes drastically depending on distance, which results in a periodic force. For steep potentials between fluid molecules and the wall and for short-range interactions between the fluid molecules, the force is maximal for dense, ordered structures. Thus, maxima occur at , , , and so on.
Figure 10.2 (a) Normalized force versus distance across liquid OMCTS between two mica surfaces measured upon approach with an SFA (, adapted from Ref. [1136]). The continuous line is a fit with Eq. (10.7) and N , nm, nm, and . Only those parts are plotted where the force increases with decreasing distance. Regions in between are inaccessible because the gradient of the attractive force exceeds the spring constant of the SFA. (b) Normalized force between a microfabricated silicon nitride tip of an atomic force microscope and a planar mica surface in 1-propanol at room temperature [1164]. The tip had a radius of curvature of nm. The different symbols were recorded during approach () and retraction () of the tip. For comparison, the calculated van der Waals force is plotted as a continuous line.
Figure 10.3 Schematic of an isotropic liquid and nematic and smectic liquid crystals.
Figure 10.4 Normalized force versus distance curve measured in 8CB between a glass sphere (radius 8.5–10 m) and a glass plate with an AFM [1196]. The force curve was recorded in the nematic phase at C, that is, 0.7 K above the nematic to smectic phase transition. It was fitted with Eq. (10.8) using N, , , nm, and nm. Both glass surfaces were coated with
N
,
N
-dimethyl-
N
-octadecyl-3-aminopropyltrimethoxychlorosilane so that 8CB completely wets the surfaces.
Figure 10.5 Normalized hydration forces measured on approach between two mica and two silica surfaces in aqueous solution on a linear (a) and logarithmic (b) scale. The measurements between mica were carried out with an SFA in 1 M KCl [1208] () and in 1 M [5] (squares) at pH 5.7. The continuous line is to guide the eye. The measurements between a silica sphere (5 m diameter) and an oxidized silicon wafer were performed with an AFM in 1 M NaCl at pH 7.0 [1215].
Figure 10.6 Attractive force measured with the SFA between two mica surfaces that had been coated with the double-chain cationic surfactant
N
-(-trimethylammonioacetyl)-
O
, -bis(1
H
,1
H
,2
H
,2
H
-perfluorodecyl)-
l
-glutamate chloride by Langmuir–Blodgett transfer [1252]. Force curves were recorded with different spring constants. The jump-in was prevented by the drainage method. Forces are negative because they are attractive. The line is a fit with two exponentials (Eq. (10.16)) with N , nm, m N , nm.
Figure 10.7 Nanobubbles on a silicon wafer, which had been hydrophobized by OTS (octadecyltrichlorosilane), in water [1285]. The macroscopic contact angle was . Nanobubbles were created by first immersing the wafer in ethanol. Ethanol dissolves more nitrogen and oxygen than water. Then, the ethanol was exchanged by water. Temporarily, an oversaturation was created and bubbles of 15–35 nm height formed on the wafer surface. Once formed, they are stable for many hours. The image of 20 m size was recorded with an AFM in tapping mode.
Figure 10.8 Typical approaching force versus distance curves recorded in normal and degassed water (redrawn from Ref. [1263]). Force curves were measured between a glass sphere of 8.8 m diameter attached to an AFM cantilever of 0.15 N spring constant and a naturally oxidized silicon wafer. Both surfaces were hydrophobized by a fluorinated alkyl silane.
Figure 10.9 Typical approaching force curves measured between a silanated silica sphere of 4.7 m diameter and a silicon wafer in distilled water using an AFM [1266]. The silicon oxide surfaces were made hydrophobic by vapor silanization with hexamethyldisilazane, leading to an advancing contact angle of . The first three approaches are plotted. The cantilever, to which the sphere was attached, had a spring constant of 0.2 N .
Figure 10.10 Normalized approaching and retracting force versus distance curves measured between a silicon wafer and a silica microsphere in 1 mM aqueous solution with an AFM [1268]. The silicon oxide surfaces were made hydrophobic by silanization with propyltrichlorosilane, leading to an advancing contact angle of . The cantilever, to which the sphere was attached, had a spring constant of 0.2 N .
Figure 10.11 Chemical structure of DMPC, important head groups of phospholipids, cholesterol, dioleoyloxypropyl trimethylammonium (DOTAP), and sphingolipid.
Figure 10.12 Schematic of the osmotic stress method.
Figure 10.13 Schematic of two opposing bilayers in aqueous medium illustrating undulation and protrusion forces. Undulation forces occur between any pair of flexible membranes and are described by continuum theory. Protrusion forces are caused by individual lipid molecules jumping up and down normal to the bilayer.
Figure 10.14 Force per unit area versus distance between two lipid bilayers of EPC at 30 C, DMPC at 30 C, and DPPC at 50 C (redrawn from Ref. [1298]). The force curves were fitted with a sum of hydration forces (Eq. (10.18)), van der Waals attraction (Eq. (10.20)), and undulation forces (Eq. (10.23)). To obtain the van der Waals pressure we applied (see Exercise 10.4). The parameters were J, Pa, nm, J, and a bilayer thickness of nm for EPC (continuous line). For DMPC (dashed line), the parameters were J, Pa, nm, J, and nm. For DPPC (dotted line), we used J, Pa, nm, J, and nm.
Chapter 11: Surface Forces in Polymer Solutions and Melts
Figure 11.1 Picture of a linear polymer in the ideal freely jointed chain model.
Figure 11.2 Retracting force curve in which a single polymer chain is stretched in tetrahydrofuran. The curve was measured with the atomic force microscope (redrawn from Ref. [1390]). The gray line is a fit with Eq. (11.11) using nm, nm, and N . The black dotted line is a fit with .
Figure 11.3 Isotropic, elastic polymer as in the worm-like chain model.
Figure 11.4 Structure of polymers on surfaces.
Figure 11.6 Force distance profiles between two curved mica sheets with polystyrene brushes in toluene (redrawn from Ref. [1418]). Experimental results () were fitted with the model of Milner
et al
. [1407] using a step (Eq. (11.17)) and a parabolic (Eq. (11.18)) profile for the segment concentration, and the model of de Gennes [1406] (Eq. (11.20)). In each case, the normalized force was calculated from .
Figure 11.5 Interaction energy per unit area versus distance between two similar parallel plates coated with grafted polymer in a good solvent. For the calculation, we used a segment length nm and a chain length of segments. For low grafting density (), the interaction energy was calculated with Eq. (11.22). For the high grafting densities, we applied Eq. (11.20) with , and , .
Figure 11.7 Schematic of a polyelectrolyte brush in water.
Figure 11.8 Force distance profiles between a silica sphere of 3 m radius and a microscope slide both covered with physisorbed PSS ( kDa) measured with an AFM [1439]. The force was normalized by dividing it with the radius of the sphere. Force curves were recorded in aqueous medium containing 1, 10, and 100 mM NaCl. Experimental results were fitted with the model of de Gennes [1406] (Eq. (11.20), and , 76, and 47 nm at 1, 10, and 100 mM NaCl, respectively) and the salted brush model (Eq. (11.31), and , 60, and 29 nm and , 0.8, and 2.0 at 1, 10, and 100 mM NaCl, respectively) with . In addition, the distance was offset by 1–5 nm.
Figure 11.9 Overview of forces between solid surfaces in polymer solutions.
Figure 11.10 Approaching force versus distance curves between curved mica surfaces in 15 mg polystyrene dissolved in cyclopentane. Curves were recorded 12, 15, and 31 h after immersing the mica surfaces in the solution.
Figure 11.11 Normalized rate of flocculation of an aqueous dispersion of cellulose particles versus concentration of KCl (a) and versus the concentration of added PEO + tannic acid and PEO + folic acid plus 1 mM KCl (b). The rates were normalized by dividing them by the rate measured in 100 mM KCl polymer-free solution.
Figure 11.12 Force versus distance curve obtained with a biomimetic DOPA-containing polymer in aqueous solution of 1 mM measured with an atomic force microscope between two titanium surfaces [1486]. Approaching () and retracting () parts are plotted.
Figure 11.13 Schematic experimental setup and retracting force versus distance curves for contact adhesion (a), and bridging adhesion (b,c) between two surfaces. For bridging adhesion, two different scenarios are plotted: that of a polymer chain fixed at two points and that of a polymer that adheres to the surfaces and is peeled off like a tape. The adhesion forces and the work of adhesion (hatched area) are indicated.
Figure 11.14 Schematic of two particles dispersed in a polymer solution. The reduced osmotic pressure in the zone depleted of polymer in the gap between the particles leads to an effective attractive force, the depletion force.
Figure 11.15 Normalized force versus distance curves measured between a silica microsphere of 6.7 m diameter and a silica wafer in an aqueous dispersion of silica nanoparticles (26 nm diameter) with an AFM [1507]. The concentration of the nanoparticles was 2.0, 4.5, and 13.2 vol%. Results are vertically offset for better viewing. Curves were fitted with Eq. (10.7) (gray lines) with nm, nm, and N (2.0 vol%); nm, nm, and N (4.5 vol%); nm, nm, and N (13.2 vol%).
Figure 11.16 Force curves measured in normal methyl-terminated polyisoprene (PI, kDa) and hydroxyl-terminated polyisoprene (PI-OH, kDa). The experiments were carried out with an AFM between a silicon nitride tip and a silicon wafer. Forces were normalized by dividing them by the radius of curvature of the tip.
Chapter 12: Solutions to Exercises
Figure 12.1 Potential versus distance for high surface potentials.
Figure 12.2 Plot of the left-hand side of Eq. (12.1). The horizontal line at
y
= 1 intersects the curve at a value of
d/D
= 11.3 and the line at
y
= 10 intersects the curve at a value of
d/D
= 2.67.
Figure 12.3 Capillary force versus the humidity of two similar spheres (m, contact angle with respect to water). The spheres are kept at an effective distance of , 1, and 2 nm by an asperity.
Figure 12.4 Schematic for a sphere in contact with a plane in presence of a liquid meniscus.
Figure 12.5 Spherical particle at a water–air interface.
Figure 12.6 Distance versus time (a) and velocity versus distance (b) for a sphere falling toward a planar surface as calculated with Eqs. (12.2) and (12.3) (continuous lines) and the results of Exercise 6.1 (dashed lines). For comparison the result for a free falling sphere is also shown (dotted line).
Figure 12.7 Shear rate at
z
= 0 along the
x
-axis. The shear rate increases with decreasing distance. It shows a maximum, which shifts to larger radial distances with increasing distance
D
.
Figure 12.8 Diffusion coefficient of a sphere of 50 nm radius at a planar wall.
Figure 12.9 Bubble underneath a plate (Solution 7.3).
Figure 12.10 Forces acting on a body sliding on an inclined plane.
Figure 12.11 Force versus distance profile for the presmectic force between a sphere and a plate.
Figure 12.12 Force between DLPC bilayers.
Figure 12.13 Steric repulsion between brushes (solution 11.1).
Figure 12.14 Interaction energy per unit area calculated with
N
= 60, nm, , and for the simple osmotic brush (Eq. (12.19)), the Milner, Witten, Cates model with a step (Eq. (11.18)) and parabolic segment profile (Eq. (11.19)), and the model of de Gennes (Eq. (11.21)).
Figure 12.15 Excluded volume for two approaching spheres.
List of Tables
Chapter 1: Introduction
Table 1.1 Types of dispersions
Chapter 3: van der Waals Forces
Table 3.1 Contributions of the Keesom, Debye, and London interaction energies to the total van der Waals interaction between similar molecules as calculated with Eqs. (3.20), (3.22) and (3.23) using .
Table 3.2 Permittivity , refractive index
n
, and main absorption frequency in the UV for various solids, liquids, and polymers at C (Refs [267, 274, 275], handbooks, and own measurements)
Table 3.3 Hamaker constants for medium 1 interacting with medium 2 across medium 3
Chapter 4: Electrostatic Double-Layer Forces
Table 4.1 Dielectric permittivity of various liquids
Chapter 5: Capillary Forces
Table 5.1 Saturation vapor pressure , surface tension , density , capillary constant , and Kelvin length of liquids at
Table 5.2 Different geometries for which capillary forces have been calculated using the circular approximation [560, 561]
Chapter 6: Hydrodynamic Forces
Table 6.1 Viscosities of various liquids in mPa s or 10 Pa s
Chapter 8: Contact Mechanics and Adhesion
Table 8.1 Internal surface energies
u
σ
of noble gas crystals
Table 8.2 Calculated surface tensions and surface stresses of ionic crystals for different surface orientations compared to experimental results [879] [880]
Chapter 9: Friction
Table 9.1 Examples of coefficients of friction between different materials for dry and lubricated friction [967]
Chapter 10: Solvation Forces and Non-DLVO Forces in Water
Table 10.1 Surface tension of various hydrophobic solids, the interfacial tensions between the solid and water , and advancing contact angles of water on these solids at 20 C (from own measurements, Refs [1238–1240], and references therein)
Table 10.2 Saturation concentration , standard entropy , enthalpy , and free energy of solvation for various nonpolar gases and liquids in water at 25 [1241] [1242]
Table 10.3 Names of important acyl chains occurring in lipids and the melting points of the corresponding fatty acid
Table 10.4 Bending moduli of various phosphatidylcholine bilayers in water at different temperatures
T
and melting temperature
Chapter 11: Surface Forces in Polymer Solutions and Melts
Table 11.1 Structure of common polymers, their glass transition , melting temperature , molar mass of the repeat unit in g , and examples for good solvents
Table 11.2 Structure of common water-soluble polymers and the molar mass of the repeat unit in g
Table 11.3 Characteristic ratio [1370, 1375–1378] and lengths of repeat units
Surface and Interfacial Forces
Hans-Jürgen Butt and Michael Kappl
Second Edition
Authors
Prof. Hans-Jürgen Butt
MPI for Polymer Research
Ackermannweg 10
55128 Mainz
Germany
