Lee R. White

Electrophoretic mobility of a spherical colloidal particle

The equations which govern the ion distributions and velocities, the electrostatic potential and the hydrodynamic flow field around a solid colloidal particle in an applied electric field are reexamined. By using the linearity of the equations which determine the electrophoretic mobility, we show that for a colloidal particle of any shape the mobility is independent of the dielectric properties of the particle and the electrostatic boundary conditions on the particle surface. The mobility depends only on the particle size and shape, the properties of the electrolyte solution in which it is suspended, and the charge inside, or electrostatic potential on, the hydrodynamic shear plane in the absence of an applied field or any macroscopic motion.New expressions for the forces acting in the particle are derived and a novel substitution is developed which leads to a significant decoupling of the governing equations. These analytic developments allow for the construction of a rapid, robust numerical scheme for the solution of the governing equations which we have applied to the case of a spherical colloidal particle in a general electrolyte solution. We describe a computer program for the conversion of mobility measurements to zeta potential for a spherical colloidal particle which is far more flexible than the Wiersema graphs which have traditionally been used for the interpretation of mobility data. Furthermore it is free of the high zeta potential convergence difficulties which limited Wiersemas calculations to moderate values of ζ. Some sample computations in typical 1:1 and 2:1 electrolytes are exhibited which illustrate the existence of a maximum in the mobility at high zeta potentials. The physical explanation of this effect is given. The importance of the mobility maximum in testing the validity of the governing equations of electrophoresis and its implications for the colloid chemists picture of the Stern layer are briefly discussed.

Method for the calibration of atomic force microscope cantilevers

The determination of the spring constants of atomic force microscope (AFM) cantilevers is of fundamental importance to users of the AFM. In this paper, a fast and nondestructive method for the evaluation of the spring constant which relies solely on the determination of the unloaded resonant frequency of the cantilever, a knowledge of its density or mass, and its dimensions is proposed. This is in contrast to the method of Cleveland et al. [Rev. Sci. Instrum. 64, 403 (1993)], which requires the attachment of masses to the cantilever in the determination of the spring constant. A number of factors which can influence the resonant frequency are examined, in particular (i) gold coating, which can result in a dramatic variation in the resonant frequency, for which a theoretical account is presented and (ii) air damping which, it is found, leads to a shift of -4% in the resonant frequency down on its value in a vacuum. Furthermore, the point of load on the cantilever is found to be extremely important, since a small variation in the load point can lead to a dramatic variation in the spring constant. Theoretical results that account for this variation, which, it is believed will be of great practical value to the users of the AFM, are given.

The calculation of hamaker constants from liftshitz theory with applications to wetting phenomena

Abstract The physical signifance of the function ϵ (iξ) for a dielectric material is discussed together with the rationale for the construction of ϵ (iξ) from dielectric data. The utility of refractive index/wavelength data in the visible region in providing the crucial unltraviolet relaxation frequency and oscillator strength is demonstrated. A variety of real systems is analysed to obtain the data necessary for the construction of ϵ(iξ). Hamaker constants for these materials are computed. The use of these Hamaker constants in describing wetting phenomena is discussed with examples.

The influence of eroding topography on steady-state isotherms. Application to fission track analysis

The influence of surface topography on the form of steady-state isotherms during erosion-driven denudation is investigated. This is of particular interest to the interpretation of low-temperature geochronological data, for example fission track data, because this rests generally on the untested assumption that isotherms are not perturbed by topography and, therefore, that the data can be interpreted with one-dimensional models. In order to calculate isotherms and investigate the potential errors introduced by this assumption we use an approximate analytical solution of the diffusion-advection equation in two dimensions. It is shown that, for realistic geothermal gradients and a topography amplitude of H = 3 km and wavelength of w = 20 km, the 100°C isotherm may be perturbed to an amplitude of 1000 m/Ma. The effect increases for larger H and smaller w. For the interpretation of denudation histories derived from fission tracks, these relationships imply that the denudation rate of a terrain may be substantially overestimated, if it is only inferred from the slope of data points in a plot of sample elevation versus fission track age and if the sampling profile is not exactly vertical. A simple relationship is discussed that can be used to estimate the real denudation rate from the apparent denudation rate in an elevation-age plot.

Ionizable surface group models of aqueous interfaces

Abstract The classical electrochemistry of the electrical double layer appropriate to materials that operate as electrodes is shown to be a limiting description for non-electrode materials such as clays, inorganic oxides, insoluble salts, latex colloids and biosurfaces. The following unified treatment is valid in the first instance for double layers where the surface charge results from ionization of discrete identifiable surface sites. The major surface and bulk parameters that control such models are illustrated and principal trends and effects, in particular the non-Nernstian behaviour of such surfaces, are illustrated.

Accurate analytic expressions for the surface charge density/surface potential relationship and double-layer potential distribution for a spherical colloidal particle

Approximate analytic expressions of the surface charge density/surface potential relationship and double-layer potential distribution are derived for a spherical colloidal particle in 1-1 and 2-1 electrolyte solutions and a mixed 1-1 and 2-1 electrolyte solution. These expressions are shown to be excellent approximations to the exact numerical values obtained by Loeb, Overbeek, and Wiersema (“The Electrical Double Layer Around a Spherical Colloidal Particle.” MIT Press, Cambridge, Mass., 1961) over a wide range of ϰa values (ϰ = Debye-Huckel parameter and a = particle radius) for all values of the surface potential and are considerably better approximations than those obtained previously by White (J. Chem. Soc. Faraday Trans. 2 73, 577 (1977)). The mathematical basis for an empirical surface charge density/surface potential relationship for 1-1 electrolytes quoted by Loeb et al. is ellucidated. The method is extended to the case of a cylinder in 1-1 electrolyte. In addition, a more accurate expression for the interaction energy of two spherical colloidal particles at large distance is also derived.

The consolidation of concentrated suspensions. Part 1.—The theory of sedimentation

The concentration or consolidation of suspensions of fine particles under the influence of a gravitational field has been analysed. The rate and extent of consolidation depends upon a balance of three forces, the gravitational driving force, the viscous drag force associated with flow of liquid in the sediment and a particle or network stress developed as a result of direct particle–particle interactions. In the case of colloidally stable suspensions, this particle stress is the osmotic pressure of the particles; in the case of flocculated or coagulated suspensions, it is the elastic stress developed in the network of particles. A constitutive equation is suggested for irreversibly flocculated suspensions undergoing consolidation which embodies the concept of a concentration-dependent yield stress Py(ϕ). This is then used to analyse the sedimentation behaviour of flocculated sediments and to derive expressions for the initial sedimentation rate. The initial rate of change of sediment height with time in a uniform gravitational or centrifugal field is given approximately by: [graphic ommitted] where B=Δρgϕ0H0/Py(ϕ0), u0 is the sedimentation rate of an isolated particle, ϕ0 is the initial (uniform) volume fraction of solids, r(ϕ0) is a dimensionless hydrodynamic interaction parameter, Δρ is the difference in density between solid and liquid, g is the gravitational or centrifugal acceleration and H0 is the initial sediment height. The theory accounts correctly for the equilibrium consolidation behaviour of strongly flocculated suspensions, and preliminary experimental data suggest that it is not inconsistent with their dynamic behaviour. The estimation of the yield stress Py(ϕ) from a batch centrifuge experiment is also described.

Theoretical analysis of the static deflection of plates for atomic force microscope applications

The analysis of the static deflection of cantilever plates is of fundamental importance in application to the atomic force microscope (AFM). In this paper we present a detailed theoretical study of the deflection of such cantilevers. This shall incorporate the presentation of approximate analytical methods applicable in the analysis of arbitrary cantilevers, and a discussion of their limitations and accuracies. Furthermore, we present results of a detailed finite element analysis for a current AFM cantilever, which will be of value to the users of the AFM.

The translational and rotational drag on a cylinder moving in a membrane

The translational and rotational drag coefficients for a cylinder undergoing uniform translational and rotational motion in a model lipid bilayer membrane is calculated from the appropriate linearized Navier–Stokes equations. The calculation serves as a model for the lateral and rotational diffusion of membrane-bound particles and can be used to infer the ‘microviscosity’ of the membrane from the measured diffusion coefficients. The drag coefficients are obtained exactly using dual integral equation techniques. The region of validity of an earlier asymptotic solution obtained by Saffman (1976) is elucidated.

The interaction of colloidal particles collected at fluid interfaces

Simple approximate expressions are derived for the meniscus forces acting between spherical and cylindrical bodies at a fluid interface using a superposition approximation due to Nicolson (Proc. Cambridge Philos. Soc. 45, 288 (1949)). These expressions are correct to lowest order in the Bond number, B = (ϱB - ϱA)gR2/γAB, and are applicable to bodies that may be dissimilar in Bond number and wetting characteristics. Our results compare very favorably with the exact numerical calculations of Gifford and Scriven (Chem. Eng. Sci. 26, 287 (1971)) for parallel cylinders (for B ≲ 10−1). The small Bond number expressions derived herein are directly applicable to the interaction between particles of colloidal dimensions collected at fluid interfaces. Some sample calculations are given to illustrate the importance of capillary forces in interfacial coagulation processes. The extension of the theory to higher Bond number is discussed briefly.