Hamaker Coefficient Research Articles

Calculating van der Waals-London dispersion spectra and Hamaker coefficients of carbon nanotubes in water fromab initiooptical properties

The van der Waals-London dispersion (vdW-Ld) spectra are calculated for the [9,3,m] metallic and [6,5,s] semiconducting single wall carbon nanotubes (SWCNTs), graphite, and graphene (a single carbon sheet of the graphite structure) using uniaxial optical properties determined from ab initio band structure calculations. The [9,3,m], exhibiting metallic optical properties in the axial direction versus semiconducting optical properties in the radial direction, highlights the strong anisotropic nature of metallic SWCNTs. Availability of both efficient ab initio local density band structure codes and sufficient computational power has allowed us to calculate the imaginary parts of the frequency dependent dielectric spectra, which are then easily converted to the required vdW-Ld spectra for Hamaker coefficient calculations. The resulting Hamaker coefficients, calculated from the Lifshitz quantum electrodynamic theory, show that neither graphite nor graphene are accurate model materials for estimating the Hamaker coefficients of SWCNTs. Additionally, Hamaker coefficients were calculated between pure radial-radial, radial-axial, and axial-axial components of both SWCNTs. Analysis of these coefficients reveals that the vdW-Ld interactions will depend on both chirality and the particular orientation between neighboring SWCNTs. The minimization of energy, with respect to orientation, predicts that vdW-Ld alignment forces will arise as a result of the anisotropic optical properties of SWCNTs.

Graded interface models for more accurate determination of van der Waals–London dispersion interactions across grain boundaries

Attractive van der Waals--London dispersion interactions between two half crystals arise from local physical property gradients within the interface layer separating the crystals. Hamaker coefficients and London dispersion energies were quantitatively determined for $\ensuremath{\Sigma}5$ and near-$\ensuremath{\sum}13$ grain boundaries in $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$ by analysis of spatially resolved valence electron energy-loss spectroscopy (VEELS) data. From the experimental data, local complex dielectric functions were determined, from which optical properties can be locally analyzed. Both local electronic structures and optical properties revealed gradients within the grain boundary cores of both investigated interfaces. The results show that even in the presence of atomically structured grain boundary cores with widths of less than $1\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, optical properties have to be represented with gradual changes across the grain boundary structures to quantitatively reproduce accurate van der Waals--London dispersion interactions. London dispersion energies of the order of 10% of the apparent interface energies of $\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$ were observed, demonstrating their significance in the grain boundary formation process. The application of different models to represent optical property gradients shows that long-range van der Waals--London dispersion interactions scale significantly with local, i.e., atomic length scale property variations.

Van der Waals interactions between bodies having inhomogeneous dielectric permittivities

We present analytical results for the contribution of electromagnetic fluctuations inthe distribution of interaction energy and pressure in isotropic systems whoseproperties depend only on one spatial coordinate. If we neglect the continuousinhomogeneity introduced here and consider the simplest case of two macroscopichomogeneous bodies separated by a homogeneous film our result reduces to thewell-known Lifshitz formula. As a first application of theory, a one-dimensionalmodulated system with a homogeneous layer embedded in it is considered and asuitable perturbation theory for this system is developed. In the main part ofthis paper we limit the calculations to the non-retarded case, that is only thecalculation of van der Waals interaction energy is given. As a second applicationof theory we consider the van der Waals interaction between two semi-infinitemedia across a planar region within which there is a thin film having an arbitraryvariation of the dielectric permittivity. The importance of the precise evaluationof the transverse magnetic surface mode dispersion relation in inhomogeneousmedia is elucidated. For concreteness the influence of the transition layer betweenwater and lipid in a symmetrical configuration is considered in some detail. Thezero-frequency term and the dispersion-only contribution to the Hamaker coefficient aregiven analytically using some approximations and modelling of the dielectricconstants reasonable for these dielectrics at room temperature. As a whole theresults indicate the necessity of performing Lifshitz-type calculations on realisticinhomogeneous layered models, as are the models described here, for accurateinteraction energy modelling. Of course further work is needed for real justificationof the continuous variation of dielectric permittivities across phase interfaces.

Nonadditivity in van der Waals interactions within multilayers

Working at the macroscopic continuum level, we investigate effective van der Waals interactions between two layers within a multilayer assembly. By comparing the pair interactions between two layers with effective pair interactions within an assembly we assess the significant consequences of nonadditivity of van der Waals interactions. This allows us to evaluate the best numerical estimate to date for the Hamaker coefficient of van der Waals interactions in lipid-water multilamellar systems.

Intergranular films at metal–ceramic interfaces: Part II – calculation of Hamaker coefficients

We report here theoretical calculations indicating the stability of equilibrium nanometer-thick films at metal–ceramic interfaces. Hamaker coefficients were calculated for metal–ceramic interfaces in the presence of a SiO 2-based film, which indicated that a stronger attractive force is expected for intergranular films at metal–alumina interfaces, relative to alumina grain boundaries, correlating to the experimentally measured film thickness presented in part I of this report. From the combination of an attractive van der Waals force together with steric repulsion due to ordering induced in the amorphous film, a force balance can be obtained leading to the formation of equilibrated films at metal–ceramic interfaces.

Contact Electrification and the Interaction Force between a Micrometer-Size Polystyrene Sphere and a Graphite Surface

Two independent techniques are used to measure the interaction force between a single 3 μm radius polystyrene sphere and an atomically flat, highly oriented pyrolytic graphite substrate. The variation of the interaction force with the surface-to-surface separation between the sphere and plane is determined using both a static and a dynamic atomic force technique. The measured interaction force is dominated at long range by an electrostatic force arising from localized charges triboelectrically produced on the sphere when it makes contact with the substrate. For small sphere−substrate separations, evidence for a van der Waals force is observed. The data provide consistent estimates for both the Hamaker coefficient and the triboelectrically produced charge which can be measured to an accuracy of ±10 electrons.

Direct Measurement of Repulsive van der Waals Interactions Using an Atomic Force Microscope

The forces of interaction between a flat poly(tetrafluoroethylene) (PTFE) surface and gold spheres (of radii 3–8 μm) were measured as a function of apparent surface separation for different intervening media. For air, fluorinated alkanes, and polar liquids the interaction between the surfaces was found to be attractive. With intervening liquids of low-polarity the interaction was found to be repulsive. This repulsion is attributed to a negative composite Hamaker coefficient leading to van der Waals repulsion.

Van der Waals energies of cylindrical and spherical single layer systems

The van der Waals energies for cylindrical and spherical single layer systems are obtained, in the nonretarded limit, from the heuristic procedure of summing the normal surface electromagnetic mode frequencies. Both the zero and finite temperature results are presented. The geometrical dependence of the energies for the cylindrical and spherical systems is shown to satisfy inequalities involving simple model dielectric configurations. The exact energies have a first-order correction to the planar case, for a large inner radius in comparison to the film thickness, which is proportional to the mean curvature with a new Hamaker coefficient. It is conjectured that this applies to arbitrarily shaped smooth films with small curvature. In the two-body summation approximation the energies factorize into a geometrical factor, for which exact analytical forms are found, and the standard Hamaker constant. The effect of geometry on the van der Waals energy is shown to be important, in that the effective Hamaker constants for curved films of inner radius r and layer thickness w rises rapidly at r∼2.6w for the cylindrical and at r∼1.8w for the spherical system .

The Hamaker coefficient for titanium dioxide (rutile) in liquid media

The Hamaker coefficients for rutile across water and dodecane are calculated from optical data. At contact a value of 15.6 kT at 298 K (0.064 aJ) is obtained in water; the value for dodecane is rather similar at 14 kT. These values are less than one-half of that obtained in vacuo (38 kT). The asymmetric case of rutile/water (or dodecane)/polystyrene (or poly(methyl methacrylate) (PMMA)) is also considered (4.6 kT for water, 3.1 kT for dodecane). The effect of coatings on the Hamaker coefficient of rutile pigments is also discussed.

The Influence of Interstitial Liquids on the Cohesive Strength of Carbon-Black Agglomerates

Abstract The influence of interstitial liquids on the cohesive strength of carbon black has been investigated. Calculations of effective Hamaker coefficients for the wet materials predicted order of magnitude decreases from the Hamaker constants for the dry materials. Tensile-strength data showed slight increases over the Hamaker constants for the dry materials. It is clear that further research is needed to elucidate the effects of liquid content and wettability.

Solvent structure effects in the macroscopic theory of van der Waals forces

The theory of van der Waals forces is generalized in order to include the effects of solvent structure by explicitly dealing with the spatial correlations between the solvent molecules. This is achieved within the framework of nonlocal dielectric response theory, with the specular reflection approximation to apply the nonlocal response of the structured solvent in the bulk to give the response of such solvent intercalated between the interacting macroscopic surfaces. A closed form expression for the zero-order contribution to the van der Waals free energy is derived for two semi-infinite dielectric regions (characterized by a standard, local dielectric response and fixed dielectric constant), interacting across a layer of a structured solvent described by a nonlocal dielectric response function. The nonlocal character of the latter gives rise to several quantitatively and qualitatively new features of the dispersion interaction; the most important ones are similar in consequences to the effects of spatial variation of the Hamaker coefficient, as seen from the particular case of solvents with an exponentially decaying spatial intersolvent correlations.

Free energy potential for aggregation of giant, neutral lipid bilayer vesicles by Van der Waals attraction

Here, we report the first direct observation of Van der Waals' attraction between biomembrane capsules using measurements of the free energy reduction per unit area of membrane-membrane contact formation. In these studies, the membrane capsules were reconstituted neutral (egg phosphatidylcholine) lipid bilayers of giant (greater than 10(-3) cm diam) vesicles. Micromanipulation methods were used to select and maneuver two vesicles into proximity for contact; after adhesion was allowed to occur, the extent of contact formation was regulated through the vesicle membrane tensions that were controlled by micropipette suction. The free energy reduction per unit area of contact formation was proportional to the membrane tension multiplied by a simple function of the pipette and vesicle dimensions. The free energy potential for Van der Waals attraction between the neutral bilayers in 120 mM NaCl solutions was 1.5 X 10(-2) ergs/cm2. Also, when human serum albumin was added to the medium in the range of 0-1 mg/ml, the free energy potential for bilayer-bilayer adhesion was not affected. Using published values for equilibrium spacing between lipid bilayers in multilamellar lipid-water dispersions and the theoretical equation for van der Waals attraction between continuous dielectric layers, we calculated the value for the Hamaker coefficient of the Van der Waals attraction to be 5.8 X 10(-14) ergs.

The “equilibrium distance” between two bodies immersed in a liquid

By virtue of the space occupied by at least one monomolecular layer of a liquid (3) separating continua of two identical (1) or two different materials (1) and (2), it could be argued that the minimum equilibrium distance d 0123 and d 0123 must be greater than the equilibrium distances d 0123 or d 0123 (all of which < 2 Λ) between two identical or ion-identical continua (liquid or solid) in the absence of a third, intervening liquid. As long as actual liquid molecules are intercalated between the two continua, that is indubitably the case. However, within the same thermodynamic model used for deriving the Hamaker coefficients A123 or A??? (using d 0??? or d???), d * 0??? or d * 0??? (the asterisks serving to indicate the “equilibrium distances” as defined by the thermodynamic model) may be used for deriving A??? and A???, and within that framework it is postulated that all the above d 0 and d * 0 values may be taken to be of the same order of magnitude. The agreement among the experimental results obtained by a number of different approaches appear to confirm that all d 0??? and d 0??? are of the same order of magnitude. The agreement between A??? values found via surface tension measurements and the postulated d *??? and A??? values for the same systems derived via the Lifshitz approach, indicates that d *??? is indeed also of the same order of magnitude as d 0??? and d 0???. In real physical systems (e.g., antigen—antibody interactions of the van der Waals type) it can be shown that the interaction begins as one of the “132” type and then tends toward the “41” variety, accompanied by extrusion of liquid (3). However that process is seldom completed; the final equilibrium binding energy has a value intermediate between Δ F 123 and Δ F 13.

The concept of negative hamaker coefficients. III. Determination of the surface tension of small particles

The concept of negative hamaker coefficients. III. Determination of the surface tension of small particles

Interaction of phagocytes with other blood cells and with pathogenic and nonpathogenic microbes.

Owing to the high surface tension of blood cells and to the equally high surface tension of their liquid habitat, the Hamaker coefficients A131 of blood cells (subscript 1) in blood (subscript 3), are unusually small; they are of the order of 0.25 to 2.5 X 10(-16) ergs. The very small van der Waals attractions such low Hamaker coefficients give rise to, coupled to the medium low but still sizable negative xi-potentials (-11 to -18 mV) of the cells, which cause an appreciable mutual electrostatic repulsion between blood cells, have been used to elaborate potential energy vs. distance diagrams, which closely reflect the unusual stability of blood cells in blood. When bacteria find their way into the bloodstream, they initially form an almost equally stable suspension. However, relatively hydrophobic nonpathogenic bacteria quickly aspecifically adsorb immunoglobulin G (IgG) molecules from blood serum, whilst hydrophilic pathogenic bacteria sooner or later also become coated with specific antibody molecules of the IgG-class. Through receptor sites on the surface of phagocytic blood cells, which can specifically bind to the Fc tails of IgG molecules, bacteria are first bound and then removed from the blood circulation and surrounding tissues. These Fc-receptor bonds presumably also are of a combined van der Waals and electrostatic nature. Thus in the normal course of events and by purely physicochemical mechanisms, phagocytic leukocytes will neither interfere with other leukocytes nor with any other blood cells, whilst they specifically interact with microorganisms and other unwanted foreign particles via IgG-IgG-receptor interactions. Also discussed, in the light of the principles elaborated above, are: some of the antiphagocytic mechanisms developed by certain pathogenic bacteria; the phagocytic disposal of aged, weak, or abnormal blood cells; and the role played by immunoglobulins other than IgG, and by complement, in the removal of bacteria and viruses.

Zeta potentials, van der Waals forces and hemagglutination.

It has recently become possible to determine the van der Waals (Hamaker) coefficient of erythrocytes, whilst their zeta-potential has been known for some time. With these two data the net potential energy of interaction versus distance diagrams could be elaborated for unsensitized human erythrocytes suspended in saline water, as well as for erythrocytes monogamously sensitized with anti-D (Rh0) antibodies of the IgG class. Unsensitized erythrocytes can approach each other, to within approximately equal to 79 A of their sialoglycoprotein surfaces, leaving a distance between their actual cell membranes of approximately equal to 180 A, which is considerably more than the maximum distance between the two valencies of an IgG molecule (approximately equal to 120 A). This explains why unaided anti-D (Rh0) antibodies of the IgG class cannot cross-link two D (Rh0)-positive erythrocytes, although cross-linking can easily be achieved with IgM class antibodies. D (Rh0)-positive erythrocytes, monogamously sensitized with antibodies of the IgG class, can approach each other to within approximately equal to 60 A (between the Fc ends of the protruding antibodies), which makes cross-linking by means of anti-IgG antibodies of the IgG class feasible.

The concept of negative hamaker coefficients, II. Thermodynamics, experimental evidence and applications

Via a surface-thermodynamic approach the experimental conditions can be predicted under which the effective Hamaker coefficient A132 (of a system comprising two different materials 1 and 2, immersed in a liquid medium 3) acquires a negative value. These predictions could be confirmed by a number of different experimental approaches. The realization that experimental conditions leading to a negative value for A132 can be readily achieved, has opened the way to a number of novel separation methods, and also furnishes a clearer explanation for the mechanism of a few older separation processes. With the surface-thermodynamic approach it becomes a simple matter to determine the A132 value of a system. The precise point at which the A132 value of a system changes signs is usually accompanied by an easily observable physical phenomenon, e.g., a minimum of interaction, a maximum stability, a phase change, or a sudden detachment or elution of particles, cells or macromolecules.

Effects of monovalent ion binding and screening on measured electrostatic forces between charged phospholipid bilayers

In an effort to determine the role that monovalent ions play in the modification of intermembrane forces, we have measured these forces between charged phospholipid bilayers in monovalent ionic solutions. The osmotic stress technique allowed the net electrostatic pressure between the bilayers to be measured while their separation was concurrently determined by x-ray diffraction. Taken together, these measurements yielded electrostatic pressure as a function of bilayer separation. We have related measured pressures to the bilayer surface charge density and surface potential through an exact solution of the full nonlinear Poisson-Boltzmann equation for this system. Quantitative differences in bilayer separation amongst monovalent alkali metal cations indicated differential binding of these to phosphatidylglycerol (PG), phosphatidylserine (PS), and phosphatidic acid (PA); binding affinity series were determined for Li+, Na+, K+, Cs+, and TMA+ ions to these lipids. The anions Cl-, Br-, I-, and CH3COO- were found to have no differential effect on the repulsive forces between PS bilayers. Debye lengths for the electric double layer estimated from the slopes of the experimental pressure curves were consistently longer than predicted on the basis of classic Gouy-Chapman theory. Estimates of the van der Waals Hamaker coefficient between bilayers of PS and PG in salt solution were found to be weaker than between phosphatidylcholine bilayers in pure water, a difference possibly due to electromagnetic retardation and ionic screening.

Dynamic light scattering from cetyltrimethylammonium bromide micelles. Intermicellar interactions at low ionic strengths

Quasi-elastic light scattering measurements have been performed on aqueous solutions of CTAB (cetyltrimethylammonium bromide). Theoretic fits to the measured diffusion coefficients at low NaBr were obtained by using the generalized Stokes-Einstein equation expanded to first order in the surfactant concentration (including both thermodynamic and hydrodynamic terms), with a pair-interaction potential taken from Dlvo theory. Good fits are obtained at low NaBr concentrations by assuming a constant micellar radius for each temperature and assuming that the remaining parameters (aggregation number, fractional ionization, and Hamaker coefficient) are independent of temperature. This procedure yields an unambiguous value for the fractional ionization alpha for CTAB micelles: alpha = 0.22. The limits of validity of the theoretic fitting procedure (previously applied by Corti and Degiorgio to the system sodium n-dodecyl sulfate (SDS) + NaCl) are discussed. 65 references.

Influence of non-ionic surfactants on the equilibrium distance of approach of emulsion droplets

The equilibrium distances between two n-octane droplets in aqueous solutions of non-ionic sucrose surfactants were measured as a function of surfactant and of added electrolyte concentration. From the equilibrium film thickness, values of the Hamaker coefficient and the surface potential were calculated. Increasing the electrolyte concentration at constant surfactant concentration reduced the film thickness. Increasing the surfactant concentration at constant electrolyte concentration increased the film thickness. The results were analysed in terms of a simple three-layer oil–water–oil model, which showed that increasing the concentration of surfactant increased the repulsive forces.The rate of thinning of the films was determined and used to calculate the disjoining pressure as a function of the distance between the oil droplets as they approached.