Surface force Hamaker constant

A thin film of hydrocarbon spread on a horizontal surface of quartz will experience a negative dispersion interaction. Treating these as 1 = quartz, 2 = n-decane, 3 = vacuum, determine the Hamaker constant A123 for the interaction. Balance the negative dispersion force (nonretarded) against the gravitational force to find the equilibrium film thickness. 

Determine the net DLVO interaction (electrostatic plus dispersion forces) for two large colloidal spheres having a surface potential 0 = 51.4 mV and a Hamaker constant of 3 x 10 erg in a 0.002Af solution of 1 1 electrolyte at 25°C. Plot U(x) as a function of x for the individual electrostatic and dispersion interactions as well as the net interaction. 

SFA has been traditionally used to measure the forces between modified mica surfaces. Before the JKR theory was developed, Israelachvili and Tabor [57] measured the force versus distance (F vs. d) profile and pull-off force (Pf) between steric acid monolayers assembled on mica surfaces. The authors calculated the surface energy of these monolayers from the Hamaker constant determined from the F versus d data. In a later paper on the measurement of forces between surfaces immersed in a variety of electrolytic solutions, Israelachvili [93] reported that the interfacial energies in aqueous electrolytes varies over a wide range (0.01-10 mJ/m-). In this work Israelachvili found that the adhesion energies depended on pH, type of cation, and the crystallographic orientation of mica. 

Hamaker [32] first proposed that surface forces could be attributed to London forces, or the dispersion contribution to van der Waals interactions. According to his model, P is proportional to the density of atoms np and s in the particle and substrate, respectively. He then defined a parameter A, subsequently becoming known as the Hamaker constant, such that... 

Typical forces profdes measmed between glass surfaces in ethanol-cyclohexane mixtures are shown in Fig. 2. Colloidal probe atomic force microscopy has been employed. In pure cyclohexane, the observed force agrees well with the conventional van der Waals attraction calculated with the nometarded Hamaker constant for glass/cyclohexane/glass. 

The attractive force (F) is dependent on the Hamaker constant and the shortest distance between the particles, z. F may be decreased by decreasing A or increasing z. Theoretically, the Hamaker constant can be decreased by decreasing the densities of the two interacting particles. Since the separation distance plays a significant role in van der Waals attraction, any means to increase this distance will reduce the attractive force and increase the ease of dispersion. Surface roughening and the use of spacer particulates can increase interparticulate separation with the improved particle dispersion. 

The Hamaker constant A can, in principle, be determined from the C6 coefficient characterizing the strength of the van der Waals interaction between two molecules in vacuum. In practice, however, the value for A is also influenced by the dielectric properties of the interstitial medium, as well as the roughness of the surface of the spheres. Reliable estimates from theory are therefore difficult to make, and unfortunately it also proves difficult to directly determine A from experiment. So, establishing a value for A remains the main difficulty in the numerical studies of the effect of cohesive forces, where the value for glass particles is assumed to be somewhere in the range of 10 21 joule. 

Historical deveiopment of van der Waais forces. The Lennard-Jones potentiai. intermoiecuiar forces. Van der Waais forces between surfaces and coiioids. The Hamaker constant. The DLVO theory of coi-loidal stability. 

Once we have established reasonable values for the Hamaker constants we shonld be able to calculate, for example, adhesion and surface energies, as well as the interaction between macroscopic bodies and colloidal particles. Clearly, this is possible if the only forces involved are van der Waals forces. That this is the case for non-polar liquids such as hydrocarbons can be illustrated by calculating the surface energy of these liqnids, which can be directly measured. When we separate a liquid in air we mnst do work Wc (per unit area) to create new surface, thus ... 

This relation is clearly very simplified, being based on a number of approximations, such as the validity of the use of the Hamaker constant at such close distances and the particle and surface being of the same material. Also, the relationship between surface force and van der Waals forces does not hold for many solids, in particular for metals where metallic bonding is important. Nonetheless, if taken as an indication of the forces holding particles to each other and to snr-faces, it does give a feel for these forces. 

Finally, even if these criteria are satisfied, there remains the question of whether the product will adhere to form a film or just precipitate homogeneously in the solution. This is the most difficult criterion to answer a priori. The hydroxide and/or oxy groups present on many substrates in aqueous solutions are likely to be quite different in a nonaqueous solvent (depending on whether hydroxide groups are present or not). Another factor that could conceivably explain the general lack of film formation in many organic solvents is the lower Hamaker constant of water compared with many other liquids this means that the interaction between a particle in the solvent and a solid surface will be somewhat more in water than in most other liquids (see Chapter 1, van der Waals forces). From the author s own experience, although slow precipitation can be readily accomplished from nonaqueous solutions, film formation appears to be the exception rather than the rule. The few examples described in the literature are confined to carboxylic acid solvents (see later). 

An expression for the Hamaker constant analogous to Equation (67) had been proposed by Fowkes (1964) for the case when only dispersion forces determine the surface tension. The Fowkes equation... 

In the new version, Chapter 10 focuses exclusively on van der Waals forces and their implications for macroscopic phenomena and properties (e.g., structure of materials and surface tension). It also includes new tables and examples and some additional methods for estimating Hamaker constants from macroscopic properties or concepts such as surface tension, the parameters of the van der Waals equation of state, and the corresponding state principle. 

Equation (6.25) not only allows us to calculate the Hamaker constant, it also allows us to easily predict whether we can expect attraction or repulsion. An attractive van der Waals force corresponds to a positive sign of the Hamaker constant, repulsion corresponds to a negative Hamaker constant. Van der Waals forces between similar materials are always attractive. This can easily be deduced from the last equation for 1 = e2 and n = n2 the Hamaker constant is positive, which corresponds to an attractive force. If two different media interact across vacuum ( 3 = n3 = 1), or practically a gas, the van der Waals force is also attractive. Van der Waals forces between different materials across a condensed phase can be repulsive. Repulsive van der Waals forces occur, when medium 3 is more strongly attracted to medium 1 than medium 2. Repulsive forces were, for instance, measured for the interaction of silicon nitride with silicon oxide in diiodomethane [121]. Repulsive van der Waals forces can also occur across thin films on solid surfaces. In the case of thin liquid films on solid surfaces there is often a repulsive van der Waals force between the solid-liquid and the liquid-gas interface [122],... 

In general, the forces which contribute to the net force exerted on the tip can be divided into three groups ( Fig. lb) (i) surface forces, Fs, (ii) forces due to the sample deformation, Fd> and (iii) the deflection force of the cantilever, Fc. It is important to note that all three forces can be of either sign. The van der Waals force is determined by the Hamaker constant, which depending on the dielectric properties of the contacting phases can be positive or negative. Also the deformation force can be repulsive or attractive, when the sample is pressed or stretched by the tip, respectively. 

For a typical value of the Hamaker constant in vacuum, A=10 19 J, the attractive force emerging between a tip with an apex radius of 10 nm and a surface separated by 1 nm distance will be F=1 nN. This value sets an approximate scale of the forces which are sensed by the scanning force microscope. 

The van der Waals forces between surfaces covered with adsorbed layers can be calculated using the effective Hamaker constant A given by (18)... 

The effect of salt concentration on the force is presented in Fig. 3 for a Hamaker constant of 0.5 x 10 20 J. As the salt concentration increases, the force between plates at constant surface potential becomes more attractive at large distances. 

Figure 6.8. Variation with separation distance r of the interaction energy due to van der Waals forces as calculated by the pairwise interaction model, between two different atoms (a) and between two surfaces (b). A, the Hamaker constant, equals n2C p,p2 where C is the constant of the atom-atom pair potential in equation (6.11) (case a) and p 1 and p2 are the number of atoms per unit volume in the...

There arc two principal ways to estimate the Hamaker constant using AFM. direct form fitting of the experimental data to a theoretical expression (e.g.. an inverse-square relationship for nonretarded systems), and (when confident that there are no other contributions to the surface force, for instance. EDI. interactions), by the surface separation at jump-in Hlump) according to... 

FIGURE 10.4 The force measured between two curved mica surfaces in solutions of 2 1 electrolytes Ca Sr and Ba ) at pH 5.8. The solid lines are based on the theory for potentials and concentrations shown along with the van der Waals attraction corresponding to a Hamaker constant of 2.2 x 10 J. Redrawn from Pashley and Israelachvili [17]. Reprinted with permission from Academic Press. 

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