Hamaker equation

Figure 12 Residuals plot for the nonlinear model 5.2.1 Hamaker equation...

In studies of steric stabilizers too little attention is generally paid to the dispersion force attractions between particles and the critical separation distance (H ) needed to keep particles from flocculating. Adsorbed steric stabilizers can provide a certain film thickness on each particle but if the separation distance between colliding particles is less than H the particles will flocculate. The calculation of H is not cr difficult and measurements to prove or disprove such calculations are not difficult either. For equal-sized spheres of substance 1 with radius or in medium 2 the Hamaker equation for the dispersion force attractive energy (Uj2i) at close approach is (7) ... 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

Surfactants are used for stabilization of emulsions and suspensions against flocculation, Ostwald ripening, and coalescence. Flocculation of emulsions and suspensions may occur as a result of van der Waals attraction, unless a repulsive energy is created to prevent the close approach of droplets or particles. The van der Waals attraction Ga between two spherical droplets or particles with radius R and surface-to-surface separation h is given by the Hamaker equation,... 

It may be added here that the four basic laws of capillarity, i.e., the equations of Gibbs [(10.2)], Laplace [(10.7)], Kelvin [(10.9)] and Young [(10.10)], all describe manifestations of the same phenomenon the system tries to minimize its interfacial free energy. (Another manifestation is found in the Hamaker equations see Section 12.2.1.) These laws describe equilibrium situations. Moreover, dynamic surface phenomena are of great importance. 

Aggregation. The interactions involved are treated in Chapter 12. It follows that the main cause is often van der Waals attraction (Section 12.2), as given by the Hamaker equations. Another important cause is depletion interaction, where the driving force is increase in mixing entropy of polymer molecules or other small species present in solution (Section 12.3.3). 

Healy (Chapter 7) and Dumont also prefer the first approach. Healy sets down a model based on the control of coagulation by surface steric barriers of polysilicate plus bound cations. Healy s electrosteric barrier model is designed to stimulate new experimental initiatives in the study of silica sol particles and their surface structure. Dumont believes that many particular aspects of the stability of silica hydrosols could be explained not only by the low value of the Hamaker constant but also by the relative importance of the static term of the Hamaker equation. 

Dumont believes that many particular aspects of the stability of silica hydrosols could be explained not only by the low value of the Hamaker constant but also by the relative importance of the static term of the Hamaker equation. 

An analysis presented of the forces contributing to the attraction and repulsion interactions between macromolecules in acrylate latices. The electrostatic repulsion forces, enthalpy and entropy effects, and the attraction forces from the expanded Hamaker equation are analysed. The influence of the structure of copolymers consisting of monomeric units of alkyl acrylate or methacrylate (methyl to n-butyl) and acryhc or methacryhc acid on the physico-chemical properties of the latices and their stabihty were determined. On the basis of experiments and calculations it was established that the stability of latices is decided by two mechanisms. The first (ionic stabilisation) consists in adsorption of anionic emulsifier particles, and the second (ionic-steric stabilisation) involves adsorption of such an emulsifier on an adsorption layer formed by the polymer macromolecules forming the latex. 25 refs. Articles from this journal can be requested for translation by subscribers to the Rapra produced International Polymer Science and Technology. 

The energy arising from the van der Waals attraction may be calculated from the simple Hamaker equation, modiHed for the presence of an adsorbed layer, i.e.. 

Derive for the Lifshitz theory, starting from the Hamaker equation for two different particles in a medium. Equation 2.8, the following equation for the Hamako- constant of particles type (1) in a medium (2) ... 

Show the reduction of Eq. VI-34 for the case of two identical materials 1 interacting through a medium 3. An analysis similar to that in Problem 12 provides a proof of your equation. Formulate this proof it is due to Hamaker [44]. 

Calculate the Hamaker constant for Ar crystal, using Eq. VII-18. Compare your value with the one that you can estimate from the data and equations of Chapter VI. 

This equation was empirically derived from 16 polar fluids and has an average error of 2.9%. A technique for estimating surface tension using nonretarded Hamaker constants (89) has also been presented. 

The equation by Hamaker is one of the most commonly used methods for describing dissipation kinetics using a linear fit. The basic computational form of the equation is... 

As mentioned previously, most agrochemicals do not exhibit linear degradation patterns. As a result, Hamaker proposed another variation of the linear-fit equation that allows better description of nonlinear data sets ... 

In a similar approach to Hamaker, Timme et al. proposed six functions that are also empirically based. However, they took the additional step of suggesting that the choice of the equation should be based on the regression correlation coefficient (r). 

Hamaker constants can sometimes be calculated from refractive igdex data by the Lifshitz equations (8), but it now appears that Y values are closely related to refractive indices and are a direct measure of the Lifshitz attractions. In Equation 1 a correction factor f for "retardation" of dispersion forces is shown which can be determined from Figure 2, a graph of 1/f at various values of H and a as a function of Xj, the characteristic wavelength of the most energetic dispersion forces, calculable and tabulated in the literature (9). 

From the above equation, the variation of equilibrium disjoining pressure and the radius of curvature of plateau border with position for a concentrated emulsion can be obtained. If the polarizabilities of the oil, water and the adsorbed protein layer (the effective Hamaker constants), the net charge of protein molecule, ionic strength, protein-solvent interaction and the thickness of the adsorbed protein layer are known, the disjoining pressure II(x/7) can be related to the film thickness using equations 9 -20. The variation of equilitnium film thickness with position in the emulsion can then be calculated. From the knowledge of r and Xp, the variation of cross sectional area of plateau border Qp and the continuous phase liquid holdup e with position can then be calculated using equations 7 and 21 respectively. The results of such calculations for different parameters are presented in the next session. 

In this version the relationship is called the Girifalco-Good-Fowkes equation. (We use a similar approach again in Chapter 10, e.g., see Equations (10.77) and (10.78), to determine the Hamaker constant for the van der Waals interaction forces.) Although we use y and yd interchangeably, it is important to recognize that yd values are determined by a particular strategy as illustrated in Example 6.5. 

We have already seen from Example 10.1 that van der Waals forces play a major role in the heat of vaporization of liquids, and it is not surprising, in view of our discussion in Section 10.2 about colloid stability, that they also play a significant part in (or at least influence) a number of macroscopic phenomena such as adhesion, cohesion, self-assembly of surfactants, conformation of biological macromolecules, and formation of biological cells. We see below in this chapter (Section 10.7) some additional examples of the relation between van der Waals forces and macroscopic properties of materials and investigate how, as a consequence, measurements of macroscopic properties could be used to determine the Hamaker constant, a material property that represents the strength of van der Waals attraction (or repulsion see Section 10.8b) between macroscopic bodies. In this section, we present one illustration of the macroscopic implications of van der Waals forces in thermodynamics, namely, the relation between the interaction forces discussed in the previous section and the van der Waals equation of state. In particular, our objective is to relate the molecular van der Waals parameter (e.g., 0n in Equation (33)) to the parameter a that appears in the van der Waals equation of state ... 

The Hamaker constant has energy units since (3 has the units energy length6 and the term in parentheses in Equation (62) has the units (volume-1)2. The potential energy of attraction between blocks as calculated by Equation (63) is expressed per unit area of the facing surfaces. Note also that this attraction grows weaker as the distance increases. This is different from the... 

Combining these results leads us to estimate the Hamaker constant to lie in the range 10 20 to 10 1<> J. If we take the midpoint value of 5 10 20 J for A, Equation (63) predicts that equals 1.33 and 1.33 10 2 mJ m 2 when d equals 1.0 and 10 nm, respectively. Table 10.3 presents Hamaker constants for some materials of practical interest. Values for many other materials and reviews of methods of calculating Hamaker constants are available in Gregory (1969) and Visser (1972). 

The same logic that we used to obtain the Girifalco-Good-Fowkes equation in Section 6.10 suggests that the dispersion component of the surface tension yd may be better to use than 7 itself when additional interactions besides London forces operate between the molecules. Also, it has been suggested that intermolecular spacing should be explicitly considered within the bulk phases, especially when the interaction at d = d0 is evaluated. The Hamaker approach, after all, treats matter as continuous, and at small separations the graininess of matter can make a difference in the attraction. The latter has been incorporated into one model, which results in the expression... 

Equations (67) and (68) provide alternatives to Equations (34) and (62) for the evaluation of the Hamaker constant. Although the last approach uses macroscopic properties and hence avoids some of the objections cited at the beginning of the section, the practical problem of computation is not solved by substituting one set of inaccessible parameters (yd and d0) for another (a and n). 

Table 10.5 shows a few numerical examples of how well this attempt at unification succeeds. Equations (28), (32), and (62) have been used to calculate values of the Hamaker constant from refractive index data at visible wavelengths. These values have then been used along with yd values from Chapter 6 to calculate d0 values according to Equation (68). The resulting values of d0 are seen to be physically reasonable. That such plausible values for d0 are obtained is especially noteworthy in view of the approximations made in the calculations. 

---