Hamaker approach interaction constant

Furthermore, these van der Waals interactions are important only near the interface, where it is unlikely that either Lifshitz or Hamaker approaches are accurate for spheres of molecular sizes. For example, the magnitude of the interaction for Na+ ions at. 5 A from the interface is only approximately 0.02kT (the values of B used in the calculation, Z Na = — 1X10 50 J m3 was obtained from fit by Bostrom et al. [17] and ZJNa= +0.8X10 511 J m3 was calculated by Karraker and Radke [18]). Eq. (8) might provide a convenient way to account for the interfacial interactions, if suitable values for Bt (not related to the macroscopic Hamaker constants) would be selected. 

The expression van der Waals attraction is widely used and is here defined as the sum of dispersion forces [9], Debye forces [17] and the Keesom forces [18]. Debye forces are Boltzmann-averaged dipole-induced dipole forces, while Keesom forces are Boltzmann-averaged dipole-dipole forces. The interaction for all three terms decays as 1 /r6, where r is the separation between the interacting particles, and they are combined into one term with the proportionality constant denoted the Hamaker constant. In order to determine the van der Waals force there are at least two approaches, either to calculate the force between two particles assuming that the interaction is additive, (this is usually called the Hamaker approach) or to use a variant of Lifshitz theory. 

The description of the van der Waals interaction based on the Lifshitz approach is now sufficiently advanced to provide accurate predictions for the complete interaction energy. For the geometry of two half-spaces, the exact theory is available in a formulation suited for computational purposes. " In parallel with work on planar systems, there has been a focus on the interaction between spheres. " These developed theories have been used as the exact solutions in the validation of the approximate predictions using the Hamaker approach. The significant contribution of the continuum approach to our understanding of the van der Waals interaction lies in the reliable prediction of the Hamaker constant. The interaction energy for two half-spaces and two spheres is summarized below. 

Adsorbed on the surface of dispersed particles, polymer chains lessen the attraction energy by steric reasons (the minimum distance to which particles can approach increases) and because they change the efficient Hamaker s constant value. The attraction energy in expressions for Ur dependence on A is the function of not only interaction constants of dispersed phase A, dispersion medium A2 and the phase with the medium A 2, but of Gamaker s constant for adsorption layer 3 too. The effect of polymer adsorption layers on molecular attraction of particles has been described theoret-ically. Below is an expression for Ur, based on the Lifshits macroscopic theory... 

In its simplest form a partitioning model evaluates the distribution of a chemical between environmental compartments based on the thermodynamics of the system. The chemical will interact with its environment and tend to reach an equilibrium state among compartments. Hamaker(l) first used such an approach in attempting to calculate the percent of a chemical in the soil air in an air, water, solids soil system. The relationships between compartments were chemical equilibrium constants between the water and soil (soil partition coefficient) and between the water and air (Henry s Law constant). This model, as is true with all models of this type, assumes that all compartments are well mixed, at equilibrium, and are homogeneous. At this level the rates of movement between compartments and degradation rates within compartments are not considered. 

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. 

The concept of additivity is unsatisfactory when applied to closely packed atoms in a condensed body. In this case, a new approach to the energy of interaction needs to be developed or a modification of Hamaker s constant is desirable. 

Note that, as the isoelectric pH approaches the pH of the surface (pH ), the deposition rate increases for a given Hamaker constant. This is most easily explained in terms of changes in the surface potential and charge density which occur as the two surfaces are brought together. Consider the interaction between two identical plane surfaces which display an isoelectric point. Prieve and Ruckenstein (7) have noted that, as the... 

Fig. 1. Effect of surface potential fa) and Hamaker s constant (b) upon the total interaction energy profile for a sphere of radius 6-7 /an approaching a flat plate in a solution having s = 74-3 and tc 1 = 8A.

Alternatively, the Hamaker constant can be calculated in the Lifshitz quantum electrodynamic continuum approach [7], which incorporates all three types of van der Waals interactions for condensed systems (Lifshitz-van der Waals interactions) through surface tension determinations with apolar liquids (e.g. diiodomethane, a-bromonaphthalene)... 

An attractive interaction arises due to the van der Waals forces between molecules of colloidal particles. Depending on the nature of dispersed particles, the Keesom forces (or the dipole-dipole interaction), the Debye forces (or dipole-induced dipole interaction), and the London forces (or induced dipole-induced dipole interaction) may contribute to the van der Waals interaction. First, the van der Waals interaction was theoretically computed using a method of the pairwise summation of interactions between different pairs of molecules of the two macroscopic particles. This method called the microscopic approximation neglects collective effects, and, as a consequence, misrepresents the Hamaker constant. For many problems of a practical use, however, specific features of the total interaction are determined by a repulsive part, and such an effective, gross description of the van der Waals interaction may often be accepted [3]. The collective effects in the van der Waals interaction have been taken into account in the calculations of Lifshitz et al. [4], and their method is known in the literature as the macroscopic approach. 

Dispersion. Dispersion or London-van der Waals forces are ubiquitous. The most rigorous calculations of such forces are based on an analysis of the macroscopic electrodynamic properties of the interacting media. However, such a full description is exceptionally demanding both computationally and in terms of the physical property data required. For engineering applications there is a need to adopt a procedure for calculation which accurately represents the results of modem theory yet has more modest computational and data needs. An efficient approach is to use an effective Lifshitz-Hamaker constant for flat plates with a Hamaker geometric factor [9]. Effective Lifshitz-Hamaker constants may be calculated from readily available optical and dielectric data [10]. 

The major disadvantage of this microscopic approach theory was the fact that Hamaker knowingly neglected the interaction between atoms within the same solid, which is not correct, since the motion of electrons in a solid can be influenced by other electrons in the same solid. So a modification to the Hamaker theory came from Lifshitz in 1956 and is known as the Lifshitz or macroscopic theory." Lifshitz ignored the atoms completely he assumed continuum bodies with specific dielectric properties. Since both van der Waals forces and the dielectric properties are related with the dipoles in the solids, he correlated those two quantities and derived expressions for the Hamaker constant based on the dielectric response of the material. The detailed derivations are beyond the scope of this book and readers are referred to other publications. The final expression that Lifshitz derived is... 

The interaction energy between two identical particles depends on the potential and retarded Hamaker constant (cf. any handbook of colloid chemistry), and Fig. 3,83 shows that a ( potential of about (pins or minus) 40 mV assures an energy barrier that prevents fast coagulation even with relatively high Hamaker constant. When the absolute value of the ( potential is lower this barrier disappears (Fig. 3.82), and the stability ratio (Fig. 3.6) approaches 1. The theory of colloid stability is discussed in detail in handbooks of colloid chemistry. 

Fig. 3 Electrostatic repulsive thick line), van der Waals attractive dotted line) and total thin line) interaction energies of two approaching spherical particles. Particle radius, R=100 nm Stern potential, Pd=10 niV Hamaker constant, A=0.5xl0" J...

The objections to the Hamaker theory were overcome by Lifshitz and his coworkers [Lifshitz 1956, Dzyaloshinskii 1961] using the bulk optical properties of the interacting bodies. The approach employed by Lifshitz uses the so-called Lifshitz-van der Waals constant h that depends only on the materials involved provided the separation distance is relatively small. Under some conditions the constant h can be related to the Hamaker constant by... 

In the original treatment, also called the microscopic approach, the Hamaker constant was calculated from the polarizabilities and number densities of the atoms in the two interacting bodies. Lifshitz presented an alternative, more rigorous approach where each body is treated as a continuum with certain dielectric properties. This approach automatically incorporates many-body effects, which are neglected in the microscopic approach. The Hamaker constants for a number of ceramic materials have been calculated from the Lifshitz theory using optical data of both the material and the media (Table 9.1) (9). Clearly, all ceramic materials are characterized by large unretarded Hamaker constants in air. When the materials interact across a liquid, their Hamaker constants are reduced, but still remain rather high, except for silica. 

Lifshitz (1956) and Dzyaloshinskii et al. (1961) developed an approach to the calculation of the Hamaker constant Ah in condensed phases, called the macroscopic theory. The latter is not limited by the assumption for pairwise additivity of the van der Waals interaction. The authors treat each phase as a continuous medium characterized by a given uniform dielectric permittivity, which depends on the frequency, V, of the propagating electromagnetic waves. 

A is the Hamaker constant (2). In real colloidal systems the particles are suspended in a medium or solvent, in which case an effective Hamaker constant has to be used to describe the force operating between the particles. When two particles 1 and 2 separated by a solvent 3 are at large distances from one another, the interactions are particle-solvent interactions and A 23. When the particles approach one another, particle-solvent and particle-particle interactions must now be considered. The effective Hamaker constant becomes... 

Colloidal approaches also frequently accoimt for van der Waals interactions, ie interactions due to fluctuating dipoles. For atomic species, these interactions vary as distance to the minus sixth power. For protein/surface systems modeled via a colloidal description, this 1/r dependence is integrated over the volumes of the interacting bodies. The result is the product of a Hamaker constant, which depends upon material properties, and a term dependent on the system s geometry. In addition, forces related to solvation (114) and donor/acceptor (115) affects may also be included. 

Figure 7 Plots of potemialenergy of interaction as a function of the distance ff between ihe Surfaces of Identical spherical particles with radius a = I pm. Top classical DLVO theory. Bottom Extended DLVO approach. The following values were used for the calculations - 30 mV 1 1 electrolyte concentralion I0 - M Hamaker constant A = 2 X lO J atrid-base component of the solid-liquid inlerl acial tension = lO mJ/m fieo text for details.

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