Hamaker function

A fuller theoretical analysis of vdW interactions requires recourse to Lifshitz theory [8[. Lifshitz theory requires a description of the dielectric behavior of materials as a function of frequency, and there are several reviews for the calculation of Hamaker functions using this theory. The method described by Hough and White (H-W) [95], employing the Ninham-Parsegian [96] representation of dielectric data, has proved to be most useful. The nonretarded Hamaker constant (for materials l and 2, separated by material 3) is given by... 

The behaviour of the interaction free energy depends strongly on geometry. Sometimes, the geometric factor can be evaluated by taking a "Hamaker function" [3-10] calculated by Lifshitz theory, multiplied by an appropriate geometric distance factor evaluated from pairwise summation. (This assumption fails for the temperature dependent contribution, i.e. in water, the most interesting liquid )... 

As seen in Fig. 1 of Ref 37 the coordinates of the secondary minimum corresponds to Khjjjjjj= 5-12 nm. Owing to this rather large distance the frequency dependence of the Hamaker constant may be of impor tance, and the Hamaker function A(h) characterizing molecular interaction should be introduced. 

The result from 36 systems in Ref. 82 are in quite good accordance with die ealeulations of odier papers. Aeeording to Churaev, the system polystyrene-water-polystyrene ean be used to estimate the Hamaker function for oil-water systems. However, with increasing droplet separation the importance of Aq increases on aeeount of [A(h) Aq]. The component Ao is screened in electrolyte eoneentrations, because of dielectric dispersion (83-85). At a distanee of khmin=3-5 nm the authors (84) found that moleeular interaction disap peared at zero frequeney. Experimental evidence con cering this statement is diseussed in Ref. 14. When evaluating the secondary minimum coagulation, Aq can be omitted, as illustrated in Ref 85. 

In Fig. 9, the domain of flocculation is located above and to the left of curve 2 the domain of coagulation is located beneath and to the right of curve 1. To characterize the sensitivity of the domain boundaries to the Hamaker function value, curves 1 and 2 are calculated using values twice as high as those of curves 1 and 2. 

Figure 9 Domains of coagulation and flocculation. Curves 1 and 2 are calculated with the Rabinovich-Churaev Hamaker function a twice higher value is used for calculation of curves 1 and 2. The domain of flocculation is located above curve 1, while the domain of coagulation is located beneath curve 2. Volume fraction 0 = 0.01 (a) (j> = 0.l (b). Particle dimension 2a = 4 (xm. (From Ref 26.)...

To illustrate the dependence of the mobility function d>y on the concentration of surfactant in the continuous phase, in Fig. 12 we present theoretical curves, calculated in Ref 138 for the nonionic surfactant Triton X-100, for the ionic surfactant SDS ( + 0.1 M NaCl) and for the protein bovine serum albumin (BSA). The parameter values, used to calculated the curves in Fig. 12, are listed in Table 4 and K are parameters of the Langmuir adsorption isotherm used to describe the dependence of surfactant adsorption, surface tension, and Gibbs elasticity on the surfactant concentration (see Tables 1 and 2). As before, we have used the approximation Dj Dj (surface diffusivity equal to the bulk dif-fusivity). The surfactant concentration in Fig. 12 is scaled with the reference concentration cq, which is also given in Table 4 for Triton X-100 and SDS + 0.1 M NaCl, cq is chosen to coincide with the cmc. The driving force, F, was taken to be the buoyancy force for dodecane drops in water. The surface force is identified with the van der Waals attraction the Hamaker function Ajj(A) was calculated by means of Eq. (86) (see below). The mean drop radius in Fig. 12 is a = 20 /pm. As seen in the figure, for such small drops 4>y = 1 for Triton X-100 and BSA, i.e., the drop sur-... 

Strictly speaking, Eq. (3.25) is valid for two half-spaces only. In the case of two spherical particles, there is a complex interrelation between geometiy parameter and material properties. The exact analytical solution of the Lifshitz approach was presented by Langbein (1974, as cited by Thennadil and Garcia-Rubio 2001). However, the necessary computations are rather laborious and converge slowly. For that reason, it seems appropriate to approximate the exact analytical solution by combining the Lifshitz-Hamaker function of two half-spaces (Eq. (3.25)) with the Hamaker geometry function for two spherical particles (Eq. (3.23)). 

The shape of the function Vi h) for the total interaction energy depends on the Hamaker function the difluse layer potentials t/ j (approximated with the... 

Both approaches merge when the material constant A132 at Hamaker-de-Boer (cf. Eq. (3.23)) is replaced by a distance dependent material function. For two parallel half spaces (i.e. thick plates) the Hamaker function Ai32(fi) is computed by ... 

The shape of the Hamaker function as defined by Eq. (B. 100) can be approximated weU with a simple three-parameter equation (Viravathana and Marr 2000, cf. Sect. 3.2.1) ... 

The non-retarded Hamaker constant Ai32,o reflects the contribution of both static as well as dispersion interactions. At large surface distances h, the latter vanish and the Hamaker function tends towards its static part Ai32,s ... 

The quantity /id from Eq. (B.105) typically amounts to several nanometres it is best derived by fitting the real Hamaker function (B.lOO). 

Bergstrom (1997) provided comprehensive dielectric data of many substances for the computation of Hamaker functions. They were used to calculate the parameters Ai32,o, Ai32,s, and for three oxides which are frequently produced as pyrogenic powders. 

Table B.l Parameters of the Hamaker function acc. to Eq. (B.105), symmetric pairs...

Equation (2.74) tells us how far the dispersion interaction effectively reaches into the material for the case D -C d, the second and third terms in the square brackets will vanish and the interaction will be the same as if the slabs had infinite thickness. This means that the van der Waals interaction between two bodies with parallel planar surface will occur essentially between surface layers with a thickness of the order of the separation D. So for very small values of D, only a very thin surface layer will contribute. As a consequence, the van der Waals interaction between layered materials will have complex dependence on distance. As an example, we will take the interaction between two half-spaces of material 1 coated with a thin layer of material 2, separated by a gap filled with material 3. The corresponding Hamaker constant Au-m, which should better be called Hamaker function, will depend on the gap thickness D. For a very small gap width D d, the van der Waals interaction will essentially occur between the two layers of material 2 and Aujn (D 0) will be equal to A232, just as if we had the interaction of two half-spaces of material 2 across material 3. For distances much larger than the film thickness of material 2, the interaction will reach far into the material 1 and A12321 (D 0) will approach the value of Am, just as if the layers of material 2 would not exist (Figure 2.9). 

A rough approximation to understand qualitatively the transition from the non-retarded van der Waals force to the retarded regime can be derived using Eq. (2.45). Note that only the London dispersion contribution is affected by retardation. When retardation comes into play, the Hamaker constant A123 becomes, in fact, a Hamaker function that will depend on separation x. Under the simplifying assump-... 

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