Hamaker parameter

The inter-particle force F can be computed as / = A]idol l2Hl), where Ah is the Hamaker parameter for the liquid-particle system and is the distance between two primary particles. The coordination number is based on experimental observation and can be calculated as kc 150p, where 0p is the volume fraction of solid within the aggregates. In the case of compact (or solid) particles 0p is close to unity, whereas in the case of fractal aggregates 0p can be determined once the fractal dimension T)f of the aggregates is known 0p = (0.414T)f - 0.21 l)(r/p/(io) , where dp is the size of the particle and do is the size of the primary particle (Vanni, 2000b). 

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... 

Calculations for Rp as a function of the relevant experimental parameters (eluant ionic species concentration-including surfactant, packing diameter, eluant flow rate) and particle physical and electrochemical properties (Hamaker constant and surface potential) show good agreement with published data (l8,19) Of particiilar interest is the calculation which shows that at very low ionic concentration the separation factor becomes independent of the particle Hamaker constant. This result indicates the feasibility of xmiversal calibration based on well characterized latices such as the monodisperse polystyrenes. In the following section we present some recent results obtained with our HDC system using several, monodisperse standards and various surfactant conditions. 

The calculations in Figure 3 were made on the assumption that the Hamaker constants of the adsorbed dispersant films are the same as the liquid media. This is an excellent assumption for polymers, since this means that the solubility parameter (6 ) of the dispersant and of the medium must be about the same, which is the basic requirement for solubility of polymers. 

The parameter A is known as the Hamaker constant. The force on sphere a then follows via... 

Figure 3. Separation factor-particle diameter behavior computed from the pore-partitioning model showing the effect of the Hamaker constant at a low eluant ionic strength (O.OOl M). Other parameters are = 0.60, interstitial capillary radius = l6 fim, pore radius = fim,...

Figure 5. Separation factor-particle diameter behavior as a function of the pore radius for the pore-partioning model. Hamaker constant = 0.05 pico-erg all other parameters are the same as in Figure 3.

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. 

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 ... 

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). 

List some reasons why it is desirable to relate Hamaker constants to measurable macroscopic properties instead of relying entirely on molecular parameters. 

It is understood that A in Equation (1) is the effective Hamaker constant A2I2 for the system. Of the variable parameters in this equation, it is the one over which we have least control its value is determined by the chemical nature of the dispersed and continuous phases. The presence of small amounts of solute in the continuous phase leads to a negligible alteration of the value of A for the solvent. 

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. 

The VDW interactions seem to have little effect on the rate of aggregation of small vesicles in a primary minimum. However, this statement may be made only because the magnitudes of Hamaker coefficients are less than 10 13 erg (5 X 10 14 erg), in contrast to much higher values frequently used in treatments in colloid science (3). Our estimates of VDW parameters for phospholipid vesicles are based on the analysis of a significant amount of recent data (33,42). 

Figure 3 summarizes the rates of deposition calculated for hydrosols depositing onto a rotating disk. The four curves correspond to the four pairs of surfaces whose interactions are characterized in Fig. 1. Surface characteristics of particle and collector are interchangeable in the calculation of the rates. Values of other parameters include a = 0.1 p,m, cx = I08 cm-3, to = 6 rev/sec, and v = 0.01 cm2/sec. Rates are presented as a function of Hamaker s constant, which characterizes the van der Waals attraction, because this parameter is most difficult to experimentally determine, and because the rate is sensitive to its value.

Calculations of deposition time constants from equation (18) were repeated, assuming lower values for the surface potential. Values of the Hamaker s constant necessary to obtain agreement with the experimental results of Weiss Harios, together with the corresponding values of other parameters are summarized by Table 1. Thus, to obtain agreement using a small Hamaker constant, the surface potential must be considerably lower. 

In Figure 7 we present the free energy for an asymmetric Gaussian distribution (a = 1.4) as a function of distance for various values of the Hamaker constant (with all the other parameters unchanged). For H > 3.825 x 10-21 J, a stable minimum is obtained at a finite distance. For H < 3.825 x 10 21 J, the stable minimum is at infinite distance however, for 3.825 x 10-21c7 > H > 3.45 x 10-21 J, a local (unstable) minimum is still obtained at finite distance. For H = 3.825 x 10-21 J, a critical unbinding transition occurs, since the minima at finite and infinite distances become equal. However, these two minima are separated by a potential barrier, with a maximum height of 1.68 x 10 7 J/m2, located at a separation distance of 90 A. The results remained qualitatively the same for any combination of the interaction parameters. 

For the values of the parameters employed (a relatively large Hamaker constant), the potential barrier is only a few kT or less hence, the apoferritin should coagulate at almost all the concentrations studied. Since experiment shows that the proteins did not coagulate, another repulsion should be present, at least al low separation distances. This repulsion, while essential for the stability of the system, did not affect much, because of its short range, the behavior of the second virial coefficient. In the calculation of the second virial coefficient, it was assumed that the distance of closest approach between apoferritin proteins cannot be less than 8 A. This value leads to a dimensionless second virial coefficient for the hard spheres repulsion of 4.8 instead of 4. 

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