Hamaker constant function

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. 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

The electrostatic repulsive forces are a function of particle kinetic energy (/ T), ionic strength, zeta potential, and separation distance. The van der Waals attractive forces are a function of the Hamaker constant and separation distance. 

Thus, the spacing of the chains relative to the neutral, free, swollen size of the buoy blocks is, for a given chemical system and temperature, a unique function of the solvent-enhanced size asymmetry of the diblock polymer and a weak function of the effective Hamaker constant for adsorption. The degree of crowding of the nonadsorbing blocks, measured by a decrease in the left-hand side of Eq. 28, increases with increasing asymmetry of the block copolymer. 

The adhesion term is a primary contribution in most cases of atomic-scale friction since a typical external pressure is in the order of 10 MPa while the internal van der Waals pressure is estimated as 1 GPa when using a t5q)ical Hamaker constant. The adhesion term should be also considered as a function of the applied normal load because it is inversely proportional to the mean space Dq, which is dependent on the external pressure. 

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. 

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). 

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.

It is extremely difficult to measure the Hamaker constant directly, although this has been the object of considerable research efforts. Direct evaluation, however, is complicated either by experimental difficulties or by uncertainties in the values of other variables that affect the observations. The direct measurement of van der Waals forces has been undertaken by literally measuring the force between macroscopic bodies as a function of their separation. The distances, of course, must be very small, so optical interference methods may be used to evaluate the separation. The force has been measured from the displacement of a sensitive spring (or from capacitance-type measurements). 

It is evident from Figure 10.7 that the measurements are consistent with both unretarded and retarded attractive forces at appropriate separation distances. It has also been possible to verify directly the functional dependence on radii for the attraction between dissimilar spheres (see Table 10.4), to determine the retardation of van der Waals forces (see Table 10.1), and to evaluate the Hamaker constant for several solids, including quartz. Values in the range of 6 10 20 to 7 10 20 J have been found for quartz by this method. This is remarkably close to the value listed in Table 10.5 for Si02. 

Table 6.3 Hamaker constants for medium 1 interacting with medium 2 across medium 3. The variation in calculated Hamaker constants is due to the fact that different dielectric functions were used in Eq. (6.23). Results were partially taken from Refs. [124,126-130].

Knowles and Turan (2000) and Knowles (2005) have used the approach of Parsegian and Weiss (1972) to examine the effect of anisotropy on Hamaker constants. Such calculations are relevant for materials such as 7 -BN and rutile, Ti02, which exhibit strong anisotropy in their refractive indices because of their crystal structure. They make little difference to predicted values of Hamaker constants as a function of grain orientation for materials such as Si3N4 and SiC which, by comparison, exhibit modest values of birefringence. 

The analysis of Knowles and Turan (2000) of 7 -BN-amorphous silica-3C SiC interfaces showed that Eq. (17.3)could be used to calculate values of the Hamaker constant as a function of the orientation of / -BN with respect to a planar interface containing a thin amorphous silica film, provided that the effective values of static dielectric constant and refractive index for / -BN, /,., and / bx respectively, were taken to be... 

The coagulation coefficient is a function of the radius of the particle Rp, its mass m , the effective particle Knudsen number k, the temperature of the medium T, and the depth of the interaction potential well between two particles. Using the expression for the overall interaction potential given in Appendix I, the depth of the interaction potential well can be calculated from the knowledge of the Hamaker constant for the particle. The friction coefficient f is related to the diffusion coefficient of the particle, D, through the Einstein equation... 

For particles of unit density, in air, at a pressure of 1 atm and temperature of 298°K, the upper and lower bounds of the dimensionless coagulation coefficients are plotted as a function of Knudsen number, for different Hamaker constants, in Fig. 3. Obviously, the upper bound for the coagulation coefficient is independent of the Hamaker constant. The upper and the lower bounds tend to the Smoluchowski expression for small Knudsen numbers. For large Knudsen numbers, the upper bound coincides with the free molecular limit, as can be seen from Fig. 3. The lower bound is found to decrease dramatically with a decrease in the Hamaker constant, for large Knudsen numbers. Both the lower and the upper bounds exhibit a maximum at intermediate values of the Knudsen number. 

The main result of the paper is shown in Fig. 3 in which the upper and lower bounds of the ratio between the coagulation coefficient and the Smoluchowski s coagulation coefficient are plotted as a function of the Knudsen number, for different Hamaker constants (for the lower bound), for particles of unit density. 

From Eq. [1.3] it is obvious dial the depth of the overall interaction potential well between two equal-sized particles is a function of the particle radius Rp and the Hamaker constant A only. 

The rates of formation and dissociation of a doublet consisting of equal-size aerosol particles, in air at 1 atm and 298 K, have been calculated for a Hamaker constant of 1CT12 erg from Eqs. [6] and [7]-[12]. The rate of formation of a doublet rf is plotted in Fig. 1 as a function of 2 for different values of 4>p. The rate of formation of a doublet is found to decrease with increasing particle size at constant (j>p because of the predominant effect of the decrease in the particle number concentration (Eq. [2]). The rate of formation of a doublet does not show as strong a dependence... 

However, for large Peelet numbers and small Hamakers constants, the quantity eP can be expressed as a function of only two dimensionless groups. 

The role of undulation on the equilibrium of lipid bilayers was also examined by Lipowsky and Leibler,18 who used a nonlinear functional renormalization group approach, and by Sornette,14 who employed a linear functional renormalization approach. It was theoretically predicted that a critical unbinding transition (corresponding to a transition from a finite to an infinite swelling) can occur by varying either the temperature or the Hamaker constant. However, the renormalization group procedures do not offer quantitative information about the systems, when they are not in the close vicinity of this critical point. 

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. 

The Lifshitz theory of dispersion forces, which does not imply pairwise additivity and takes into account retardation effects, shows that the Hamaker constant AH is actually a function of the separation distance. However, for the stability calculations that follow, only the values of the attraction potential at distances less than a few nanometers are relevant, and in this range one can consider that AH is constant. 

First, calculations will be presented regarding the interaction energy between two identical spherical particles of radius a = 100 A, Hamaker constant AH = 2.0 x 10-20 J, andpje = 2.6 D. The interaction energy is plotted as a function of distance for electrolyte concentrations ranging from 0.1 to 4.0 mol/dm3 (Figure 4a). At relatively low electrolyte concentrations, the height of the barrier... 

When plotted as functions of where j is the separation distance through water and >.dh is the Debye-Hiickel length, the Hamaker constants H calculated for both kinds of salts collapsed on a single curve, that can be very well represented by [14] ... 

The forth issue is the increase in the repulsion between bilayers at short distances. In Fig. 1, the osmotic pressure is plotted as a function of separation distance (data from Ref. [13]) for no added salt, for l M KC1 and for 1 M KBr. They reveal an increase in repulsion at short separation distances upon addition of salt. While the relatively small difference between 1 M KC1 and 1 M KBr can be attributed to the charging of the neutral lipid bilayers by the binding of Br (but not C.1-) [14], the relatively large difference between no salt and 1 M KCl is more difficult to explain. Even a zero value for the Hamaker constant (continuous line (2) in Fig. 1), in the 1 M KCl case, is not enough to explain the increase in repulsion, determined experimentally. The screening of the van der Waals interaction, at distances of the order of three Debye-Hiickel lengths (about 10 A) should lead, according to Petrache et al. calculations, to a decrease of only about 30% of the Hamaker constant (from 1.2kT to about 0.8kT, see Fig. 5C of Ref. [14]). Therefore, an additional mechanism to increase the hydration repulsion or the undulation force (or both) upon addition of salt should exist to explain the experiments. 

The above two different functions have been used by Pe-trache et al. to fit the interactions between bilayers at small (Ref. [13], Eq. (9)) and large (Ref. [14], Eq. (2)) separations. Since the ranges of validity of the above equations (Eq. (2) and Eq. (9)) are not known, it is clear that the calculation of the Hamaker constant based on either of them is not very accurate. 

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... 

TABLE 1 Accessible Force Measuring Range for vdW Interactions as a Function of Spring Constant. The Calculations Assume a Tip Radius of 5 x IO-h m and a Composite Hamaker Constant of l x lO-20 J... 

For many practical examples in the experimental literature the Hamaker constant is assumed to be a constant and not a function of separation, h, in accordance with Hamaker s original equation [1] ... 

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