Hamaker constant typical values

The dimensional coefficient A that appears in (6 84) is known as the Hamaker constant. Its value depends on the materials involved, but a typical magnitude is 10 20 to 10 19 J. Generally A is positive, which corresponds to a positive disjoining pressure and attraction between the interface and the solid substrate. However, in some circumstances, A < 0, and the surfaces repel. 

Molecules that self-assemble into reverse micelles with low surfactant properties are generally efficient extractants (such as HDEHP, TBP, malonamides, etc.). Their adsorptions at the interface permit the complexation of the aqueous solute and their low surfactant properties permits the avoidance of the formation of very stable emulsion. Hence, ions are extracted, but typically there is less than one water molecule per ion extracted. Exact determination of coextracted water is still important, however, for interpreting the conductivity values and for evaluating the polar core volumes. Typical values are found for the Hamaker constant, because polar cores are supersaturated salt solution. 

For a typical value of the Hamaker constant in vacuum, A=10 19 J, the attractive force emerging between a tip with an apex radius of 10 nm and a surface separated by 1 nm distance will be F=1 nN. This value sets an approximate scale of the forces which are sensed by the scanning force microscope. 

For a suitably high critical value of A, this theoretical model predicts a lower limit on the equilibrium thickness that can be observed. This lower limit on Z, Z n, is defined by the conditions F= 0 and <1F/<1L = 0 since for a stable film F= 0 and dF/dL > 0 (Clarke, 1987). Various solutions to these conditions have been examined by Knowles and Turan (2000). In the absence of capillary pressure and external pressure, Zmin = 2.58. Using reasonable estimates for Knowles and Turan estimate Zmin to be >6.50 A. That in practice the observed intergranular film thicknesses are typically of the order of 1-2 nm in non-oxide engineering ceramics indicates that the relevant Hamaker constants for ceramics are significantly lower than the critical value. 

Now a and the pressure (computed as the derivative of the free energy per unit area) will be calculated, using the procedure outlined in this article. For the Hamaker constant and the bending modulus, typical values from literature, namely, 1.0 x 10"20 and 1.0 x 10 l l J, respectively, will be used. For th and tc we employed the values obtained from X-rav data,1 th = 7.6 A and tc = 37.8 A for EPC and th = 7.6 A and tc = 36.4 A for DMPC, respectively. Because the hydrocarbon thicknesses tc of EPC and DMPC produced almost no difference (see eq 27), in the following only the results for the EPC are presented (tc = 37.8 A). For the degree of asymmetry a the value of 1.4, which is in agreement with the Monte Carlo simulations,17 was selected. 

The value of A, the Hamaker constant, depends on the particles material and on the properties of the solvent. The Hamaker constant is typically of the order of 10-19-10-21 J. 

Equations (32), (33) and (35) allow us to calculate C2/C2 p as a function of B and X for different values of H213 (Hamaker constant). Due to the complexity of calculation (evaluation of two triple Integrals), this is best done on a computer. Typical results of computer calculations are shown in Figure 7. The effect of Van der Waals attractive forces will lead to an accumulation of solute molecules near the walls for small values of X but at large values of X, the positions close to the pore axis start being preferred. 

Table I reports the values of the static (Agt) and dispersive (Ajisp) parts of the Hamaker constant of silica in water, calculated from Equation (2). The corresponding values for TiOa, a typical electrocratic colloidal oxide, are also included for comparison, which is probably the key to an explanation of the special behavior of the silica hydrosols. These data show that the Hamaker constant of Si02 is approximatively 35 times smaller than that of Ti02. Thus, the attraction energy between two silica particles is 35 times smaller than that between two Ti02 particles of the same size. This weakness of the attraction energy enhances the role of the afore-mentioned structural forces, which are strongly dominated by the London-van der Waals attraction in the case of Ti02.

Note that, for macroscopic particles, the interaction potential decays much more slowly than does that between molecules (oc r decaying only as the reciprocal separation for a sphere and as /LP- for two flat surfaces. The combination Kjf p is a material constant, with dimensions of energy, known as the Hamaker constant. It typically takes values of order 10 J for interactions between identical solids across a vacuum. 

TABLE 4.5. Typical Values of the Hamaker Constant (xlO J ) for Commonly Encountered Materials (A and B) and Intervening Phases (2)... 

Upon approach (or withdrawal) of objects surface forces can be determined (surface forces apparatus, AFM). To estimate forces for a typical AFM experiment, let us consider a sphere with a radius of 1 pm (replacing an AFM probe) at a distance of 1 nm from a surface. A characteristic value of the constant C = 4ae = 5 X 10 Jm with 6>i = 6>2 = 3 x 10 m . In this case a Hamaker constant A = 1 x 10 J was used and the corresponding force obtained had a value of 1.6 x 10 N (16 nN). In reality, the tip radii of AFM probes are much smaller (on the order of 10-20 nm) and the tip-sample distance varies depending on AFM operating conditions. Typical values of contact forces (at pull-off) in characteristic AFM experiments are in the range of 10 pN-10 nN. 

For the collision of this drop with another immobile one, we have = AH/(16jtApgF ). We see that hp is inversely proportional to the drop radius. For typical values of the Hamaker constant = 4 x 10" ° J, density difference Ap = 0.12 g/cm and R = 10 pm, the thickness of pimple formation is hp = 82.3 nm. Note that this thickness is quite large. The pimple formation can be interpreted as the onset of instability without fluctuations (stability analysis of the film intervening between the drops has been carried out elsewhere [62]). 

If the gap between the surfaces approaches zero, forming a direct ideal contact, that is, there is no film, then Cf= 0, and IAOf(/ o)l = 2a = W, which yields the work of cohesion in the condensed phase. The value of the parameter h = remains somewhat undefined. To generate a crude estimate for h, it is possible to work backward from the relationship for the Hamaker constant, that is, A/I2nhi = 2a. For hydrocarbons, typical values of o are around 20-25 mJ/m, from where we can obtain h. A/24nd) = (1.5-2) x 10 ° m, that is, h is on the order of 1.5-2 A, which corresponds to interatomic distances. 

For typical lyophobic colloidal systems with a complex Hamaker constant 10" J and a particle-particle separation of 0.2-1 nm, the adhesive energy in the contact is significantly larger than kT, which indicates that thermodynamics favors the formation of coagulation contacts. The primary potential energy minimum is even deeper in systems conposed of coarser particles. At the same time, for the case of low values of the complex Hamaker constant, 10 -10 J, the adhesion between the particles in systems that are not too coarse (particles with a diameter up to a micron) is overcome by the Brownian motion, and the formation of structures with coagulation contacts is impossible. 

Taking the data of Cooper et al, Ranee and Zichy (48) compared the number of charges at the plane of shear per PVC particle in VCM with the surface charge of typical aqueous anionogenic polymer latices of similar particle size. These data are shown in Table 2. Values of Lifshitz-Hamaker constants for these three systems indicate that the attractive interactions... 

When we have a colloid system in a medium othffl than vacuum or air, the Hamako constant is decreased and an effective value must be used. We thoeforc use the symbol A in the relevant equatiems. Thus, in the more realistic situation where spherical particles or surfaces are imm ed in anoth liquid or sohd medium, th i the van der Waals forces can be greatly reduced. As most industrial processes are carried out in a liquid, thCTC arc typically two effects (i) we may have adsorption phmomraia on the particle surface and thus reduction of the surface raiogy and (ii) we have reduced van der Waals forces due to the smaller value of the effective Hamaker constant in aliquid compared to vacuum or air. 

The Hamaker constant can be calculated directly from London s theory (Chapter 2, Equation 2.6). The Hamaker constant values range typically between 0.4 and 4 x 10 J. The relatively constant values for different compounds arise because the parameter C (see Equation 2.6) is roughly proportional to the square of the polarizability which is roughly proportional to the square of the volume (or the inverse of number density), as discussed in Chapter 2. Thus the whole product (the Hamaker constant) is roughly constant, which is, of course, a gross oversimplification. 

Moreover, as shown by Shaw (1992), setting in these equations a typical CCC value equal to 0.1 mol at 25 ° C and z (counter-ion) = -1 for a surface potential equal to 75 mV, we obtain a Hamaker constant equal to 8 X 10 ° J, which is within the expected values from the London/van der Waals theories (as discussed previously). 

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