Hamaker constant from surface tension

Equations 6.12 and 6.13 can be used in many ways, e.g. for estimating Hamaker constants from surface tension data and vice versa. Once reliable values of surface tensions and Hamaker constants are available from other techniques or measurements, these equations can be used to estimate the interparticle distance. Such calculations yield for many compounds like hydrocarbons, water and polystyrene values around 0.2-0.3 nm, which are physically reasonable. 

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. 

Calculate the Hamaker constant from the surface tension of heptane (7 = 20.3 mJ m ) and dodecane (7 = 25.4 mJ m ). From the density of the solvents and their molecular weights, the molecular spacing can be determined. This allows the calculation of tiie Hamaker constant using equation 10.12, giving the following results ... 

This equation was empirically derived from 16 polar fluids and has an average error of 2.9%. A technique for estimating surface tension using nonretarded Hamaker constants (89) has also been presented. 

For nonpolar materials the Hamaker constant can be calculated from the surface tension or surface free energy of the material, y, and the equilibrium spacing, d, between molecules in the material as follows [11] ... 

As the structure of the surface, and hence U° and S , are unique for each liquid and completely determined by the nature of the molecules and their interactions, it follows that this also applies to ycmd dy/dT. Therefore, it makes sense to search for molecular interpretations of both of these quantities. However, direct relationships between the surface tension and bulk properties, such as energy densities or Hamaker constants, are basically incomplete unless they take the interfacial rearrangements into account. At best one can say that bulk properties and the surface tension are different manifestations of the same interaction. From the data in sec. 1.12 and app. 1, it is concluded that neglecting the TS° term cam lead to errors of several tens of a percent. 

Recall that we already derived a similar expression from the van der Waals theory under a number of restrictive simplifications, see (2.5.44 and 45]. There the geometric mean was related to the same mean of Hamaker constants. This equation can be tested experimentally for liquids like water, in which a variety of forces are operative, y can be established by measuring interfacial tensions against organic liquids in which the interaction is dominated by the dispersion forces. This analysis can be illustrated with the data of table 2.3. In (2.11.19] a is an organic liquid (like a hydrocarbon, he) for which it was assumed that only dispersion forces determined the surface tension y = Consequently, y" is the only unknown. Its value appears to be invariant at about 22 mJ m", comprising 30% of the total tension. 

For a small App/nv, that is, better dispersion, one wants the value of App to be close to that of Amm- The Amm values calculated from the surface tension of the liquid [2,5] are listed in Table 1. If the dispersoid particles are covered by another material, the value of App approaches that of the adsorbate material. In this case, for small App/m, one can select a dispersant having a value of A close to that of the liquid medium. Unfortunately, accurate determination of the Hamaker constant for a given material system is not yet well established. 

Using a Hamaker constant value for hexane equal to 4.1 x 10 ° J and assuming a separation equal to 0.15 nm, provide an estimation of the surface tension of hexane. Compare it to the experimental value from direct measurements (18.4 mN m ). 

Moreover, dispersion surface tensions from Fowkes are in good agreement with Hamaker constants and experimentally measured van der Waals forces between surfaces (Hiemenz and Rajagopalan, 1997 Fowkes, 1964). 

A frequently used method to measure the Hamaker constant of solids is based on the idea that the interaction of a solid surface with nonpolar liquids will mainly occur through the London dispersion interaction. One can therefore define a dispersive surface tension 7° that contains only the dispersive contribution. The value of Ys can be determined from contact angle measurements with nonpolar liquids using the Fowkes equation [83]... 

Here, 0 is the contact angle (see Section 5.3) and Yl is the surface tension of the nonpolar liquid. By plotting cos 0 versus l/ / for several nonpolar liquids, one obtains from the slope of a linear fit. The Hamaker constant Ai vi for material 1 interacting with itself over a gap in vacuum is... 

With this expression and observed values of the index of refraction, assuming w = 2.63 X 10 rad sec, Israelachvili (1974) calculated the Hamaker constants for a number of liquids and also thdr surface tension from ... 

---