Problem 10.1. Hamaker Constant

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  

These values are lower by a factor of 4 than those for similar materials given in Table 10.2. 

3 Attractive Interaction Energy for Former Coated Particles 

AVhen the particles are coated with a pol3mier of thickness S, the van der Waals attractive interaction energy is calculated by [13—16] 

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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). 

From a plot of log W versus log c determine the CCC value and T0 [by means of Eq. (53)]. Use the approximation for T0 given in Problem 6 to estimate for this colloid. Use the values of the CCC and T0 determined in Eqs. (5) and (6) to estimate the effective Hamaker constant Ain for polystyrene dispersed in water. Describe how A might be estimated using a more realistic model than that used in the derivation of Eqs. (5) and (6). 

In view of the model used in Problem 15, criticize or defend the following proposition If one surface carries adsorbed rods and the other is bare, the system could be stabilized against flocculation by dispersing the particles in a medium of intermediate y. Such a system would remain dispersed indefinitely since both steric considerations and a negative Hamaker constant oppose flocculation. 

The major problem in calculating the van der Waals interaction between colloidal particles is that of evaluating the Hamaker constant, A. Two methods are available. 

The third issue is that, in determining the appropriate values of the Hamaker constant, it was supposed that both the exponential hydration (Eq. (1)) and the Helfrich repulsion (Eq. (3)) remain unchanged upon addition of a salt (and that the osmotic pressure due to the interlamellar salt deficit, discussed above, is negligible). While the exponential form for the hydration force has been determined at high osmotic pressures (corresponding to separations of around 10 A) it is not clear that this behavior will remain the same at distances of the order of 40 A. For a decay length of 2.2 A [14], the magnitude of the hydration forces decreases about 106 times between 10 and 40 A, and even a minuscule deviation from the exponential behavior will have drastic consequences in the evaluation of the Hamaker constant. The problems associated with a constant Helfrich repulsion (due to the thermal undulations of the bilayers) are even more complicated and will be addressed in detail in Section 3. 

An attractive interaction arises due to the van der Waals forces between molecules of colloidal particles. Depending on the nature of dispersed particles, the Keesom forces (or the dipole-dipole interaction), the Debye forces (or dipole-induced dipole interaction), and the London forces (or induced dipole-induced dipole interaction) may contribute to the van der Waals interaction. First, the van der Waals interaction was theoretically computed using a method of the pairwise summation of interactions between different pairs of molecules of the two macroscopic particles. This method called the microscopic approximation neglects collective effects, and, as a consequence, misrepresents the Hamaker constant. For many problems of a practical use, however, specific features of the total interaction are determined by a repulsive part, and such an effective, gross description of the van der Waals interaction may often be accepted [3]. The collective effects in the van der Waals interaction have been taken into account in the calculations of Lifshitz et al. [4], and their method is known in the literature as the macroscopic approach. 

Problem 7.4 (Worked Example) Consider a suspension of silica particles in water for which the Hamaker constant is 10" J and the dielectric constant is = 50. If the surface charge is 0.1 charges/nm, calculate how high the molarity of NaCl must be to induce flocculation. Remember, each surface charge is that of an electron, e = 1.6 x 10" C, the permittivity of space is q = 8.8 x 10 J m , and the Bjerrum length is ib — 58/s nm. Assume a weak... 

PAGE 131 In problem 3,23 change the text to A is a dimensionless parameter proportional to the Hamaker constant and also at the end effective polymer-surface interaction parameter A ,... 

The role of the medium, in which contacting and pull-off are performed, has been mentioned but not considered so far. However, the surroundings obviously influence surface forces, e.g., via effective polarizability effects (essentially multibody interactions e.g., by the presence of a third atom and its influence via instantaneous polarizability effects). These effects can become noticeable in condensed media (liquids) when the pairwise additivity of forces can essentially break down. One solution to this problem is given by the quantum field theory of Lifshitz, which has been simplified by Israelachvili [6]. The interaction is expressed by the (frequency-dependent) dielectric constants and refractive indices of the contacting macroscopic bodies (labeled by 1 and 2) and the medium (labeled by 3). The value of the Hamaker constant Atota 1 is considered as the sum of a term at zero frequency (v =0, dipole-dipole and dipole-induced dipole forces) and London dispersion forces (at positive frequencies, v >0). 

Particles immersed in a liquid medium experience a smaller attractive force. The calculation of such forces and their relationship to the nature of the particles and of the dispersion medium has been, and continues to be, a major theoretical problem. Important advances towards its solution have been made in recent years, and they will be mentioned briefly in Chapter 15. For the present purposes it is sufficient to use a relatively simple approximate equation according to which the appropriate Hamaker constant to be employed when two particles of material 1 are separated by a medium 2 is given by... 

Equation (2.1.6) fails completely for liquids such as water, methanol and formamide. This is not surprising, because in these liquids hydrogen bonds play a substantial role in providing cohesive forces whose effect is not included in the purely dispersive Hamaker constant. One approach to this problem, initiated by Fowkes (1962), has been to assume that each type of intermolecular force makes an additive contribution to the surface tension. So-called group contribution methods have been developed on the basis of this idea, which are of considerable practical use in classifying and predicting surface tension behaviour in a semi-empirical way. 

Lifshitz avoided this problem of additivity by developing a continuum theory of van der Waals forces that used quantum field theory. Simple accounts of the theory are given by Israelachvili and Adamson. Being a continuum theory, it does not involve the distinctions associated with the names of London, Debye and Keesom, which follow from considerations of molecular structure. The expressions for interaction between macroscopic bodies (e.g. Eqns. 2-4) remain valid, except that the Hamaker constant has to be calculated in an entirely different way. 

Estimate the Debye length at the conditions of the problem. How are the Debye length and the stability expected to change if the NaCl solution is replaced by an AICI3 solution of the same concentration Draw qualitatively the V-H curve in the cases that (i) the effective Hamaker constant is changed to 0.4 x 10 ° J (all other parameters have the same values) and (ii) the zeta potential is changed to -20 mV (aU other parameters have the same values). 

If the Hamaker constant is decreased to 0.4 x 10 ° J (4 x 10 J), then the stability is improved (higher V-H curve, e.g. from form (b) to form (a)). If the zeta potential changes to -20 from -40 mV, then the stability is decreased. Then, the curve will change, see figure below, from form (b) (curve of the problem) to one of the other lower ones shown in the figure taken fiom Israelachvili (1985). Without additional information we cannot conclude whether the dispersion wiU remain stable, but in all cases the stability is decreased. 

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