Model Hamaker

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.

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... 

By combining Hertz s contact theory (Eq. 1) and with Hamaker s functional form for the attractive force (Eq. 17), the Derjaguin model takes the form... 

Figure 12 Residuals plot for the nonlinear model 5.2.1 Hamaker equation...

General degradation rate models of organics in soils have been described by Hamaker,146 Larson,147 and Rao and Jessup.148 In most instances, biodegradation is the major, but not necessarily the only, process affecting the rate of degradation. 

In its simplest form a partitioning model evaluates the distribution of a chemical between environmental compartments based on the thermodynamics of the system. The chemical will interact with its environment and tend to reach an equilibrium state among compartments. Hamaker(l) first used such an approach in attempting to calculate the percent of a chemical in the soil air in an air, water, solids soil system. The relationships between compartments were chemical equilibrium constants between the water and soil (soil partition coefficient) and between the water and air (Henry s Law constant). This model, as is true with all models of this type, assumes that all compartments are well mixed, at equilibrium, and are homogeneous. At this level the rates of movement between compartments and degradation rates within compartments are not considered. 

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,...

The same logic that we used to obtain the Girifalco-Good-Fowkes equation in Section 6.10 suggests that the dispersion component of the surface tension yd may be better to use than 7 itself when additional interactions besides London forces operate between the molecules. Also, it has been suggested that intermolecular spacing should be explicitly considered within the bulk phases, especially when the interaction at d = d0 is evaluated. The Hamaker approach, after all, treats matter as continuous, and at small separations the graininess of matter can make a difference in the attraction. The latter has been incorporated into one model, which results in the expression... 

This review indicates that good solvent conditions (in terms of either x or 0) result in a positive value for AGR. This is what would be expected from a model that assumes that the first encounter between particles with adsorbed layers is dominated by the polymers. Conversely, in a poor solvent AGR is negative and amounts to a contribution to the attraction between the core particles as far as flocculation is concerned. Under these conditions the polymer itself is at the threshold of phase separation. Van der Waals attraction between the core particles further promotes aggregation, but it is possible that coagulation could be induced in a poor solvent even if the medium decreases the effective Hamaker constant to zero. 

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). 

MackorJ used the model outlined in Example 13.6 to derive the expression AGR = NkBT( — d/L) for the repulsion per unit area of particles carrying N rods of length L when the surfaces are separated by a distance d. Assuming this repulsion equals the van der Waals attraction when the particle separation is 1.5 nm, calculate the effective Hamaker constant in this system if L = 2.5 nm. Select a reasonable value for Win this calculation and justify your choice. 

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. 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

Numerous studies have shown that several factors affect pesticide degradation rates, including soil type, water content, pH, temperature, and clay and organic matter content (Rao and Davidson, 1980). Hamaker (1972) has published an excellent review on the quantitative aspects of pesticide degradation rates in soils. He consider two types of rate models ... 

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. 

The theory has been developed for two special cases, the interaction between parallel plates of infinite area and thickness, and the interaction between two spheres. The original calculations of dispersion forces employed a model due to Hamaker although more precise treatments now exist [194],... 

C. Argento and R. H. French, "Parametric tip model and force-distance relation for Hamaker constant determination from atomic force microscopy," J. Appl. Phys., 80, 6081-90 (1996). 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

Table 1 compares the dimensionless coagulation coefficient predicted by the present model with other models. Since the Hamaker constant for most of the aerosol systems is of the order of 10"12 eig, this value is used in the calculation of the lower bound. Particle diffusion coefficients based on Philips slip correction factor for an accommodation coefficient of unity are used for the calculation of the coagulation coefficients ft (the Fuchs interpolation formula) and fts (the Sitarski... 

Knudsen number was varied by varying the pressure of the medium (He) for particles of almost constant size. Their experimental data are compared with different models in Tables II and III. Comparison of the experimental data with the upper bound, the lower bound (for a Hamaker constant of 10-12 erg), and the Fuchs interpolation formula is also shown in Fig. 8. Since the Knudsen number was varied by varying the pressure of the medium for particles of constant size, the upper and the lower bounds approach the limiting values for large Knudsen numbers. The experimental data of Wagner and Kerker (12) lie between the upper bound and the Fuchs for-... 

Fig. 4. Comparison of the dimensionless coagulation coefficient y calculated through Monte Carlo simulation () with the previous models (1) for equal-sized particles of unit density, in air, at 298 K and 1 atm for a Hamaker constant of 10 12 erg. Curve 1 refers to the values of dimensionless coagulation coefficient y calculated from the equation derived previously for large particles (1). Curve 2 refers to the values of the dimensionless coagulation coefficient y calculated from the equation derived previously for sufficiently smalt particles (1) (the lower bound).

As cells start to adhere to the surface, the uniformity in its characteristics is disturbed. The altered spots on the surface, which consist of adhering cells, present new characteristics concerning the surface potential and Hamaker s constant, and consequently, the rate of cellular deposition is not uniform over the surface. The present model is based on the simplified assumption that the overall rate of deposition is the arithmetic sum of two contributions deposition on the bare surface and deposition on the altered surface. Each of them has its own time constant and depends also on the fraction of the area which is already covered, X. Therefore, by extending equation (19), the overall rate of deposition is given by ... 

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