Hamaker’s constant

The concept of additivity is unsatisfactory when applied to closely packed atoms in a condensed body. In this case, a new approach to the energy of interaction needs to be developed or a modification of Hamaker s constant is desirable. 

Solving Eq. (3.91) for the Lifshitz-van der Waals constant or Hamaker s constant usually is not trivial. Many approximations for its calculation have been proposed for various cases [Landau and Lifshitz, 1960 Hough and White, 1980 Visser, 1989 Israelachvili, 1992]. For materials 1 and 2 interacting through material 3, Hamaker s constant becomes... 

An Arrhenius type equation is obtained for the apparent reaction rate constant. Equations for the apparent activation energy and for the frequency factor are established as functions of Hamaker s Constant, ionic strength, surface potentials and particle radius. 

Fki. 4.—Ratio of the apparent activation energy to Hamaker s constant against AV and r (a) low... 

In this paper it is shown that the rate of deposition of Brownian particles on the collector can be calculated by solving the convective diffusion equation subject to a virtual first order chemical reaction as a boundary condition at the surface. The boundary condition concentrates the surface-particle interaction forces. When the interaction potential between the particle and the collector experiences a sufficiently high maximum (see f ig. 2) the apparent rate constant of the boundary condition has the Arrhenius form. Equations for the apparent activation energy and the apparent frequency factor are established for this case as functions of Hamaker s constant, dielectric constant, ionic strength, surface potentials and particle radius. The rate... 

Lifshitz more recent continuum analysis of the van der Waals interaction (Israelaehvili and Tabor, 1973) leads to approximately the same distance dependence hut to a more accurate physical interpretation of Hamaker s constant A. However, this distinction becomes unimportant to the present analysis if A is treated as a measurable property. 

Figures 1 to 3 illustrate die quantitative effect of various properties on the interaction profile (it). Four observations are noteworthy. First, diere exists a narrow range of Hamaker s constant and surface potentials for which a maximum = (hmax) ] and two minima...

The capture efficiency or Sherwood number was shown to be a function of three dimensionless groups—the Peclet number, the aspect ratio (collector radius divided by par-dele radius), and the ratio of Hamaker s constant (indicating the intensity of London forces) to the thermal energy of the particles. Calculated values for the rate of deposition, expressed as Ihe Sherwood number, are plotted in Figure 6 as a function of the three dimensionless groups. 

For large Peelet numbers and small Hamaker s constants, appreciable concentration variation occurs only very near to the collector. Then Stokes s expressions for the fluid velocity may be expanded in a Taylor series about the collector surface and higher order terms together with curvature effects may be neglected, yielding... 

Reported values of A/kT nearly always are in the range 10 4 < A/kT < 10s. Thus Figure 5 snows that, even in the best of circumstances (R = 1), this variation in Hamaker s constant by a factor of 104, results in a change in the Sherwood number by only a factor of 7. Particle transport rates in stagnant fluids are not highly sensitive to small changes in Hamaker s constant. 

In the present paper, the rate of deposition is calculated as a function of particle radius, ionic strength, surface potential, Hamaker s constant, and surface chemistry. By the last term, we mean the effect of the dissociation constant and the number density of acidic and basic groups on the surface, when the surface potential at large separations is held fixed. For the sake of simplicity, all calculations are done for deposition onto a rotating disk. 

Figures 1 through 4 illustrate the effect of Hamaker s constant, disk rotation speed, ionic strength, surface potentials, and particle radius on the rate of deposition of particles onto a rotating disk as computed from Eqs. [lj and [23. For this set of figures, the interaction energy profiles (%) were evaluated from Eqs. 41 5], and 9].

Hamaker s constant (ergs) particle concentration far from collector (partides/cm3) local diffusion coefficient (cmysec) diffusion coefficient far from collector (cms/sec)... 

Figure 3 summarizes the rates of deposition calculated for hydrosols depositing onto a rotating disk. The four curves correspond to the four pairs of surfaces whose interactions are characterized in Fig. 1. Surface characteristics of particle and collector are interchangeable in the calculation of the rates. Values of other parameters include a = 0.1 p,m, cx = I08 cm-3, to = 6 rev/sec, and v = 0.01 cm2/sec. Rates are presented as a function of Hamaker s constant, which characterizes the van der Waals attraction, because this parameter is most difficult to experimentally determine, and because the rate is sensitive to its value.

The curves of Fig, 3 display two distinct regions. For small values of Hamaker s constant, van der Waals attraction is weak and a large energy barrier E develops due to the electrostatic repulsion. This makes K small and when small enough, Eq. [12] reduces to Ny = Kcx, which corresponds to reaction -controlled deposition. Conversely, when A is large,... 

Fig. 1. Effect of surface potential fa) and Hamaker s constant (b) upon the total interaction energy profile for a sphere of radius 6-7 /an approaching a flat plate in a solution having s = 74-3 and tc 1 = 8A.

Figure 1 suggests that as Hamaker s constant becomes very small compared to kT, vanishes. For this condition equation (16) is not valid. Instead, if the contribution to double-layer forces is much smaller than kT for x > 3, then only the gravitational contribution needs to be considered ... 

Calculations of deposition time constants from equation (18) were repeated, assuming lower values for the surface potential. Values of the Hamaker s constant necessary to obtain agreement with the experimental results of Weiss Harios, together with the corresponding values of other parameters are summarized by Table 1. Thus, to obtain agreement using a small Hamaker constant, the surface potential must be considerably lower. 

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 ... 

Theoretically, the values of rt and x2 should be computed using equation (16), keeping in mind that t = 1 IP, with the appropriate values of the surface potentials and Hamaker s constants for each of the two regions. Practically, however, because of the difficulties in the direct measurement of Hamaker s constant and in determining the surface potential of the cells adhering to the surface, it is preferable to evaluate xl and r2 empirically by comparison with experimental data for cellular deposition. This will be discussed in more detail along with the presentation of the results. 

The values of the Hamaker s constant A are of a reasonable order of magnitude (Table 3.7). In Fig. 3.38 are shown plots of Eq. (3.86) calculated with the value for water (3.7-10 13 erg [246]) and for polymers (8-10" 3 erg [246]). In view of the experimental scatter the general agreement is acceptable. Note that such behaviour has not been detected in SFA experiments [242]. Electrostatic repulsion cannot be responsible for film stabilisation and by inference steric forces are operative [127], i.e. the conclusion reached within the single layer treatment remains unaltered. Though seemingly firm, this qualitative hypothesis evades quantitative treatment. 

Obviously, this peculiarity can be explained with the decrease in the intermolecular attraction in emulsion films and the resulting increase in the barrier of the 11(h) isotherm (see Section 3.3.1). Hamaker s constant A for emulsion films is by a decimal order of magnitude lower than that of aqueous foam films and is within the range of 1.5-10-10 21 J depending on the surfactant kind and the nature of the organic phase [31]. It should be noted that in the... 

With respect to the molecular interactions the simplest asymmetric films are these from saturated hydrocarbons on a water surface. Electrostatic interaction is absent in them (or is negligible). Hence, of all possible interactions only the van der Waals molecular attraction forces (molecular component of disjoining pressure) can be considered in the explanation of the stability of these films. For films of thickness less than 15-20 nm, the retardation effect can be neglected and the disjoining pressure can be expressed with Eq. (3.76) where n = 3. When Hamaker s constants are negative the condition of stability is fulfilled within the whole range of thicknesses. 

The calculation of Hamaker s constant in [523] is performed with the integral formulae of the macroscopic theory of molecular interactions... 

It should be noted that the analogous calculations for water films on organic substrate using Eq. (3.155) give positive values of the Hamaker s constant. 

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