Hamake s equation

From knowledge of the hydrodynamic force required for particle removal one could calculate the force of adhesion. The force of adhesion could be compared with the attractive force calculated from Hamaker s equation. The results showed that the force of adhesion was about two orders of magnitude lower than the theoretical value calculated from the van der Waals attraction. 

Among dispersed magnetic particles, four different interparticle interactions exist— these are the van der Waals forces, magnetic attraction, steric repulsive forces, and electric repulsive forces. Van der Waals forces or London dispersion forces originate fi-om die interaction between orbital electrons or induced vibrating dipoles. For the equivalent two spherical particles, Hamaker s equation holds [95]. This force is strong only within short distances. 

The rate of deposition of material can be determined theoretically from Hamaker s equation ... 

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. 

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... 

Predicted values of the rate for all four speculations in Table I are much more sensitive to ionic strength than observed values. At high ionic strengths, the energy barrier disappears and the rate becomes equal to the maximum value (6110 particles/cm2 sec) predicted from Levich s equation (Eq, 3J). One possible explanation for the discrepancy in ionic strength dependence is that the Hamaker constant has a different value in each of the five electrolyte solutions tested. The Hamaker constant can be.affected by adsorbed layers of surfactant (18). Since the concentration of surfactant used in solutions of different ionic strengths varied between 1 X 10 4 and 4X 10-i 37/liter, the Hamaker constant could be affected differently. However, to obtain agreement between predicted and observed rates under speculation 1, the Hamaker constant would have to vary from 0.98 X10 13 erg... 

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. 

For many practical examples in the experimental literature the Hamaker constant is assumed to be a constant and not a function of separation, h, in accordance with Hamaker s original equation [1] ... 

When the Hamaker constant is positive, it corresponds to attraction between molecules, and when it is negative, it corresponds to repulsion. By definition, 3 = 1 and n3 = 1 for a vacuum. As we know from McLachlan s equation (Equation (92)), the presence of a solvent medium (3) rather than a free space considerably reduces the magnitude of van der Waals interactions. However, the interaction between identical molecules in a solvent is always attractive due to the square factor in Equation (567). On the other hand, the interaction between two dissimilar molecules can be attractive or repulsive depending on dielectric constant and refractive index values. Repulsive van der Waals interactions occur when n is intermediate between nx and n2 in Equation (566). If two bodies interact across a vacuum (or practically in a gas such as air at low pressure), the van der Waals forces are also attractive. When repulsive forces are present within a liquid film on a surface, the thickness of the film increases, thus favoring its spread on the solid. However, if the attractive forces are present within this film, the thickness decreases and favors contraction as a liquid drop on the solid (see Chapter 9). 

In equation (6.5a) and (6.5b), ty is the peld stress, n is ttie power law index, K is the consistency index, C is a positive constant representing ttie total number of nearest neighbours of each sphere, Ah is Hamaker s constant, the maximum volume fraction, Cq tihe dielectric constant of the matrix, ttie thickness of the electrostatic interaction layer, lf the surfece potential of the particulates, D tite particle diameter and y the shear rate. Basically, equation (6.4) along with (6.5b) is identical to the empirical Herschel-Bulkley [91] model given by equation (2.45), other theories which relate yield stress to volume fraction and particle size, and these are available in Rajaiah [93]. 

From equation (6.5a), it can be seen that fi e yield stress increases when volume fraction or surface potential or Hamaker s constant increases and when particle size decreases. The usefulness of equation (6.5a) is limited by the fact that most filler particles are neither monodisperse particles nor are values of Hamaker s constant, the... 

Lifshitz (1955) approached the problem of the van der Waals interaction by examining the macroscopic properties of materials, as opposed to the Hamaker treatment of summing individual atomic interactions. The derivation is based on Maxwell s equations modified to allow rapid temporal fluctuations (Rytov, 1959). This gives an approximate expression for the free energy of interaction between two different (1 and 2) semi-infinite surfaces separated by a third material (3) of thickness ... 

Equation (74) with m = 6 and n = oo has been tested for a range of a s by Croucher (1981) and leads to very good estimates for the Hamaker constant for a number of liquids. Moreover, the temperature dependence of the Hamaker constant is also accounted for explicitly. Example 10.4 illustrates the use of this procedure. 

The last equation shows that the energy in PBS-dFFF is a function of the size and of the surface potential of the particle, of the Hamaker constant, and of the ionic strength of the carrier solution, as the reciprocal double-layer thickness is immediately related to the ionic strength of the suspending medium. Thus, selectivity in PBSdFFF results from differences in particle size or chemical composition of the particles and of the suspending medium, where the latter will affect the surface potential and the Hamaker constant of the particle, as well as the medium s ionic strength. 

The two system-specific parameters in the LJ equation encompass a and s. If their values, the number density of species within the interacting bodies, and the form/shape of the bodies are known, the mesoscopic/macroscopic interaction forces between two bodies can be calculated. The usual treatment of calculating net forces between objects includes a pairwise summation of the interaction forces between the species. Here, we neglect multibody interactions, which can also be considered at the expense of mathematical simplicity. Additivity of forces is assumed during summation of the pairwise interactions, and retardation effects are neglected. The corresponding so-called Hamaker summation method is well described in standard texts and references [5,6]. Below we summarize a few results relevant for AFM. 

The quantity Ap in equation (4.85) comprises two partial pressures. One of which resulted from the difference in the radii of the curves and was described by 2a/r (where r is the bubble radius). The pressure in the film (and in the bubble) is Po -F 2a/r. where pg is the pressure in the liquid bulk. The other partial pressure arose from the van der Waal s forces, but was, however, negligible for film thicknesses <5 < 1000 A. The sum was given by Ap = 2alr- - A/ G-n5 ), where A is the Hamaker constant, The pressure effects could be ignored due to the electric double layer. 

Twenty years after Marrucci s work, Prince and Blanch [446] showed that equation (4.86) predicted values which were a factor of 4 too low, because in the range of the film thicknesses occurring here (<)" > 100 A) the Hamaker relationship Fh = A/(67i<5 ) does not apply and that the van der Waal s relationship for delayed attraction forces Fh = B/d applied instead (B is the delay coefficient). With this expression and taking into consideration the inertia of the liquid film at the beginning of the thinning process in the force equilibrium equation, a new relationship is obtained ... 

It is essential to know the relation of e(iv) with v, in order to calculate the Hamaker constant from the sum over many frequencies. The static dielectric constants, , e2 and e3 are the values of this function at zero frequency. The integral in Equation (564) has a lowest value of vl = 2nkTlh = 3.9 x 1013Hz (s-1) at 25°C. This corresponds to a wavelength of 760 nm. If we assume that the major contribution to the Hamaker constant comes from the frequencies in the visible light or UV region, the relation of e(iv) with v can be given as... 

Healy (Chapter 7) and Dumont also prefer the first approach. Healy sets down a model based on the control of coagulation by surface steric barriers of polysilicate plus bound cations. Healy s electrosteric barrier model is designed to stimulate new experimental initiatives in the study of silica sol particles and their surface structure. Dumont believes that many particular aspects of the stability of silica hydrosols could be explained not only by the low value of the Hamaker constant but also by the relative importance of the static term of the Hamaker equation. 

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