Direct Calculation of Interaction Force

The interaction force P can be calculated by integrating the excess osmotic pressure An and the Maxwell stress tensor T over an arbitrary closed surface 21 enclosing either one of the two interacting particles (Fig. 8.3), which is written as [8] 

FIGURE 8.2 Overlapping of the electrical double layers around two interacting particles. 

FIGURE 8.3 Calculation of the interaction force between two particles by integrating the excess osmotic pressure AH and the Maxwell stress T over an arbitrary surface 2 enclosing 

The potential energy V of the double-layer interaction is then obtained by integrating the force P with respect to the particle separation. 

We assume that the potential J/(x) in the region outside the plates obeys the one-dimensional planar Poisson-Boltzmann equation  

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DIRECT CALCULATION OF INTERACTION FORCE 191 On multiplying d J//dx on both sides of Eq. (8.11), we have... 

When two charged colloidal particles approach each other, their electrical double layers overlap so that the concentration of counterions in the region between the particles increases, resulting in electrostatic forces between them (Fig. 8.2). There are two methods for calculating the potential energy of the double-layer interaction between two charged colloidal particles [1,2] In the first method, one directly calculates the interaction force P from the excess osmotic pressure tensor All and... 

The force between particles is the sum of a pH-independent van der Waals component, which is always attractive, and a pH-dependent electrostatic component, which can be attractive or repulsive. In Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the potential is used to calculate the interaction force or energy as a function of the distance between the particles. Atomic force microscopy (AFM) makes it possible to directly measure the force between the particles as a function of the distance, and commercial instruments are available to perform such measurements. Different approaches have been proposed to utilize the results obtained by AFM to determine the pHq. The quantity obtained by AFM corresponds to the lEP rather than the PZC. AFM was used to measure the force between SiO2 (negative potential over the entire studied pH range) and Si,N4 (lEP to be determined) in [681]. The pH at which the force at a distance of 17 nm was equal to zero was identified with the lEP. The van der Waals forces are negligible at such a distance, and the force is governed by an electrostatic interaction. The experimental results were consistent with DLVO theory. 

In Fig. (7.8) are shown direct calculations of the planar spin spirals as a function of moment and of volume, the moment is seen to increase monotonically with the volume. One can see that the spin spiral qxw is stable for low volumes and moments, while the spin spiral qrx is stable at higher volumes and moments. Now, we calculate the spin spirals for fixed moments, both with the BGFM and with with the Heisenberg model with both FM and DLM interactions, just as described above for bcc Fe, fee Ni, and fee Co. We also calculate the spin spirals directly with the force theorem without the Heisenberg expansion by doing one calculation with the potentials from either the ferromagnetic or anti-ferromagnetic solution and the new spin spiral structure. The results can be seen in Fig.(7.9). 

Thus, when the vibrational frequency, v, of a bond has been measured, most commonly by observation of the infrared or Raman spectrum, the force constant can be calculated. In polyatomic molecules there are often strong interactions between the vibrations of different bonds, so that the calculation of the force constant for a particular bond cannot be made directly from any one of the observed frequencies. However, there are well-established methods for dealing with this problem. 

In a recent study Raj et al. presented the first direct study of adhesion forces, by colloidal force microscopy, between smooth PLA films representing the polymer matrix, and a microbead of cellulose that mimics the cellulose material in flax fibers [65]. Normalized adhesion force measurements demonstrated the importance of capillary forces when experiments were carried out under ambient conditions. Experiments, conducted under dry air allowed for the deduction of the contribution of pure van der Waals forces, and the results, through the calculation of the Hamaker constant, show that these forces, for the PLA/cellulose/air system, were lower than those obtained for the cellulose/cellulose/air system and hence underlined the importance of optimizing the interface among these materials. The study demonstrated the capacity of AFM to probe direct interactions in complex systems by adjusting the nature of the surface and... 

The second method [109] allows direct calculation of 8 through the correlation between interaction energy due to dispersion forces and the polarizability a of the injected components. 

From the theoretical calculations of the hydrodynamic force, we can estimate the distance dependences of the interaction coefficients in the x- and y-directions. According to ref [29], the interaction coefficients P /y and Py/y are calculated from the following approximated equations, assuming that the two particles are spherical in shape with the same diameter. 

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