Repulsion, charged particles

Often the van der Waals attraction is balanced by electric double-layer repulsion. An important example occurs in the flocculation of aqueous colloids. A suspension of charged particles experiences both the double-layer repulsion and dispersion attraction, and the balance between these determines the ease and hence the rate with which particles aggregate. Verwey and Overbeek [44, 45] considered the case of two colloidal spheres and calculated the net potential energy versus distance curves of the type illustrated in Fig. VI-5 for the case of 0 = 25.6 mV (i.e., 0 = k.T/e at 25°C). At low ionic strength, as measured by K (see Section V-2), the double-layer repulsion is overwhelming except at very small separations, but as k is increased, a net attraction at all distances... 

Here we consider the total interaction between two charged particles in suspension, surrounded by tlieir counterions and added electrolyte. This is tire celebrated DLVO tlieory, derived independently by Derjaguin and Landau and by Verwey and Overbeek [44]. By combining tlie van der Waals interaction (equation (02.6.4)) witli tlie repulsion due to the electric double layers (equation (C2.6.lOI), we obtain... 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

A hexapole assembly of rods (poles) is built similarly to the quadrupole, but now there are three sets of opposed rods evenly spaced around a central axis. The hexapole cannot act as a mass filter by applying a DC field and is used only in its all-RF mode. It is therefore a wide band-pass filter and is used to collimate an ion beam. (Like-charged particles repel each other, and an electrically charged beam will tend to spread apart because of mutual repulsion of ions unless steps are taken to reduce the effect.)... 

Because the electrically charged droplets retain their charge but get smaller, their electric field increases. At some point, mutual repulsion between like charges causes charged particles (ions) to leave the surface of the droplet (ion evaporation). These ions can be detected by the mass spectrometer. 

Response to Electric and Acoustic Fields. If the stabilization of a suspension is primarily due to electrostatic repulsion, measurement of the zeta potential, can detect whether there is adequate electrostatic repulsion to overcome polarizabiUty attraction. A common guideline is that the dispersion should be stable if > 30 mV. In electrophoresis the appHed electric field is held constant and particle velocity is monitored using a microscope and video camera. In the electrosonic ampHtude technique the electric field is pulsed, and the sudden motion of the charged particles relative to their counterion atmospheres generates an acoustic pulse which can be related to the charge on the particles and the concentration of ions in solution (18). 

Electrostatic force A field in which stationary electrically charged particles are subjected to a force of attraction or repulsion, as the result of another stationary electric charge. 

The layers of opposite charges formed on the surface of individual electrically-charged particles in water, causing mutual repulsion. 

FIGURE 17.17 When a positively charged particle approaches a nucleus, it is repelled strongly. However, a very fast particle can reach the nucleus before the repulsion turns it aside, and a nuclear reaction may result. 

New elements and isotopes of known elements are made by nucleosynthesis the repulsive electrical forces of like-charged particles are overcome when very fast particles collide. 

Neutrons readily induce nuclear reactions, but they always produce nuclides on the high neutron-proton side of the belt of stability. Protons must be added to the nucleus to produce an unstable nuclide with a low neutron-proton ratio. Because protons have positive charges, this means that the bombarding particle must have a positive charge. Nuclear reactions with positively charged particles require projectile particles that possess enough kinetic energy to overcome the electrical repulsion between two positive particles. 

Verwey and Hamaker (10) have modified the Morse curve to take into account the approach of two charged bodies, as shown in Figure 4. Here, as one moves outward toward increasing distance of separation, an electrostatic repulsion is encountered because the charges are similar. A secondary minimum is then encountered as a result of the concentration of counter ions around each charged particle. The shallowness of the secondary minimum shows that the deflocculated system is metastable. The importance of the Verwey and Hamaker concept lies in its ability to show graphically the correlation between the secondary minimum and the metastable position. A, Figure 1. 

The individual terms in (5.2) and (5.3) represent the nuclear-nuclear repulsion, the electronic kinetic energy, the electron-nuclear attraction, and the electron-electron repulsion, respectively. Thus, the BO Hamiltonian is of treacherous simplicity it merely contains the pairwise electrostatic interactions between the charged particles together with the kinetic energy of the electrons. Yet, the BO Hamiltonian provides a highly accurate description of molecules. Unless very heavy elements are involved, the exact solutions of the BO Hamiltonian allows for the prediction of molecular phenomena with spectroscopic accuracy that is... 

However, billiard balls are a pretty bad model for electrons. First of all, as discussed above, electrons are fermions and therefore have an antisymmetric wave function. Second, they are charged particles and interact through the Coulomb repulsion they try to stay away from each other as much as possible. Both of these properties heavily influence the pair density and we will now enter an in-depth discussion of these effects. Let us begin with an exposition of the consequences of the antisymmetry of the wave function. This is most easily done if we introduce the concept of the reduced density matrix for two electrons, which we call y2. This is a simple generalization of p2(x1 x2) given above according to... 

Fig. 1 Illustration of the DLVO theory interaction of two charged particles as a function of the interparticle distance (attractive energy curve, VA, repulsive energy curve, VR and net or total potential energy curve, Vj).

The first two tenns in Eq. (2) represent the kinetic energy of the nuclei and the electrons, respectively. The remaining three terms specify the potential energy as a function of the interaction between the particles. Equation (3) expresses the potential function for the interaction of each pair of nuclei. In general, this sum is composed of terms that are given by Coulomb s law for the repulsion between particles of like charge. Similarly, Eq. (4) corresponds to the electron-electron repulsion. Finally, Eq. (5) is the potential function for the attraction between a given electron (<) and a nucleus (j). 

Figure 9.59 One of the first methods of preparing water-soluble QDs was to use thioacetic acid modification of the nanocrystal surface. This resulted in a negative charge on the surface of each dot that provides like charge repulsion of particles suspended in aqueous solution. The carboxylate group also could be used for conjugation with amine-containing molecules.

Figure 9.61 QDs containing carboxylate groups can be coupled to amine-containing proteins or other molecules using the EDC/sulfo-NHS reaction to form amide bond linkages. The intermediate sulfo-NHS ester is negatively charged and will help maintain particle stability due to like charge repulsion between particles.

Space-Charge Effect Result of mutual repulsion of particles of like charge that limits the current in a charged-particle beam or packet and causes some ion motion in addition to that caused by external fields. 

The DLVO theory, a quantitative theory of colloid fastness based on electrostatic forces, was developed simultaneously by Deryaguin and Landau [75] and Verwey and Overbeek [76], These authors view the adsorptive layer as a charge carrier, caused by adsorption of ions, which establishes the same charge on all particles. The resulting Coulombic repulsion between these equally charged particles thus stabilizes the dispersion. This theory lends itself somewhat less to non-aqueous systems. 

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