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22 Results This sign determines in turn if the force on an adatom on the surface is attractive to- wards the tip or repulsive. Thus there exist a threshold field E0 for mound formation that we can calculate from our model. 2.3 Temperature dependence of triboelectricity The existence of cyclic triboelectric series, as we saw in the introduction chapter, indicates more than one charging mechanism. For metals it is clear that electrons are transferred, but for non-metals the charge carriers could be electrons or ions. McCarty and Whitesides [4] have developed a model where the charge of the ions on a dielectric plate is neutralized by bound counter ions close to the surface of this material. The ions thus ”belong” to this plate making it neutral. When another plate is brought in contact, some of the ions are transferred to the other plate. The ion transfer then creates an electric field between the plates which counteract more ion transfer. In this way the number of ions on the two surfaces can be calculated and hence the net charge on the plates. We have developed a model [Paper III] that does not assume that the ions belong to one plate: we are assuming that they come from both plates. They are then dis- tributed between the plates depending on the difference in work function for ions for the two plates creating a charge difference between the electrodes. Because this model then has two constant energy levels that do not change as the charging takes place we call it a two-level Schottky model in analogy with the Schottky model for heat capacity that gives rise to the Schottky anomaly. A two-level Schottky system [37] is a well known model with only two energy levels, 1 and 2, separated by the energy E. The ratio of the number of particles on the two levels is given by the Boltzmann factor N2 = e−E/kBT. (2.26) N1 If the number of particles is fixed to N = N1 + N2 the number of particles on the lower energy level is given by N1 = N . (2.27) 1+e−E/kBT We see that at low temperatures T all the particles are on the lowest level N1 = N, N2 = 0 and at high temperatures they are equally distributed N1 = N2 = N/2. From the partition function Z = 1 + exp(−E/kBT) for the two-level system many thermodynamical functions of state like internal energy, heat capacity and entropy can be calculated [38]. When two materials with a difference E in binding energy for ions are brought into contact, a two-level system will develop. We assume that the materials have N ions in common that can be distributed between them. The net charge of the capacitor is then Q = q(N1 − N2), see Fig. 2.5. We obtain 2 Q = qN 1+e−E/kBT −1 . (2.28)PDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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