Bulk limits

The experimental data presented confirm that - at least after the disappearance of the smallest elements at the beginning of the sputtering - the particle sizes can indeed be adjusted by the applied ion dose, which allows us to follow the size dependent changes of the physical properties of the particles from the bulk limit. 

The increase of the cohesive energy per atom with cluster size is fast up to n = 7 and then rather slow, and we observe that the value for Pbu is still far from the bulk limit (about 20% lower). In Fig. 3 we plot the energies required... 

Fig. 3.29(b) shows the published J-V curves of an Al/OCiCio/ITO diode by Jain et al. [40], [OCiCio is poly(2-methoxy-5-(3,7-dimethyloctyloxy)-p-phenylene vinylene).] In this paper [40] Jain et al. attributed the extremely fast rise of the hole current at low voltages to Shockley like current due to the forward biased Al Schottky contact. In their later paper [56] Jain et al. showed that by including the PFE they can fit the theory of bulk limited trap controlled space charge currents with the same experimental results (see Fig. 3.29(b)). Jain et al. [40] suggested that PFE induces the high injection effect earlier, which presumably makes the Schottky barrier at the Al contact small. 

A.J. Campbell, M.S. Weaver, D.G. Lidzey, D.D.C. Bradley, Bulk limited conduction in electroluminescent polymer devices, J. Appl. Phys. 84 (1998) 6737-6746. 

Figure 5.7 exhibits the prediction of the SS-LMBW/RISM-KHM theory for the site density profiles of the liquid/Iiquid interface in the binary system with partially miscible n-hexane and methanol at ambient conditions. The bulk limiting values of the n-hexane density in the phase rich in n-hexane (z —oo) and of the methanol density... 

This quantity tends to the bulk limit for long interaction distances (r/f - 00). 

Consider now the relative variance rr., / (s ). Clearly, it does not decrease if wo make the system larger in the x- or j/-directions but will only decrease as we approach the bulk limit by letting approach infinity. However, tliis inevitably changes the physical nature of the confined fluid. 

The coefficients Ai i = 1-3) are determined by the condition that the energy density, along with the chemical potential and compressibility from Eq. (48) in the bulk limit (A oo), corresponds to these properties for the macroscopic liquid helium at zero temperature and pressure [20, 232, 248]. 

As clusters become much larger in size, the collective mode can be tracked from the quantum limit to the bulk [712], so that the evolution of the many-body resonances is known over an enormous range. In the bulk limit, it has been shown that the resonances tend towards the surface plasmon of the solid, except that, since the solid does not have spherical symmetry, the -y/3 factor of the Mie-Drude theory does not appear and... 

Fig. 12.19. Trend of the giant resonance or surface plasmon frequencies towards the bulk limit as a function of the number of atoms in the cluster (after J.H. Parks and S.A. Donald [712]).

Solids and surfaces can be modeled as large clusters, as embedded clusters, and as infinite periodic systems. Each of these strategies has been used in conjunction with semiempirical MO methods. The most straightforward semiempirical approach employs clusters of increasing size, taking advantage of the fact that semiempirical calculations can be extended easily to systems with more than 1000 atoms which facilitates convergence studies to the bulk limit. Semiempirical work in this field has been reviewed recently [45]. 

The structure of bulk Na is body-centered-cubic. Consequently, the transition to the bulk structure has not yet occurred for N 20000, in the Na clusters formed in Martin s experiments. Alonso et al. [119] have proposed that the reason why the bcc phase is not yet formed at these large sizes is that the screening cloud, r (r), around a Na ion in a finite Na cluster depends so strongly on cluster size, that "( ) has not yet converged to its bulk limit even for clusters with ten thousand atoms. Since determines the effective... 

Pig. 6. A schematic representation of the free energy of a clxister as a function of an order parameter for nonrigidity, for a case of a phase that is observable for clusters but not for bulk matter. The reason is that in the range of conditions under which this phase is locally stable and present in observable concentrations for small systems, it is never the most stable phase, and hence, in the bulk limit, is never present in observable concentrations. 

The second way we may link our knowledge of phase changes in clusters to our concepts of macroscopic phase transitions is by using information about transitions that are presumably second-order or continuous transitions in the bulk limit. Here the information is still limited, but enough is known to permit us to gain some new insights. The systems for which small system counterparts of second-order transitions have been studied are molecular clusters, particularly structural (solid-solid) transitions of octahedral molecules such as TeFe, in clusters and in bulk. 

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