Cluster radius

The interaction among the clusters via the common diffusion held leads in general to a coarsening of the clusters with time t. One denotes this by Ostwald ripening [58,96] (see Sec. HID). According to the Lifshitz-Slyozov theory [58] on this process, the typical cluster radius R increases as... 

The nucleation stage is followed by two different kinetic regimes of cluster growth (i) a diffusional one (occurring at the earlier stage of growth) which is characterized by a time dependence of cluster radius scaling as where... 

From Figure 4(f), we observe that below 700-800 °C the cluster radius increases very slowly with the annealing... 

Figure 4. Cross-sectional bright-field TEM views of Au-implanted silica samples at 3 x lO Au /cm, 190 keV, aimealed for 1 h at (a) 400 °C in air, (b) 700 °C in air, (c) 900 °C in air, and (d) 900 °C in Ar, respectively (e) the histograms of the size distribution of the samples annealed 1 h in air at different temperatures (f) Arrhenius plot of the squared average cluster radius after 1 h annealing in air (filled circles) or argon (empty triangles). Solid lines are linear fit to the experimental data.

Fig. 1. Schematic of cluster growth during crosslinking. At p = 0, only the monomer is present. With increasing crosslinking index, p, the connectivity increases and the molecular clusters (radius R) grow in size. In the solid state, the network spans the entire sample, 2R > H...

Figure 4-1 Extra Gibbs free energy of clusters as a function of cluster radius. The critical cluster size is when the extra free energy reached the maximum. A5m c = 56 J/K/mol, Vc = 46 cm /mol, a = 0.3 J/m, Te = melting temperature = 1600 K, and system temperature = 1500 K. AGc mX ASm ( (r—Tg) = —5600 J/mol. The radius of the critical cluster is r = 2aVg/(AGm c) = (2) (0.3) (46 x 10 )/5600m = 4.93 nm. The Gibbs free energy of the critical cluster relative to the melt is AG = (16/3)7tCT /(AGm c/l g)2 = 3.05 x lO i J.

Note that the critical radius is when the free energy of the cluster is at maximum, not when AGr = 0. The latter occurs when the cluster radius is 1.5r. ... 

There is nothing in the work equations to indicate the existence of a critical size for a cluster. However, if resort is made to thermodynamics, the free energy of the entire system may be found and this value will yield helpful information. The system is imagined to be subjected to a small change of cluster radius and the laws of equilibrium are applied to the system during this imposed change. 

The correlation length in the thermosetting polymer is defined by the z-average cluster radius, which scales with conversion x as... 

Fig. 3.10 SQUID fuzzy pharmacophore model for COX-2 inhibitors. Using a larger cluster radius results in more general models. From left to right 1 A, 1.5 A, 2.5 A, 3.5 A.

Fig. 12.2. 2-D calculations of axial radial profiles for the parameters of a two-phase Ar jet at the cross-section positioned 1.5-mm downstream from the nozzle outlet. The solid, dotted, and broken lines represent argon gas pressures of 60, 40, and 20 bar, respectively a the mean cluster concentration, nciust b the gas-phase concentration nat c the mean cluster radius (r) d the mean distance between clusters (d)...

Light-scattering experiments on flocculating suspensions of silica colloids provided the data in the following table. (R is the average cluster radius.) Estimate the fractal dimension of the clusters formed and indicate whether the flocculation process is transport or reaction controlled. Hint Apply Eq. 6.1 and the concepts in Section 6.1.)... 

Figure 13.8 Comparison of the enrichment obtained with the combined pharmacophore model (PPP model) resulting from a 1,5-A PPP cluster radius with the results of retrospective screening using the COX-2 inhibitors Rofecoxib, M5, and SC-558 as seed structures (courtesy S. Renner).

Figure 1. Ionization potential (IP(N)) and electron affinity (EA(N)), both in eV, for clusters of Na atoms Nan plotted vs. N (roughly proportional to the reciprocal of the cluster radius). Also shown are bulk work function and limiting metal sphere values which the IP(N) and EA(N) curves must approach as N —> oo.

The present measurements agree quite closely with the earlier measurements of Buffat and Borel up to a cluster radius of 1 nm at which point the melting temperature becomes constant at 0.4 of the melting temperature of bulk... 

The ionization potentials (IPs) of ammonia clusters containing alkali metal atoms, such as Li [10], Na [8] and Cs [9], have been reported by Hertel s and Fuke s groups. These clusters have been prepared by pickup sources coupled with a heated oven (Na and Cs) or a laser-vaporization source (Li). The IP(n) values decrease almost linearly with (n-f 1) , which is approximately proportional to the inverse of the cluster radius. Although the IPs of free atoms are different (5.392, 5.139 and 3.894 eV for Li, Na and Cs, respectively), those of the clusters (n > 5) are almost the same irrespective to the metal atoms. The intercept at (n + 1) 0... 

For the idealized cubic arrays found as the crystal structures of simple metals, the lattice vector is determined solely as the square root of the sums of squares of the allowed m, n and p coefficient assuming that a = b = c = 1. So the problem to find the distribution of the Oh orbits as a function of cluster radius is reduced to the 3-squares problem in mathematics . 

Figure 13 Photoemission spectra of free Cu clusters for n = 1-36 (top) and onset of the 4s and 3d bands as a function of the reciprocal cluster radius (down). (From K.J. Taylor et al. [153].)...

Figure 1. Gibbs free energy of formation as a function of cluster radius for typical vapor at deflned temperature and supersatured pressure.

S T) is the reaction probability that is assumed to be the same as on a Pd extended surface, Nq the density of surface atoms, R the cluster radius, P x, y) is given by (3.13) and Xg by (3.11). When the capture zones overlap (at low temperature) the TON becomes maximum and has a simple expression ... 

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