Disk model

The flux, amount per unit time per unit surface area, is a valuable quantity to be able to estimate a priori. The rotating disk model enables this calculation... 

Figure 4.3. The rod (or disk) model for torsion and flexure of DNA. The DNA is modeled as a string of rods (or disks) connected by Hookean twisting and bending springs which oppose, respectively, torsional and flexural deformations. The instantaneous z and x axes of a subunit rod around which the mean squared angular displacements , j = x, z, take place are indicated. The filament is assumed to exhibit mean local cylindrical symmetry in the sense that for any pair of transverse x- and y-axes. Twisting = mean squared angular displacement about body-fixed x -axis = (/)y(t)2) (assumed).

Anion exclusion volumes calculated on the disk model [34] for a particle with a large radius (ro > 250 nm) are similar to those obtained from the infinite-plane model because, for a large particle, the anion exclusion volume contributed from the region outside the edge surface is relatively smaller than that from above the basal plane. Very large values of the anion exclusion volume are found on the disk model for small particles (r0 = lOnm) in dilute electrolyte solutions because anion depletion outside the edge surface becomes very important for small particles [34]. 

The dependence of anion exclusion volume on particle thickness in the infinite-plane model arises solely from the geometric decrease in exclusion-specific surface area with an increase in particle thickness via quasicrystal formation [Eq. (13)]. Thus, on the infinite-plane model, the anion exclusion volume simply has an inverse relation to particle thickness. Reduction of the anion exclusion volume from an increase in particle thickness is, however, more complicated and more significant on the disk model (Fig. 6). An increase in particle thickness (or the number of unit layers... 

FIG. 6 Comparison between MGC theory (open symbols) and the disk model (filled symbols) for the dependence of Vcx on the electrical double layer thickness [Eq. (14)]. The number of unit layers in a quasicrystal is a fixed parameter for each set of curves [34],... 

Mid- and far-infrared, submillimeter, and radio-wavelength observations allow probing the presence and abundance of simple molecules in these zones and provide constraints and boundary conditions for coupled disk evolution and chemical network models. The observed abundances and predictions from the disk models can be directly compared to constraints derived from the early Solar System. 

Figure 3.2 Radial runs of various disk variables according to the simple steady-state toy disk model described in the main text for three different global accretion rates. The central star is Sun-like. We have assumed a gray opacity of k = 1 in regions where Tm < 1500K. In regions where Tm > 1500 K we switched to k = 0.01 to mimic the effect of dust evaporation. Since dust evaporation reduces the mid-plane temperature there will be a region where dust is only partly evaporated to keep Tm at 1500K. Dust evaporation acts as a thermostat here.

In short, there is still much to learn about the specific processes that drive disk evolution. Currently, the two best candidates, the MRI and gravitational torques, likely produce values of a that vary with time and location in a protoplanetary disk. In order to account properly for these variations more detailed models than those discussed in this chapter are required, and due to their complexity and computational rigor, the amount of model time that could be investigated is limited to 103-104 yr. While adopting a constant or effective value of a overlooks the details of these variations, it greatly reduces the numerical complexity of disk models, allowing the evolution of disks for times > 106 yr to be calculated. This provides a way for the timescales that are needed to study meteoritic materials or dust around other stars to be modeled. 

The innermost radius in a protoplanetary disk where temperatures are low enough for a particular chemical species to exist as a solid has traditionally been termed the condensation front of that species. The most commonly used example of such a front is the snowline, which represents the point inside of which water exists as a vapor and outside of which it exists as ice. This term largely was used in static disk models where dynamical processes were ignored. 

Analytic calculations and simple disk models reinforce the notion that the apparently higher crystallinity around cooler objects is not due to the observations of a different ratio of surface area between the cool outer and hot inner disks, but they reflect real differences in the dust properties (Apai et al. 2005 Kessler-Silacci etal. 2007). 

Evolution of the Solar Nebula. 04.4.2.1 Viscous accretion disk models... 

Figure 4 Midplane temperature as a function of heliocentric radius for a solar nebula with varying mass (inside 10 AU) undergoing mass accretion at a rate of a solar mass in —0.1-1 Myr, compared to various cosmochemical constraints, and the results of a viscous accretion disk model (dashed line) with a mass of 0.24 solar masses (source Boss, 1998).

Figure 9. Surface photometry and color profiles for M 81. (a) Surface photometry in the IRAC bands along with bulge plus disk model fit to the 3.6 pm data, (b) Color profile in [3.6] — [4.5] which shows +0.4 mag redder colors in the nucleus, indicative of an AGN. (c) and (d) Color profile in [3.6] — [5.8] and [3.6] — [8.0], which are both redder than starlight alone due to PAH emission at the long wavelengths, (e) Color profiles in [5.8] — [8.0] for both the combined starlight and warm dust and the warm dust alone. The color of the warm dust is roughly consistent with PAH color [5.8] — [8.0] = 2.06 mag predicted by Li Draine (2001).

In the past four decades, we have witnessed the significant development of various methods to describe microporous solids because of their important contribution to improving of adsorption capacity and separation. Various models of different complexity have been developed [5]. Some models have been simple with simple geometry, such as slit or cylinder, while some are more structured such as the disk model of Segarra and Glandt [6]. Recently, there has been great interest in using the reverse Monte Carlo (MC) simulation to reconstruct the carbon structure, which produces the desired properties, such as the surfece area and pore volume [7, 8]. Much effort has been spent on studies of characterization of porous media [9-15]. In this chapter we will briefly review the classical approaches that still bear some impact on pore characterization, and concentrate on the advanced tools of density functional theory (DFT) and MC, which currently have wide applications in many systems. 

The third popular approach was based on a cut-off-disk model. According to it the electrostatic repulsion between the ions leads to the formation of an area around each ion practically free from other adsorbed ions. The size of this area was identified with the average distance between the ions, so that it was dependent on the 2D density, that is, on the ion adsorption, F (as in the previous Sects. 2.1.11.2-2.1.11.4, this symbol is used for the amount of specifically adsorbed ions, unlike in Sect. 2.1.11.1). The results on... 

Spatially, the science datacube is a disk model with a diameter of approximately 400 AU and with a gap of radius = 10 AU situated 130-140 parsecs from our Solar System, where the nearest regions of ongoing star formation are. At this distance, 1 AU subtends an angle of 7 milliarcseconds. 

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