Momentum Transfer Model

To describe the nano-cBN deposition four models have been proposed the compressive stress model [190, 191], the sub-plantation model [192, 193], the selective sputter model [194], and the momentum transfer model [195], 

For the c-BN formation a stress threshold was observed in the deposited layers. The h-BN intermediate layer shows a preferred orientation, where the c-axis of the h-BN is parallel to the substrate. Both effects are explained by the compressive biaxial stress induced by the ion bombardment. The mechanism for the conversion of h-BN into c-BN is explained by rather high temperatures originated during thermal spikes (direct h-BN — c-BN transformation). The stress caused by the bombardment with high energetic ions is considered to be a reason for the growth of the c-BN crystals [190, 191]. A stress within the layer of up to 10 GPa has been observed. This biaxial stress causes a hydrostatic pressure up to the values usual in HP-HT synthesis. 

Another possibility to explain the ion assisted c-BN deposition is the subplantation model. The nucleation of c-BN crystals takes place under the surface of the substrate caused by sub-plantation of the ions and stress. The sub-plantation and high nucleation rates result in the nano-cBN coatings. 

The essential mechanism in the sputter-model is that h-BN can be removed more easily by selective sputtering than c-BN (if the BN mixtures are deposited simultaneously and the h-BN is selectively etched, the c-BN layer remains) [187, 196]. 

A correlation between the total momentum of impinging ions per deposited boron atom and the c-BN deposition has been observed. In this model, c-BN formation is correlated with the momentum-drive process, such as the formation of point defects in conjunction with the stress-induced phase transformation. 

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The difference of our model and that of Chui et al (10) and Grest et al (11) can be best seen if we calculate the second order vertex corrections. Two such diagrams are shown in Fig. 2. In the momentum transfer model the analytic contribution corresponding to these diagrams is... 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

These are the flux relations associated with the dusty gas model. As explained above, they would be expected to predict only the diffusive contributions to the flux vectors, so they should be compared with equations (2.25) obtained from simple momentum transfer arguments. Equations (3,16) are then seen to be just the obvious vector generalization of the scalar equations (2.25), so the dusty gas model provides justification for the simple procedure of adding momentum transfer rates. 

First, any analysis must be coupled with a technically correct interpretation of the equipment performance soundly rooted in the fundamentals of mass, heat, and momentum transfer rate processes and thermodynamics. Pseudotechnical explanations must not be substituted for sound fundamentals. Even when the development of a relational model is the goal of the analysis, the fundamentals must be at the forefront. 

The discrepancy may also be caused by the approximations in the calculation of the EEDF. This EEDF is obtained by solving the two-term Boltzmann equation, assuming full relaxation during one RF period. When the RF frequency becomes comparable to the energy loss frequencies of the electrons, it is not correct to use the time-independent Boltzmann equation to calculate the EEDF [253]. The saturation of the growth rate in the model is not caused by the fact that the RF frequency approaches the momentum transfer frequency Ume [254]. That would lead to less effective power dissipation by the electrons at higher RF frequencies and thus to a smaller deposition rate at high frequencies than at lower frequencies. 

In this contribution we will deal with electron-electron correlation in solids and how to learn about these by means of inelastic X-ray scattering both in the regime of small and large momentum transfer. We will compare the predictions of simple models (free electron gas, jellium model) and more sophisticated ones (calculations using the self-energy influenced spectral weight function) to experimental results. In a last step, lattice effects will be included in the theoretical treatment. 

The boundary conditions to be satisfied are that the lateral pressure difference between subchannels should be zero at the channel inlet and exit. Having passed once along the channel, this implies that iteration over the channel length may be necessary by using improved guesses of flow division between subchannels at the inlet. In practice, only one pass may be necessary, particularly for hydraulic model, in which lateral momentum transfer is neglected or only notionally included. Rowe (1969) has shown that for a single-pass solution to be stable and acceptable,... 

For different momentum transfers the dynamic structure factors are predicted to collapse to one master curve, if they are represented as a function of the Rouse variable. This property is a consequence of the fact that the Rouse model does not contain any particular length scale. In addition, it should be mentioned that Z2/ or the equivalent quantity W/4 is the only adjustable parameter when Rouse dynamics are studied by NSE. 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Fig. 6. Dynamic structure factor as observed from PI for different momentum transfers at 468 K. ( Q = 0.038 A"1 Q = 0.051 A-1 A Q = 0.064 A-1 O Q = 0.077 A"1 Q= 0.102 A-1 O Q = 0.128 A 1 Q = 0,153 A "" 11. The solid lines display fits with the Rouse model to the initial decay. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Figure 10 Intermediate incoherent scattering function for the bead-spring model at T = 0.48 for different values of momentum transfer given in the legend.

Since the Rouse model does not contain an explicit length scale, for different momentum transfers the dynamic structure factors are predicted to collapse... 

In Fig. 3.3 the corresponding NSE spectra are plotted against the scaling variable of the Rouse model. As predicted by Eq. 3.18 the results for the different momentum transfers follow a common straight line. 

Fig. 3.3 Self-correlation for a PDMS melt T=100 °C. The data at different momentum transfers are plotted vs. the scaling variable of the Rouse model. (Reprinted with permission from [42]. Copyright 1989 The American Physical Society)...

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