Fourier diffusion

In this manuscript, we briefly describe the BTE and MD approaches and show how the two methods can be coupled. We validate both the BTE and MD approaches separately by comparing with experimental thermal conductivity data for bulk and thin-film silicon. Finally, we look at the problem of self-heating in silicon-on-insulator transistors. Differences in the predictions from the different BTE-based modeling approaches and Fourier diffusion are pointed out. The impact of boundary conditions on the thermal predictions of silicon-on-insulator transistors is also discussed. 

In Fig. 9, we show the temperature contours in the domain represented by Fig. 8, obtained by using the full phonon dispersion BTE model discussed in section 2.3. The maximum temperature occurs in the hotspot. The silicon layer is isothermal in the y-direction as a consequence of the ballistic phonon transport in the silicon thin film layer. Qualitatively, the results look similar when Fourier diffusion or other BTE models are applied in the silicon layer. However, quantitatively there are significant differences in the hotspot temperature obtained from the different models. Table 1 shows the maximum temperature in the hotspot, obtained by applying different BTE models in the silicon layer. There is a large difference between the results from Fourier diffusion and the BTE-based models [46]. Fourier diffusion underpredicts the temperature rise in the hotspot since it cannot capture the non-equilibrium effects at these small scales. This is the reason why subcontinuum modeling approaches are essential. 

It can therefore be concluded that irrespective of the precise boundary conditions, it is important to consider sub-continuum modeling approaches in the silicon layer. Notice that applying Fourier diffusion in the silicon channel layer leads to a substantial underprediction of the maximum hotspot temperature. 

Clearly, mechanical, diffusive, thermal and chemical processes all act together. This section attempts to clarify some of the processes. Darcian, Fickian and Fourier diffusion processes take place simultaneously, and interact intimately. They are affected by geochemically- and mechanically-induced changes in shale properties (S0nsteb0 Horsrud 1996, Fam and Dusseault 1998, and many others). 

DRIFTS Diffuse reflectance infrared Fourier-transform Same as IR Same as IR... 

Willey R R 1976 Fourier transform infrared spectrophotometer for transmittance and diffuse reflectance measurements Appl. Spectrosc. 30 593-601... 

Lennard C J, Mazzella W D and Margot P A 1993 Some applioations of diffuse refleotanoe infrared Fourier transform speotrosoopy DRiFTS in forensio soienoe Analysis 21 M34-7... 

Kazayawoko M, Balatineoz J J and Woodhams R T 1997 Diffuse refleotanoe Fourier transform infrared speotra of wood fibers treated with maleated polypropylenes J. Appl. Polymer Sci. 66 1163-73... 

Zeine C and Grobe J 1997 Diffuse refleotanoe infrared Fourier transform DRIFT speotrosoopy in the preservation of historioal monuments studies on salt migration Mikrochim. Acta 125 279-82... 

Diffuse reflectance infrared Fourier transform spectroscopy... 

Diffuse reflection iavolves reflecting the iafrared beam off of a soHd sample, as ia specular reflectioa, but it is the aoaspecular portioa of the reflected radiatioa that is coUected. Whea an ftir spectrometer is used, diffuse reflection is caUed DRIFTS (diffuse reflectance iafrared Fourier-transform... 

Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by D g and is defined by Tick s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-181). It is analogous to the thermal diffusivity in Fourier s law and to the kinematic viscosity in Newton s law. These analogies are flawed because both heat and momentum are conveniently defined with respec t to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires information about bulk motion of the medium in which diffusion occurs. For hquids, it is common to refer to the hmit of infinite dilution of A in B using the symbol, D°g. 

In order to find the effect of broadening of the surface on the structure parameters H and K, we first study the ordered phases with the diffusive interfaces. The ordered phases can be described by the periodic surfaces (0(r)) = 0 and we can compare and with H and K. The numerators in the definitions (76) and (77) in the Fourier representation assume the forms [68]... 

This equation is the first term of an infinite series which appears in the rigorous solution of the quasi-diffusion. This equation describes the regular process of quasi-diffusion. For the low values of the Fourier number (irregular quasi-diffusion) it is necessary to use Eq. (5.1) or Boyd-Barrer approximation [105, 106] for the first term in Eq. (5.1)... 

The resolution of infra-red densitometry (IR-D) is on the other hand more in the region of some micrometers even with the use of IR-microscopes. The interface is also viewed from the side (Fig. 4d) and the density profile is obtained mostly between deuterated and protonated polymers. The strength of specific IR-bands is monitored during a scan across the interface to yield a concentration profile of species. While in the initial experiments on polyethylene diffusion the resolution was of the order of 60 pm [69] it has been improved e.g. in polystyrene diffusion experiments [70] to 10 pm by the application of a Fourier transform-IR-microscope. This technique is nicely suited to measure profiles on a micrometer scale as well as interdiffusion coefficients of polymers but it is far from reaching molecular resolution. 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

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