Nonuniformities

Lebowitz J L and Percus J 1961 Long range correlations in a closed system with applications to nonuniform fluids Phys. Rev. 122 1675... 

Yethira] A and Woodward C E 1995 Monte Carlo density functional theory of nonuniform polymer melts J Chem. Phys. 102 5499... 

Cahn J W and Hilliard J E 1958 Free energy of a nonuniform system I. Interfaoial free energy J. Chem. Phys. 28 258... 

Jin X, Wang NHL, Tarjus G and Talbot J 1993 Irreversible adsorption on nonuniform surfaoes the random site model J. Phys. Chem. 97 4256-8... 

From the earliest days, the BET model has been subject to a number of criticisms. The model assumes all the adsorption sites on the surface to be energetically identical, but as was indicated in Section 1.5 (p. 18) homogeneous surfaces of this kind are the exception and energetically heterogeneous surfaces are the rule. Experimental evidence—e.g. in curves of the heat of adsorption as a function of the amount adsorbed (cf. Fig. 2.14)—demonstrates that the degree of heterogeneity can be very considerable. Indeed, Brunauer, Emmett and Teller adduced this nonuniformity as the reason for the failure of their equation to reproduce experimental data in the low-pressure region. 

Precision For absorbances greater than 0.1-0.2, the relative standard deviation for atomic absorption is 0.3-1% for flame atomization, and 1-5% for electrothermal atomization. The principal limitation is the variation in the concentration of free-analyte atoms resulting from a nonuniform rate of aspiration, nebulization, and atomization in flame atomizers, and the consistency with which the sample is heated during electrothermal atomization. 

Particularly in mass spectrometry, where discharges are used to enhance or produce ions from sample materials, mostly coronas, plasmas, and arcs are used. The gas pressure is normally atmospheric, and the electrodes are arranged to give nonuniform electric fields. Usually, coronas and plasmas are struck between electrodes that are not of similar shapes, complicating any description of the discharge because the resulting electric-field gradients are not uniform between the electrodes. 

Micropore Diffusion. In very small pores in which the pore diameter is not much greater than the molecular diameter the diffusing molecule never escapes from the force field of the pore wall. Under these conditions steric effects and the effects of nonuniformity in the potential field become dominant and the Knudsen mechanism no longer appHes. Diffusion occurs by an activated process involving jumps from site to site, just as in surface diffusion, and the diffusivity becomes strongly dependent on both temperature and concentration. 

Difficulties of Moving-Bed Operation. The use of a moving bed iatroduces the problem of mechanical erosion of the adsorbent. Obtaining uniform flow of both soHd and Hquid ia beds of large diameter is also difficult. The performance of this type of operation can be gready impaired by nonuniform flow of either phase. 

The nonuniformity of drop dispersions can often be important in extraction. This nonuniformity can lead to axial variation of holdup in a column even though the flow rates and other conditions are held constant. For example, there is a tendency for the smallest drops to remain in a column longer than the larger ones, and thereby to accumulate and lead to a locali2ed increase in holdup. This phenomenon has been studied in reciprocating-plate columns (74). In the process of drop breakup, extremely small secondary drops are often formed (64). These drops, which may be only a few micrometers in diameter, can become entrained in the continuous phase when leaving the contactor. Entrainment can occur weU below the flooding point. 

The desctiption of SF electtical properties needed for practical appHcations is more complex than knowledge of BDV. Corona-onset voltages (COV) must be considered, particularly for the more usual nonuniform fields. Figure 2 illustrates this for a point-to-plane electrode configuration, where extensive prebreakdown corona occurs before sparkover (36). 

The theory and appHcation of SF BDV and COV have been studied in both uniform and nonuniform electric fields (37). The ionization potentials of SFg and electron attachment coefficients are the basis for one set of correlation equations. A critical field exists at 89 kV/ (cmkPa) above which coronas can appear. Relative field uniformity is characterized in terms of electrode radii of curvature. Peak voltages up to 100 kV can be sustained. A second BDV analysis (38) also uses electrode radii of curvature in rod-plane data at 60 Hz, and can be used to correlate results up to 150 kV. With d-c voltages (39), a similarity rule can be used to treat BDV in fields up to 500 kV/cm at pressures of 101—709 kPa (1—7 atm). It relates field strength, SF pressure, and electrode radii to coaxial electrodes having 2.5-cm gaps. At elevated pressures and large electrode areas, a faH-off from this rule appears. The BDV properties ofHquid SF are described in thehterature (40—41). 

Aluminum, the most common material used for contacts, is easy to use, has low resistivity, and reduces surface Si02 to form interfacial metal-oxide bonds that promote adhesion to the substrate. However, as designs reach submicrometer dimensions, aluminum, Al, has been found to be a poor choice for metallization of contacts and via holes. Al has relatively poor step coverage, which is nonuniform layer thickness when deposited over right-angled geometric features. This leads to keyhole void formation when spaces between features are smaller than 0.7 p.m. New collimated sputtering techniques can extend the lower limit of Al use to 0.5-p.m appHcations. 

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