Thickness of the Channel

Because molecules located in the same layer are much closer to each other than those situated in different layers, charge transport is expected to be much more efficient in the direction along the layers than across them. This has largely been confirmed by X-ray diffraction measurements on sexithiophene-based devices [27, 28], which indicated that the highest performance is attained when molecules are standing upright on the insulator. Similar behavior was found for pentacene [29, 30]. This is illustrated in Fig. 1.7, in which the molecules are depicted as short rods. 

Let us now turn to the thickness of the conducting channel. The concept of thickness is not that obvious, because the actual distribution of charge-carriers decreases continuously from the insulator-semiconductor interface to the semiconductor bulk, so one can more sensibly speak of an effective thickness. The distribution can be estimated by resolving Poisson s equation (Eq. 3)  

Let d be the thickness of a monolayer and n the total number of layers (that is, the thickness of the film divided by d). The layers are numbered starting from the insulator-semiconductor interface. To estimate the density n (per unit area) of charge-carriers in the ith layer we apply Gauss s law to a cylinder of unit cross section limited by the boundaries between the ith layer and each of its neighboring layers. For a long channel device, the electric field F is perpendicular to the film, and we have  

Because the density of charge is constant in each layer, the electric field varies linearly with distance between two boundaries, which implies, in turn, that the variation of the potential is quadratic within the same limits. Furthermore, we assume that charge transfer between adjacent layers is sufficiently efficient that the distribution of charge-carriers in the whole film is at thermodynamic equilibrium. Assuming Boltzmann s statistics holds, which is true as long as the charge density remains much lower than the density of molecules, this yields  

After some manipulations, the following series of equations is obtained  

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Once the pressure is known, we can compute the velocity profile across the thickness of the channel using... 

Figure 8.38 Velocity distributions across the thickness of the channel for various Brinckman numbers [9].

When the size of the separated species is commensurable with the thickness of the channel, the limit retention ratio in this mode of steric FFF is [6]... 

Reduced thermal mass came from changes to the catalyst support. The catalyst materials are deployed as a thin coating on a ceramic honeycomb substrate. The surface area for the deployment of the catalytic coating, and the mass of the substrate both contribute to the overall performance of the catalytic converter. The surface area was increased by reducing the thickness of the channel walls, thereby increasing the flow channel density and reducing the thermal mass of the substrate. 

Co is the concentration of reactant in the bulk eluent (mole/cm ), D is the diffusion coefficient (cmVsec), L is the length of the electrode (cm), b is thickness of the channel (cm), Vy is the volume flow rate (cm /sec), is the wiilth of the I hunnel (t lu), and 11 1 (he wiilth t-i (he vIeciuHh (Fig.. S). Fioiu tins cxpres.siun, it is evident that the diffusion-limited current should be proportional to the concentration of the analyte and the width of the electrode, to the one-third power of the velocity through the cell, and to the negative one-third power of the channel thickness. 

Flow FFF is perhaps most promising in the area of water-soluble polymers. These polymers, which as a class are very difficult or impossible to separate by thermal FFF, can be fractionated according to diffusion coefficient or Stokes radius (which translate to molecular mass) in a flow FFF system using a water-compatible membrane such as cellulose acetate. Such a fractionation is shown in Figure 8.15, illustrating the programmed field separation of three sulphonated polystyrene components in a 510-//m-thick channel. The fact that the time of separation is somewhat longer than desired can be related to the excessive thickness of the channel, ten times thicker than the thinnest thermal FFF channel utilized. Recently we have been able to work successfully with a... 

In Brownian elution mode, the retention ratio R is indirectly dependent on both the applied force F and the thickness of the channel w, and independent on the flow rate. It can be expressed in an approximate form ... 

To optimize the fraction of the slipping area, we should now exploit results for effective slip lengths obtained above. Figure 2.15b shows the computed value of Qy/Qx vs. H/L for several 4>2- The calculations are made using the value of 9 defined by Eq. 2.23. The data suggests that the effect of 02 on a transverse flow depends on the thickness of the channel. For a thick gap the increase in the gas fraction, transverse flow. This result has a simple explanation. For an infinite channel bl /H 1 (see Fig. (2.14b)), which gives... 

Defining the current under the condition wherein the depletion region at the drain end of the channel penetrates through the entire thickness of the channel as the saturation current or the current at the drain pinch-off condition Ip) ... 

The thickness of the channel is adjusted by putting calibrated shims 4 between insert 3 and the base 2. For measuring the pressure gradient, two narrow openings 6 with connections to join a pressure gauge are provided in the channel (the connections are not shown in Figure 3.20). 

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