Constant Field Approximation

If the spatial dependence of the electric potential across the pore is linear with distance, the electric field E x) = -d lr/dxissL constant. With a constant electric field, the PNP equations can be solved analytically (Weiss 1996, Jackson 2006). The closed-form solutions of the PNP equations with constant electric field are known as the GHK equations. For a constant electric field, given by Vm/L, across the pore, the electric potential ir x) is 

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A significant modification of the model is the addition of an additional term to include the convective contribution to the total iontophoretic transport. This is achieved by adding a linear term (v xQ [58], where v is the average velocity of the solvent and C is the concentration of the drug. Because the skin has a net negative charge, this term is positive for cations and negative for anions. From the constant field approximation, the EF is predicted to be... 

The Nernst-Planck equation [Eq. (88)] is suitable to describe the direct anion passage quantitatively in the constant field approximation. The mobile carrier mechanism is treated in the same way. The relevant diagram is shown... 

Figure 8.5 Concentration profile given by PNP for a concentration gradient of 0.4 M in monovalent salt concentration and a membrane potential of 100 mV. The constant field approximation gives the solid curves for anions (filled circles) and cations (open circles). The numerical PNP results are open triangles (cations) and filled triangles (anions). (From Corry, B. etal., Biophys.J., 78, 2364, 2000. With permission.)...

Taking the constant field approximation given by Equation 8.49, Equation 8.62... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

Hence, a series of measurements with several Tcp values will provide a data set with variable decays due to both diffusion and relaxation. Numerical inversion can be applied to such data set to obtain the diffusion-relaxation correlation spectrum [44— 46]. However, this type of experiment is different from the 2D experiments, such as T,-T2. For example, the diffusion and relaxation effects are mixed and not separated as in the PFG-CPMG experiment Eq. (2.7.6). Furthermore, as the diffusion decay of CPMG is not a single exponential in a constant field gradient [41, 42], the above kernel is only an approximation. It is possible that the diffusion resolution may be compromised. 

As previously discussed, we expect the scaling to hold if the polydisper-sity, P, remains constant with respect to time. For the well-mixed system the polydispersity reaches about 2 when the average cluster size is approximately 10 particles, and statistically fluctuates about 2 until the mean field approximation and the scaling break down, when the number of clusters remaining in the system is about 100 or so. The polydispersity of the size distribution in the poorly mixed system never reaches a steady value. The ratio which is constant if the scaling holds and mass is conserved,... 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

The optical properties of organic conductors may be described by the simplest model, which assumes noninteracting electrons (one-electron model). In this approximation the infrared (IR) properties may be derived in the self-consistent field approximation. Assuming a frequency-independent relaxation rate, y, and a background dielectric constant arising from virtual high-frequency transitions, e0, the result takes the Drude form [12] ... 

As shown in Fig. 28.2, the potential gradient in the membrane core is slightly concave (see curve 3, in particular). However, deviation from linearity is seen to be small in Fig. 28.2. This means that the usual assumption of constant field within the membrane is not a bad approximation. If we employ this approximation, Eq. (28.9) can easily be integrated in the following two limiting cases. When > Hki, l/jCn, in which case the potential far inside the membrane surface layer is in practice the Donnan potential, the constant field assumption yields [3]... 

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