Decay length

FIG. 11 Force profiles between poly(glutamic acid), 2C18PLGA(44), brushes in water (a) at pH = 3.0 (HNO3), (b) at pH 10 (KOH) 1/k represents the decay length of the double-layer force. The brush layers were deposited at tt = 40 mN/m from the water subphase at pH = 3.0 and 10, respectively. 

TABLE 1 Interaction Parameter w, Decay Length X (in Lattice Constants), and Bulk Composition /j for a Lattice Gas. Other System Parameters as in Fig. 2... 

The interactions between bare mica surfaces in 10 and 10 M KNO solutions were determined at pH = 3.5. In both cases an exponential type relation F(D) = 0-lcD was indicated, with decay lengths 1/k = 1.4 nm and 8 nm for the two salt concentrations, respectively, but with an effective surface potential tp = 40 mV, considerably lower than its value at the higher pH used in the PEO experiments (figure 6a, curve (a)). The lower value of p is probably the result of a lower net degre of ionization of the mica surface in the presence of the large H1" concentration (the low pH was used to ensure full ionization and polyelectrolyte). 

Another open question is the relationship between the H-induced radiative recombination centers and the H-induced platelets. Controlled layer removal of the plasma-processed silicon surface reveals that the density of luminescence centers decays nearly exponentially with a decay length that is comparable to the depth over which the platelets form (Northrop and Oehrlein, 1986 Jeng et al., 1988 Johnson et al., 1987a). However, the defect luminescence has also been obtained from reactive-ion etched specimens in which platelets were undetectable (Wu et al., 1988). Finally, substantial changes in the luminescence spectra occur at anneal temperatures as low as 250°C (Singh et al., 1989), while higher temperatures... 

Consider two different metals in contact and assume that both are well described by the Thomas-Fermi model (see Problem 3.3) with a decay length of Ltf 0.5 A. (a) Calculate the dipole potential drop at the contact if both metals carry equal and opposite charges of 0.1 C m 2. (b) If the work functions of the two metals differ by 0.5 eV, how large is the surface-charge density on each meted ... 

Table 12.1 Bulk composition and decay length at the interface for various values of a.

From Eq. (16.17) show that in the presence of a surface charge density a the decay length a is determined from ... 

Conventional evanescent sensing works exceedingly well for relatively small biomolecules such as proteins and DNA molecules whose size is much smaller than the decay length. However, it becomes less sensitive when detecting biospecies, such as cells, with dimensions over 1 pm. In Chap. 15, deep-probe waveguide sensors are developed to overcome this limitation, which have a decay length comparable to the size of the biospecies of interest. 

One of the main assumptions of the Donnan partition model is that two well-defined phases (polymer and solution) exist and the electrostatic potential presents a sharp transition between them. This approximation is fulfilled when the typical decay length of the electrostatic potential (Debye length) is much shorter than the film thickness. The other limiting situation is that where all the redox sites are located in a plane and thus the Debye length is larger than the film thickness. This situation can be described by the surface potential model ... 

The authors deduce the decay length (X) from the slope and the first interaction length or onset of repulsion (2Lq) defined as the distance at which the magnitude of force is 2 10 A. One of the main findings of this study is that the force profiles are strongly dependent upon the sequence of adsorption of polymer and surfactant and three cases are envisaged. 

Case I (see Fig. 2.17) corresponds to the situation such that the emulsion is initially stabilized with SDS at 8 10 mold (CMC). The repulsive force as a function of distance between the ferrofluid droplets, stabilized with SDS alone is referred as 0% PVA. Then, PVA-Vac is introduced at different concentrations varying from 0.002 to 0.5 wt%. After each addition, the emulsion is incubated for 48 h to reach equilibrium. It can be seen that the force profiles remain almost the same as in the case of 0% PVA. As the surfactant concentration is equal to CMC, the expected decay length is 3.4 nm. The experimental value of the decay length obtained from the force profile, 2.9 nm (solid line), is in good agreement with the predicted value. Thus, if the emulsion is preadsorbed with surfactant molecules, the introduction of polymer does not influence the force profile significantly. 

Case II corresponds to an emulsion preadsorbed with surfactant molecules at very low concentration (stabilized with SDS at a concentration of 0.27 10 mold or CMC/30). Polymer and surfactant were premixed separately and later added to the emulsion. In all cases, the polymer concentration was fixed at 0.6 wt%. Premixed polymer/surfactant mixture was incubated sufficiently (>2 h) before adding to the emulsion. The force profiles are again repulsive (Fig. 2.17) and exponentially decaying with a characteristic decay length comparable to the Debye length, corresponding to the equivalent amount of surfactant concentration present in the premixed system. 

Rgure 2.18. Decay length and first interaction length (2Lo) values deduced from the force curves in Fig. 2.17 (cases II and III), as a function of surfactant concentrations. Solid lines are a visual guide. (Adapted from [75].)... 

On the basis of the above experimental results, the expected conformations of polymer-surfactant complexes at the oil-water interface are depicted in Fig. 2.19. In case I, the added polymer associates with excess surfactants present in the bulk solution, but the complexes prefer to remain in the bulk phase. Alternately, the polymer-surfactant complexes are unable to displace the adsorbed surfactant molecules from the liquid-liquid interface. Irrespective of the amount of polymer-surfactant concentration in the bulk, the experimental decay length values remain comparable to the Debye lengths, corresponding to the concentration of ion species in the bulk solution (Eq. (2.11)). This means that the force profile is... 

The Bloch functions near the K points have a long decay length and contribute to the second term of Eq. (5.10). Following Eq. (5.12), in general, a surface Bloch function at that point has the form ... 

In addition to the term with n = 0, the term with n=-l have the same decay length, and thus have the same magnitude. Also, the Bloch function that generates the symmetric charge density must also be s metric. The lowest-order symmetric Fourier sum of the Bloch function near K is ... 

This result shows us that the repulsive double-layer pressure (for the case of low potentials) decays exponentially with a decay length eqnal to the Debye length and has a magnitnde which depends strongly on the surface potential. 

Figure 7.12 compares the theoretical predictions with the experimental values across the 4d series, assuming one valence s electron per atom and taking x = 12 corresponding to close-packed lattices. The experimental values of the bandwidth are taken from the first principles LDA calculations in Table 7.1. The ratio b2 a is obtained by fitting a bandwidth of 10 eV for Mo with Nd = 5, so that from eqn (7.42) b2/a = eV. The skewed parabolic behaviour of the observed equilibrium nearest-neighbour distance is found to be fitted by values of the inverse decay length that vary linearly across the series as... 

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