Propagation constant, variation

In the presence of 0.1 to 0.4M Tetralin hydroperoxide the measured termination constants for toluene, ethylbenzene, and cumene are in good agreement with the value 3.8 X 10° Mole"1 sec."1 obtained with pure Tetralin. Since the variations in the measured values are within the limits of experimental error, the cross-propagation constants, kp, were obtained using this termination constant and the measured values of 1 /(2It )1/2. 

Figure 3. Styrene emulsion polymerization—variation of the propagation constant with temperature during adiabatic polymerization of 395-A latex particles (kp in...

In practice, we can use the impedance tube to find the complex coefficient of reflection and then vary the two propagation constants in the theory to produce the same complex reflection coefficient. These variations are not easy to perform as the equation is transcendental however, there are computer programs available to do this (7). 

Thus, the rate of transfer with a PO monomer with a much higher activation energy varies more with the temperature than the PO propagation reaction. As an immediate consequence, by the decreasing the polymerisation temperature from 110-120 °C to 80 °C [69], polyether polyols with much lower unsaturation are obtained. In order to get convenient reaction rates, the catalyst concentration was increased. Table 4.3 shows the variation of propagation constant Kp of PO anionic polymerisation as a function of temperature. 

Table 4.3 Variation of propagation constant in PO anionic polymerisation, as a function of temperature [97] ...

Polymerization of styrene initiated in cumyl-methyl ether by cumyl sodium is hardly affected by variation of concentration of living polymers or by the addition of sodium tetraphenyl boride269. This is not surprising since the dielectric constant of that solvent is also very low, 3.7 at 20 °C, although higher than of dioxane (2.2). The bimolecular propagation constant at -20°C is 1 M 1s 1, a value expected for a propagation of contact ion-pairs. 

In Chap. 1 by J. Homola of this volume [1] surface plasmons were introduced as modes of dielectric/metal planar waveguides and their properties were estabhshed. It was demonstrated that the propagation constant of a surface plasmon is sensitive to variations in the refractive index at the surface of a metal film supporting the surface plasmon. In this chapter, it is shown how this phenomenon can be used to create a sensing device. The concept of optical sensors based on surface plasmons, commonly referred as to surface plasmon resonance (SPR) sensors, is described and the main approaches to SPR sensing are presented. In addition, the concept of affinity biosensors is introduced and the main performance characteristics of SPR biosensors are defined. 

The fundamental and HEi , modes of a fiber of circular cross-section are formed from the scalar wave equation solution with no azimuthal variation. Hence in Eq. (13-8) depends only on the radial position r. There is no perferred axis of symmetry in the circular cross-section. Thus, in this exceptional case, the transverse electric field can be directed so that it is everywhere parallel to one of an arbitrary pair of orthogonal directions. If we denote this pair of directions by x- and y-axes, as in Fig. 12-3, then there are two fundamental or HEi , modes, one with its transverse electric field parallel to the x-direction, and the other parallel to the y-direction. The symmetry also requires that the scalar propagation constants of each pair of modes are equal. 

The fundamental modes of the infinite parabolic profile fiber have a Gaussian spatial variation it is the exact solution of the scalar wave equation. Thus, the essence of the Gaussian approximation is the approximation of the fundamental-mode fields of an arbitrary profile fiber by the fundamentalmode fields of some parabolic profile fiber, the particular profile being determined from the stationary expression for the propagation constant in Table 15-1. Clearly this approach can be generalized to apply to higher-order modes, by fitting the appropriate solution for the infinite parabolic profile [9]. 

Table 15-4 Low-order modes of the Gaussian-profile fiber. Approximations for the spot size rQ = pR and propagation constant /8, based on the variational solution of Eq. (15-18). 

When the perturbed and unperturbed fibers are weakly guiding, the variation in both profiles n and n is small. The modal fields can then be constructed from solutions of the scalar wave equation, as described in Chapter 13. If T and jS denote the unknown solution and propagation constant for the perturbed fiber. 

As a local mode propagates, its phase increases across each section by the product of j8y(zj and the section length 5z. Consequently, the phase at an arbitrary position along the nonuniform fiber is a sum of such products. However, the slow variation of the fiber means that the propagation constant /Sy (Zj) varies only slightly from one section to the next Hence we can accurately... 

The complete spatial variation of each radiation mode is given by Eq. (25-1), and the range of values of the propagation constant satisfies 0 p < kn. For a particular value of j8, the modal fields can be regarded as a Fourier superposition of the fields of a family of plane waves, all inclined at angle 6 to the waveguide axis. In the uniform cladding, 6 is related to j8 by... 

Efficiency of Intermediate Formation. The variation of the efficiency of a primary intermediate with conversion of the feed hydrocarbon can be calculated (22). Ratios of the propagation rate constants ( 2 / i) reactor type (batch or plug-flow vs back-mixed) are important parameters. 

The propagation rate constant did not depend on the monomer concentration which corresponds to the first-order propagation step. The activation energy of the propagation calculated according to the variation of Kp with temperature was found to be 6.5 0.5 kcal/mole. 

More recent work has shown that the observed variation in propagation rate constants with composition is not sufficient to define the polymerization rates.5" 161,1152 There remains some dependence of the termination rate constant on the composition of the propagating chain. Thus, the chemical control (Section 7.4.1) and the various diffusion control models (Section 7.4.2) have seen new life and have been adapted by substituting the terminal model propagation rate constants (ApXv) with implicit penultimate model propagation rate constants (kpKY -Section 7.3.1.2.2). 

Very similar variations in average copolymer composition with conversion have recently been observed in the styrene methyl methacrylate system by both Johnson et al ( and by Dionisio and O Driscoll (. The reason for the variation may be due to a viscosity effect on propagation rate constants QO). 

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