Diffusion fitting parameters

These results have been fit to experimental data obtained for the reaction between a diisocyanate and a trifunctional polyester polyol, catalyzed by dibutyltindilaurate, in our laboratory RIM machine (Figure 2). No phase separation occurs during this reaction. Reaction order, n, activation energy, Ea, and the preexponential factor. A, were taken as adjustable parameters to fit adiabatic temperature rise data. Typical comparison between the experimental and numerical results are shown in Figure 7. The fit is quite satisfactory and gives reasonable values for the fit parameters. Figure 8 shows how fractional conversion of diisocyanate is predicted to vary as a function of time at the centerline and at the mold wall (remember that molecular diffusion has been assumed to be negligible). 

Table 5.15. Refinement of the fit parameters used to parameterize the Ce diffusion profile of...

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

Using the value obtained for cpfh and the known value of a, and considering the diffusion length Lp as a fitting parameter, one can obtain good agreement of the theory with experiment for the entire polarization curve (Fig. 9)... 

The tracer diffusion data shown in Fig. 1.55 correlate well with this equation. The thick curves in this figure, (a) and (b), are calculated ones using fitting parameters listed in Table 1.5. The difference between eqn (1.183) and eqn (1.197) is in the number of exponential terms. The deviation from eqn (1.183) indicates that the correct form of the diffusion equation must contain more than one exponential term. 

The fitted parameters are the amplitude at zero frequency (I/2Q), the diffusion time constant through the film 02/DE, and the capacitive time constant (7 E+ ti)A.F (Table 6-2). 

In contrast to the critical temperature Tc, the spinodal temperature Tsp is well below the binodal temperature for off-critical mixtures and can hardly be reached due to prior phase separation. The diffusion coefficients in the upper left part of Fig. 8 have been fitted by (23) with a fixed activation temperature determined from Dj. The binodal points in Fig. 8 mark the boundary of the homogeneous phase at the binodal. The spinodal temperatures Tsp are obtained as a fit parameter for every concentration and together define the (pseudo)spinodal line plotted in the phase diagram in Fig. 7. The Soret coefficient is obtained from (11) and (23) as... 

In Figure 2.10 we show a selection of results, in which experimental and calculated spectra are compared at 292 and 155K. The results are quite satisfactory, especially when considering that no fitted parameters, but only calculated quantities (via QM and hydrodynamic models) have been employed. The overall satisfactory agreement of the spectral line shapes, particularly at low temperatures, is a convincing proof that the simplified dynamic modelling implemented in the SLE through the purely rotational stochastic diffusive operator f, and the hydrodynamic calculation of the rotational diffusion tensor, is sufficient to describe the main slow relaxation processes. 

The parameter t is given in Eq. A-9 in the Appendix, as a function of the correlation time, t associated with internal motion. One of the input parameters is the angle j3, formed between the relaxation vector (C—H bond) and the internal axis of rotation (or jump axis), namely the C-5—C-6 bond. The others are correlation times t0 and r, of the HWH model, obtained from the fit of the data for the backbone carbons. The fitting parameters for the two-state jump model are lifetimes ta and tb, and for the restricted-diffusion model, the correlation time t- for internal rotation. The allowed range of motion (or the jump range) is defined by 2x for both models (Eqs. A-4 and A-9). 

Depending on the distribution chosen, as few as three fitting parameters may be required to define a distribution of diffusion rates. In some cases, a single distribution was used to describe both fast and slow rates of sorption and desorption, and in other cases fast and slow mass transfer were captured with separate distributions of diffusion rates. For example, Werth et al. [42] used the pore diffusion model with nonlinear sorption to predict fast desorption, and a gamma distribution of diffusion rate constants to describe slow desorption. 

If the estimated fitting parameters are compared to the predicted values of percolation theory, one finds that all three exponents are much larger than expected. The value of the conductivity exponent ji=7A is in line with the data obtained in Sect. 3.3.2, confirming the non-universal percolation behavior of the conductivity of carbon black filled rubber composites. However, the values of the critical exponents q=m= 10.1 also seem to be influenced by the same mechanism, i.e., the superimposed kinetic aggregation process considered above (Eq. 16). This is not surprising, since both characteristic time scales of the system depend on the diffusion of the charge carriers characterized by the conductivity. 

A simple fit of the data with the product of an exponential association and an exponential decay to estimate the escape depth, overestimates the escape depth by folding the positron implantation profile and diffusion into the fitting parameters [30], A more appropriate numerical fitting method based on the diffusion equation was used to take both the implantation profile and diffusion into account [31]. When it is applied to the 3-to-2 photon ratio data suitable absorbing boundary conditions need to be included. The results for the escape depth are shown in Figure 7.8 [30]. In addition to the diffusionlike motion of positronium in connected pores, positrons and positronium diffuse to the pores. 

The work of Larson et al. (62) represented the first detailed study to show agreement between AFM-derived diffuse layer potentials and ((-potentials obtained from traditional electrokinetic techniques. The AFM experimental data was satisfactorily fitted to the theory of McCormack et al. (46). The fitting parameters used, silica and alumina zeta-potentials, were independently determined for the same surfaces used in the AFM study using electrophoretic and streaming-potential measurements, respectively. This same system was later used by another research group (63). Hartley and coworkers 63 also compared dissimilar surface interactions with electrokinetic measurements, namely between a silica probe interacting with a polylysine coated mica flat (see Section III.B.). It is also possible to conduct measurements between a colloid probe and a metal or semiconductor surface whose electrochemical properties are controlled by the experimenter 164-66). In Ref. 64 Raiteri et al. studied the interactions between... 

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