Diffusion response functions

The use of Equation (15.40) is limited to closed systems like that illustrated in Figure 15.10(a). Measurement problems arise whenever /), > 0 or Dgut > 0. See Figure 15.10(b) and suppose that an impulse is injected into the system at z = 0. If Din > 0, some of the tracer may enter the reactor, then diffuse backward up the inlet stream, and ultimately reenter. If Dgut > 0, some material leaving the reactor will diffuse back into the reactor to exit a second time. These molecules will be counted more than once by the tracer detection probes. The measured response function is not f t) but another function, g i), which has a larger mean ... 

The kinetics data of the geminate ion recombination in irradiated liquid hydrocarbons obtained by the subpicosecond pulse radiolysis was analyzed by Monte Carlo simulation based on the diffusion in an electric field [77,81,82], The simulation data were convoluted by the response function and fitted to the experimental data. By transforming the time-dependent behavior of cation radicals to the distribution function of cation radical-electron distance, the time-dependent distribution was obtained. Subsequently, the relationship between the space resolution and the space distribution of ionic species was discussed. The space distribution of reactive intermediates produced by radiation is very important for advanced science and technology using ionizing radiation such as nanolithography and nanotechnology [77,82]. 

The response function, which is normally a peak and may be distorted to some extent by the electronics, clearly is the difference between the formation and removal functions at that time. Atoms leave the atom cell partly by diffusion and according to the velocity of the purge gas. The rate of formation of atoms is more difficult to identify. 

Simple diffusion is a linear process. Therefore, the original step function may be broken into a series of spatially separated impulses (Fig. 2.5c). Each impulse can be treated independently according to the results described above (Fig. 2.5d). The total concentration profile will be described by the summation of response functions to the original excitation pulses. Thus, making use of... 

When used in the time-invariant mode (i.e., in equilibrium), it is a first-order chemical sensor that can yield qualitative and quantitative information based on the LSER paradigm about composition of the vapor mixtures (Fig. 10.13). By acquiring the data in the transient regime, it becomes a second-order sensor and in addition to the composition, information about diffusion coefficients in different polymers is obtained. This is then the added value. It is possible only because the model describing the capacitance change included diffusion. In spite of the complexity of the response function, a good discrimination and quantification has been obtained. 

In this contribution, the experimental concept and a phenomenological description of signal generation in TDFRS will first be developed. Then, some experiments on simple liquids will be discussed. After the extension of the model to polydisperse solutes, TDFRS will be applied to polymer analysis, where the quantities of interest are diffusion coefficients, molar mass distributions and molar mass averages. In the last chapter of this article, it will be shown how pseudostochastic noise-like excitation patterns can be employed in TDFRS for the direct measurement of the linear response function and for the selective excitation of certain frequency ranges of interest by means of tailored pseudostochastic binary sequences. 

Fig. 4.23 also indicates a slight decrease of the signal plateau which, at a first glance, was unexpected. In the following, a reactive dispersion model given in ref. [37] is applied to deduce rate constants for different reaction temperatures. A trapezoidal response function will be used. The temperature-dependent diffusion coefficient was calculated according to a prescription by Hirschfelder (e.g., [80], p. 68 or [79], p. 104] derived from the Chapman-Enskog theory. For the dimensionless formulation, the equation is divided by M/A (with M the injected mass and A the cross-section area). This analytical function is compared in Fig. 4.24 with the experimental values for three different temperatures. The qualitative behavior of the measured pulses is well met especially the observed decrease of the plateau is reproduced. The overall fit is less accurate than for the non-reactive case but is sufficient to now evaluate the rate constant. 

Here, the response functions of the diffusion equation for a number of discrete input signals were calculated based upon the solution for a Dirac impulse input signal. 

Here, the response functions of the diffusion equation for a number of discrete input signals were calculated based on the solution for a Dirac impulse input signal. A set of transformation relations for the injected mass M and the axial coordinate was used to obtain the solutions for a reacting gas directly from the solutions of a non-reacting gas ... 

An alternative perspective on third-order responses of N coupled vibrators, which will be particularly helpful to describe spectral diffusion processes in such coupled systems (see Section IV. D), can be developed by assuming that Ri and R2 are the same and writing the total response function as ... 

Relation Between the Displacement Response Function and the Time-Dependent Diffusion Coefficient... 

Exactly like its classical analog Eq. (94), Eq. (125) allows one to express the displacement response function in terms of the time-dependent diffusion coefficient. However, contrary to the classical case in which Xxx( 0 is directly proportional to D(t — t ), in the quantum formulation Xxv( 0 is a convolution product, for the value t — t of the argument, of the functions D(t ) and logcoth(7i fi /2 h). Inverting the convolution equation (125) yields an expression for D(t) in terms of the dissipative part of the displacement response function ... 

Let us consider again the particular case of a particle diffusing in a stationary medium, in order to see how the generalized Langevin equation (22) can be deduced from the more general equation (169). When the medium is stationary, the response function x (M0 reduces to a function of t — t (% (t, t ) = X (f — t j). Introducing then the causal function y(f) as defined by... 

Figure 2 shows the spectral response functions (5,(r), Eq. 1) derived firom the spectra of Fig. 1. In order to adequately display data for these varied solvents, whose dynamics occur on very different time scales, we employ a logarithmic time axis. Such a representation is also useful because a number of solvents, especially the alcohols, show highly dispersive response functions. For example, one observes in methanol significant relaxation taking place over 3-4 decades in time. (Mdtiexponential fits to the methanol data yield roughly equal contributions from components with time constants of 0.2, 2, and 12 ps). Even in sinqrle, non-associated solvents such as acetonitrile, one seldom observes 5,(r) functions that decay exponentially in time. Most often, biexponential fits are required to describe the observed relaxation. This biexponential behavior does not reflect any clear separation between fast inertial dynamics and slower diffusive dynamics in most solvents. Rather, the observed spectral shift usually appears to sirrply be a continuous non-exponential process. That is not to say that ultrafast inertial relaxation does not occur in many solvents, just that there is no clear time scale separation observed. Of the 18 polar solvents observed to date, a number of them do show prominent fast components that are probably inertial in origin. For example, in the solvents water [16], formamide, acetoniuile, acetone, dimethylformamide, dimethylsulfoxide, and nitromethane [8], we find that more than half of the solvation response involves components with time constants of 00 fs.

Experimentally it has been shown that the threshold pressure at which combustion instability can be induced artificially in composite proplnts by pulsing is a function of the burning rate of the proplnt (in a motor size of 5-inch diameter and 40-inch length) (Ref 45). This relationship is shown in Figs 17 and 18 for both aluminized and non-aluminized composite proplnts. It was also found that potassium perchlorate, lithium perchlorate and AN proplnts were resistant to this induced instability. Since AP composites were the only proplnts, other than double-base, which were driven unstable, the rate controlling reactions and response function are those related to AP decompn and perhaps the diffusion flame between oxidizer and binder... 

More recently an alternative explanation of the complex excimer behaviour observed in polymer systems has been proposed whereby the close proximity of some fluorophores leads to a time dependent rate of quenching analogous to that predicted by the Einstein-Smoluchowski diffusion theory for low molecular weight systems. This predicts a fluorescence response function of the form... 

The substrate generation/tip collection (SG/TC) mode with an ampero-metric tip was historically the first SECM-type measurement performed (32). The aim of such experiments was to probe the diffusion layer generated by the large substrate electrode with a much smaller amperometric sensor. A simple approximate theory (32a,b) using the well-known c(z, t) function for a potentiostatic transient at a planar electrode (33) was developed to predict the evolution of the concentration profile following the substrate potential perturbation. A more complicated theory was based on the concept of the impulse response function (32c). While these theories have been successful in calculating concentration profiles, the prediction of the time-de-pendent tip current response is not straightforward because it is a complex function of the concentration distribution. Moreover, these theories do not account for distortions caused by interference of the tip and substrate diffusion layers and feedback effects. 

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