Relaxation —continued calculation

Fig. 5.8. Top Transverse relaxation curve calculated for a given cross-link density (diamonds) and multiparameter fits The Gauss-Lorentz fit (broken line) agrees well at short times. The biexponential fit (continuous line) agrees well at long times. Bottom T2 relaxation curves of the CH group for nine differently cross-linked samples of unfilled SBR samples. The effective number N,. of Kuhn segments per cross-link chain varies between 9.43 and 15.56 corresponding to vulcameter moments between 16.5 and 1 dNm. By renormalization of the time axis the master curve (top) has been obtained.

One would prefer to be able to calculate aU of them by molecular dynamics simulations, exclusively. This is unfortunately not possible at present. In fact, some indices p, v of Eq. (6) refer to electronically excited molecules, which decay through population relaxation on the pico- and nanosecond time scales. The other indices p, v denote molecules that remain in their electronic ground state, and hydrodynamic time scales beyond microseconds intervene. The presence of these long times precludes the exclusive use of molecular dynamics, and a recourse to hydrodynamics of continuous media is inevitable. This concession has a high price. Macroscopic hydrodynamics assume a local thermodynamic equilibrium, which does not exist at times prior to 100 ps. These times are thus excluded from these studies. 

In an early work by Mertz and Pettitt, an open system was devised, in which an extended variable, representing the extent of protonation, was used to couple the system to a chemical potential reservoir [67], This method was demonstrated in the simulation of the acid-base reaction of acetic acid with water [67], Recently, PHMD methods based on continuous protonation states have been developed, in which a set of continuous titration coordinates, A, bound between 0 and 1, is propagated simultaneously with the conformational degrees of freedom in explicit or continuum solvent MD simulations. In the acidostat method developed by Borjesson and Hiinenberger for explicit solvent simulations [13], A. is relaxed towards the equilibrium value via a first-order coupling scheme in analogy to Berendsen s thermostat [10]. However, the theoretical basis for the equilibrium condition used in the derivation seems unclear [3], A test using the pKa calculation for several small amines did not yield HH titration behavior [13],... 

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

Using (5.14) and the determined value of aabs, we can estimate 8 if ab is known, and vice versa. Two examples of thermal bistability data, fit to a calculated tuning curve based on (5.12), are shown below. Figure 5.7 is for the bare sphere, and Fig. 5.8 is for the PDDA-coated sphere. In the figures, the laser scans slowly across a TM-polarized WGM dip (taking several thermal relaxation times to scan Av), first down in frequency, then reversing at the vertical dashed line, and scanning back up in frequency across the same mode. The continuous smooth lines are the theoretical fits. 

To continue the investigation, carbon detected proton T relaxation data were also collected and were used to calculate proton T relaxation times. Similarly, 19F T measurements were also made. The calculated relaxation values are shown above each peak of interest in Fig. 10.25. A substantial difference is evident in the proton T relaxation times across the API peaks in both carbon spectra. Due to spin diffusion, the protons can exchange their signals with each other even when separated by as much as tens of nanometers. Since a potential API-excipient interaction would act on the molecular scale, spin diffusion occurs between the API and excipient molecules, and the protons therefore show a single, uniform relaxation time regardless of whether they are on the API or the excipients. On the other hand, in the case of a physical mixture, the molecules of API and excipients are well separated spatially, and so no bulk spin diffusion can occur. Two unique proton relaxation rates are then expected, one for the API and another for the excipients. This is evident in the carbon spectrum of the physical mixture shown on the bottom of Fig. 10.25. Comparing this reference to the relaxation data for the formulation, it is readily apparent that the formulation exhibits essentially one proton T1 relaxation time across the carbon spectrum. This therefore demonstrates that there is indeed an interaction between the drug substance and the excipients in the formulation. 

Equation (15.23) displays the feature of locality that the blending functions should possess in order to be computationally advantageous that is, during the process of matrix inversion, one wishes the calculation to proceed quickly. As mentioned earlier, the use of linear approximation functions results in at most five terms on the left side of the equation analogous to (15.23), yielding a much crader approximation, but one more easily calculated. The current choice of Bezier functions, on the other hand, is rapidly convergent for methods such as relaxation, possesses excellent continuity properties (the solution is guaranteed to look and behave reasonably), and does not require substantial computation. 

Simulations of the experimental signal were performed using Equation 1 without adjustable parameters. The spectrum of the pulse and the absorption spectrum of HPTS were measured experimentally. An examination of the molecular structure of HPTS shows that it has no center of symmetry. Since parity restrictions may be relaxed in this case, the similarity between one-photon and two-photon absorption spectra is expected. The spectral phase

phase mask was the same used for the simulations. Both experimental and theoretical data were normalized such that the signal intensity is unity and the background observed is zero. The experimental data (dots) generally agree with the calculated response (continuous line) of the dyes in all pH environments (see Fig. 2). 

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