Model-free method

Hayward, S., Kitao, A., Berendsen, H.J.C. Model-free methods to analyze domain motions in proteins from simulation A comparison of normal mode analysis and molecular dynamics simulation of lysozyme. Proteins 27 (1997) 425-437. 

A model-free method for the analysis of lattice distortions is readily established from Eq. (8.13). It is an extension of Stokes [27] method for deconvolution and has been devised by Warren and Averbach [28,29] (textbooks Warren [97], Sect. 13.4 Guinier [6], p. 241-249 Alexander [7], Chap. 7). For the application to common soft matter it is of moderate value only, because the required accuracy of beam profile measurement is rarely achievable. On the other hand, for application to advanced polymeric materials its applicability has been demonstrated [109], although the classical graphical method suffers from extensive approximations that reduce its value for the typical polymer with small crystal sizes and stronger distortions. 

The kinetics of the CTMAB thermal decomposition has been studied by the non-parametric kinetics (NPK) method [6-8], The kinetic analysis has been performed separately for process I and process II in the appropriate a regions. The NPK method for the analysis of non-isothermal TG data is based on the usual assumption that the reaction rate can be expressed as a product of two independent functions,/ and h(T), where f(a) accounts for the kinetic model while the temperature-dependent function, h(T), is usually the Arrhenius equation h(T) = k = A exp(-Ea / RT). The reaction rates, da/dt, measured from several experiments at different heating rates, can be expressed as a three-dimensional surface determined by the temperature and the conversion degree. This is a model-free method since it yields the temperature dependence of the reaction rate without having to make any prior assumptions about the kinetic model. 

An important group of methods relies on the inherent order of the data, typically time in kinetics or chromatography. These methods are often based on Evolving Factor Analysis and its derivatives. Another well known family of model-free methods is based on the Alternating Least-Squares algorithm that solely relies on restrictions such as positive spectra and concentrations. 

Most, but not all model-free methods are based on Factor Analysis and we start this chapter with a fairly detailed and comprehensive discussion of this topic. 

The spectrum of B is fairly well defined while the spectrum of A is not, there is a long extrapolation required until the its spectrum turns negative at 400nm. This exemplifies the limitation of model-free methods they rely on very simple constraints but in certain cases the range of feasible answers can be very wide, sometimes too wide to be useful. This will be discussed later in Chapter 5.4.3, Rotational Ambiguity. 

Model-free methods do neither supply absolute information about the concentrations or about the spectra. Essentially they only deliver the shapes for the profiles. In this and future examples, we normalise the concentration profiles in C to a maximum of 1 and adjust the species spectra of A in such a way that the product CA is correct. This is done in the function norm max, m. 

An important issue needs to be discussed next. Multiplying a column of C with any number and its corresponding row of A with the inverse of that number, does not affect the product CA and thus this factor is not determined at all. It can be freely chosen. Due to this multiplicative ambiguity, only the shapes of the concentration profiles (and component spectra) can be determined by any model-free method and only additional quantitative information allows the absolute determination of C and A. 

First we will discuss measurement methods that do not require postulation of a reaction model such as the determination of the reaction enthalpy by integration of qt01, which is the simplest of the model-free methods that still leads to a physically meaningful result. The... 

Nuclear relaxation in paramagnetic lan- 2.4.2. The model-free methods 376... 

Application of the model-free methods to axial bimetallic lanthanide complexes with sterically rigid ligands... 

This review focuses on the use and limitations of the model-free methods in axial paramagnetic supramolecular mono- and polymetallic lanthanide-containing complexes. A comprehensive survey of the systems with three- and fourfold symmetry for which these techniques have been applied is proposed together with a detailed discussion of the limitations of Bleaney s approach for the modeling of paramagnetic anisotropies and the detection of structural changes along the lanthanide series. 

By applying multiple heating rate DSC measurements and Ozawa s isoconver-sional model free method, an activation energy of 34.2 kcal mol1 and pre-exponential factor of 1.99 1012 s 1 were calculated from the DSC peak maximum temperature - heating rate relationship. 

By applying nonisothermal TGA experiments and the Flynn Wall isoconver-sional model free method, it was found that the activation energy changes with conversion and lies between 34.7 and 43.3 kcal mol1, while the pre-exponential factor ranges from 9.81 1011 to 1.79 1016 s-1. 

In order to assess the activation energy for development of a reasonable model for kinetic analysis of pristine PE and PE-n-MMT thermal degradation processes, a few evaluations by model-free methods have been done as the starting point. As an example, the results of a model-free Friedman analysis for thermal degradation of PE, where the activation energy is a function of partial mass loss change [24], are shown in Figure 5. 

Eor evaluation of experimental DEER data several software packages are available [59, 61]. They cater either for data analysis based on a model of the distance distribution [97-99] or for model-free methods, e.g., Tikhonov regularization [57, 59]. Eor the model-free approach, the underlying mathematical problem is (moderately) ill-posed, i.e., quality of the analyzed data is very cmcial. Incomplete labeling of double mutants results in (1) lower signal to noise of the primary data with increasing number of completely unlabelled molecules and (2) reduced modulation depth with decreasing number of doubly labeled molecules. 

Li and coworkers [50] studied the zinc hydroxide carbonate precursor, Zii4C03(0H)6-H20, synthesized from zinc sulfate using ammonium carbonate as a precipitating agent. TG), DSC, transmission electronic microscopy (TEM), infiared spectroscopy (IR) and XRD were used to characterize die precursor and the decomposition product, while the non-isothermal kinetics of the thermal decomposition of zinc hydroxide carbonate were studied in nitrogen. The kinetic parameters were obtained using a model-free method and the reaction model was then derived by means of non-linear regression. The results showed that the decomposition of zinc hydroxide carbonate is a two-step reaction a reversible reaction of two-dimensional diffusion (D2), followed by an irreversible one of... 

Another model-free method for active damage detection is based on directional transducers. For passive sensing. Section 16.9 has already discussed the use of directional strain rosettes (both piezo and optical) to effectively detect an acoustic source without the need for a structural model. For active sensing, directional transmission as well as reception of guided wave can be achieved with phased-array transducers. 

Zhang Bao-sheng, Liu Jian-zhong, Zhou Jun-hu Cen Ke-fa (2007). New model-free method based approach to mechanism of limestone decomposition. Journal of Chemical Industry and Engineering (China) 58 (5) 1204-1209. 

Model-free methods evaluations were chosen as the starting points in kinetic analysis of neat PP and PP-mPP with 7% Cloisite 20A for determining the activation energy in the development of the model. Figure 6 shows a corresponding Friedman analysis, where the activation energy is a function of partial mass loss change [16]. 

ANNs represent one of the most powerful model free methods. It has been shown recently that the stability constants can be calculated even without solving mass balance equations applying a multivariate calibration approach and the ANN method. The procedure is in some way analogous to Sillen s so-called projection maps, described in the 1960s [82,83]. The projection maps method has even been computerized, but the evaluation is done graphically [84]. 

The isoconversional methods are also known as model-free methods. Therefore, the kinetic analysis using these methods is more deterministic and gives reliable values of activation energy E, which depends on degree of transformation, a. However, only activation energy... 

Since this method does not take any mathematical approximation for the temperature integral, it is considered to give accurate estimate of E. Thus the method does not require any assumption on f(a), i.e. it is a so-called model-free method. However, being a differential method, its accuracy is limited by the signal noise (Dhurandhar et al, 2010). 

Sz > from equations of this type, in cases where data for at least three nuclei in Ln + complexes of the same ligand are available. The applicability of the various model-free methods in the structural analysis has been tested for a large series of sterically rigid ligands ... 

Equations (21.8) and (21.9) are the fundamental expressions of the analytical methods used to calculate the kinetic parameters based on TGA data [28,35,36]. The most common model-free methods used are summarized in Table 21.1. 

Calculation Procedures Based on Isoconversional (Model-Free) Methods... 

However, there arise acute problems when the reaction mechanism undertake a more complex, multi-step pathway [531-535]. It is worth noting that such a composite process might not be detected be mere variation in the apparent values of E but a more multifaceted kinetics is necessarily applied or the other, often non-parametric (model-free methods) ought to be considered as more convenient to such application [533]. 

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