Peak shape models

Most GPC columns are provided with vendor estimates of the plate count of the column and a chromatogram of a series of test peaks. These plate count estimates are usually obtained using small molecule analytes that elute at the total permeation volume (Vp) of the column. The Gaussian peak shape model... 

It is often convenient modeling the peak shape assuming some analytical functions [25]. The most commonly used functions are, at present, the Voigt and pseudo-Voigt functions, a combination of a Gaussian (G) and a Lore-ntzian (L) function centered at 20(y. An expression for Gaussian and Lorentzian contributions is ... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

Comparison of the measured peak shape with simulations based on Equations (2-5) and (2-6) reveals that a nucleation and growth model best describes the reduction... 

The modeling of electron diffraction by the pattern decomposition method, for which no structural information is required, can be successfully applied for extraction of the diffraction information from the pattern. Several parameters can be refined during the procedure of decomposition, including the tilt angle of the specimen the unit cell parameters peak-shape parameters intensities. The procedure consists of fitting, usually with a least-squares refinement, a calculated model to the whole observed diffraction pattern. 

The calculated intensity z Xi, yj) at any point x, y, of a dififaction pattern is expressed as a function of the integrated intensity h of the reflections contained in the pattern and a normalized analytical peak shape function PS x, y) is used to model the individual profiles. It is given by... 

When all the phases present were identified, we can quantify their volume fraction in the analyzed volume similarly to the way the Rietveld-method is used for phase analysis in XRD. A whole profile fitting is used in ProcessDifraction, modeling background and peak-shapes, and fitting the shape parameters, thermal parameters and volume fractions. Since the kinematic approximation is used for calculating the electron diffraction intensities, the grain size of both phases should be below 10 nm (as a rule of... 

Following the eibove rationalisation it becomes apparent that o is the only significant dimensionless group in our model. Its influence is vividly demonstrated in Figure 7 which shows that by altering a alone it is possible to cover cill known peak shapes. 

The ideal model and the equilibrium-dispersive model are the two important subclasses of the equilibrium model. The ideal model completely ignores the contribution of kinetics and mobile phase processes to the band broadening. It assumes that thermodynamics is the only factor that influences the evolution of the peak shape. We obtain the mass balance equation of the ideal model if we write > =0 in Equation 10.8, i.e., we assume that the number of theoretical plates is infinity. The ideal model has the advantage of supplying the thermodynamical limit of minimum band broadening under overloaded conditions. 

The ICLS model contains pure spectra of caustic, salt, water, and temperature. The conclusion of the model validation is that the ICLS model adequately describes the water peak shape changes due to caustic, salt, and temperature. Furthermore, it meets the required performance criterion for the prediction of caustic. The measures of performance are as follows (see Table 5.1 for a description of these figures of merit) ... 

The success of this modeling can be ascertained by the ability to replicate the observed peak shapes using Gaussian peaks centered at peak positions suggested by the modeling (Fig. 3). For the case where n = 3/2, five peaks were needed 1 for the monomeric A1 and 2 peaks each for each of the two dinuclear A1 species. These peaks were combined to successfully replicate the observed NMR peaks recorded for this sample. When n = 2, the data were reproduced using only 4 peaks, two each for each of the two dinuclear A1 species. Our earlier predictions (11) showed that HCl could combine with either of the dinuclear Al-species in three different positions, which showed different acid strengths. 

A chromatogram without noise and drift is composed of a number of approximately bell-shaped peaks, resolved and unresolved. It is obvious that the most realistic model of a single peak shape or even the complete chromatogram could be obtained by the solution of mass transport models, based on conservation laws. However, the often used plug flow with constant flow velocity and axial diffusion, resulting in real Gaussian peak shape, is hardly realistic. Even a slightly more complicated transport equation... 

The alternative is the use of a descriptive mathematical model without any relation with the solution of the transport equation. On the analog of the characterization of statistical probability density functions a peak shape f(t) can be characterized by moments, defined by ... 

A transfer function, defined as the Laplace transfer of the impulse response of a linear system, can be obtained from the model. This can be very useful, because with a transfer function the influence of extra-column effects (detector, amplifier, filter) on the peak shape can be easily calculated. The transfer function is ... 

Cassidy and Frei [23] designed a microflow cell for the Turner Assoc. Model III fluorimeter for use with HPLC. Nanogram quantities of fluorescent materials could be detected. The volume of the flow cell was only 7.5 jul. The detector was unaffected by the flow-rate or composition of the solvent. This gives this detector a decided advantage over refractive-index or UV detectors. The peak shapes were symmetrical and the linear range of response was 2-3 orders of magnitude. 

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