Multimodal distributions

The resulting active end-groups have a covalent structure, rather than ionic, with tetrahedrally bound ligands persisting unaltered during the lifetime of the polymeric chains 46). This persistence of the various initially formed structures accounts for the multimodal distribution of the molecular sizes of polymers initiated by Grignard compounds46). 

The algebraic calculations of the distribution parameters included all cells of the histogram while those obtained from the cumulative plots were sometimes based on only a part of the cells. When the distribution was multimodal, the latter calculations included only the values from the distribution with the smaller diameter, and no attempt was made to adjust the percentage count for the number of points in the larger dicuneter distribution(s). Other factors reported are the number of particles per 0.1 ft, the dust concentration measured with the Vertical Elutriator Cotton Dust Samplers (VE), the nature of the cumulative plots, and for multimodal distributions, the percent of particles in the distribution with the smaller dieuneter. 

Often bimodal, or even multimodal, distributions are encounted. Dallavalle, Orr, and Blocker 2L) discuss procedures for analyzing such distributions. 

Be fore using parametric methods, the data have to be transformed and tested to see if normal distribution occurs. Sometimes multimodal distributions exist, as illustrated in Fig. 1-8 for the occurrence of particulate matter in the atmosphere. A further difficulty in this case is that the type of distribution may change during the transmission process in the air. If normal... 

However, TEM measurements performed on a Pt0 83Sn017/C (Figure 9.15) indicated that the increase of the metal loading on the carbon support led to the formation of a multimodal distribution of the particle size. Then, to overcome this problem, colloidal methods were also developed in our laboratory. 

The subdistribution method is extremely sensitive to operator input and consistently yielded the poorest results of the three methods compared. In general CONTIN yields good results but tends to shift MWDs to higher molecular weights and sometimes produces artifact peaks or shoulders for broad unimodal or multimodal distributions. Of the three methods, the proposed GEX fitting technique seems to provide results that are most consitent with the input distributions and is the most operator independent. CONTIN and GEX fitting are not significantly affected by noisy data. 

Thus, given gparticle size distribution. For narrow size distributions, the autocorrelation function is satisfactorily analyzed by the method of cumulants to give the moments of the particle size distribution.(7) However, the analysis of QELS data for samples with polydisperse or multimodal distributions remains an area of active research.(8)... 

Our objective in this paper is to compare all three methods in the overlap region of 0.5 pm to 5 pm. We are particularly interested in accuracy, noise tolerance, ease of usage, and ability to deal with multimodal distributions. 

Automated repetition of the sample injection after one fractionation cycle which is indicated by a signal decrease of a concentration detector to the baseline signal. If samples with multimodal distributions are to be fractionated, the threshold logic for sample reinjection needs to be modified accordingly. 

Our picture of the transitions between centres is very incomplete so far, based on studies of distribution curve shapes in the products. When a monomer is polymerized by a living mechanism on two or more centres of widely differing reactivity, chains of characteristic legth are produced on each centre type. In a strictly living medium where centres of one type are not transformed to another, a product with a bi- or multimodal distribution curve of degrees of polymerization is formed. When the various centre types are in a dynamic equilibrium where the centre type changes in the course of propagation, the distribution curve of the product will be broader than the width of either of the peaks in the previous case, but narrower than the overall... 

Chains with monodisperse molecular weight distribution (Mw/Mn = 1.00) can occur in idealized conditions when all polymerizing centers initiate instantaneously and chain termination is absent. In these cases the catalyst is actually an initiator. These living polymerizations are quite rare among transition metal catalysts. More often, random chain termination leads to many chains formed per metal atom. A Schulz-Flory most probable distribution of polyalkene molecular weights (Mw/Mn = 2.00) is the result. In cases when more than one type of active site is present, bimodal or multimodal distributions of molecular weights result (Mw/Mn > 2.00). 

The goal of milling, in most cases, is to meet a size specification, which may be expressed as an acceptable range of mean particle sizes or a limit on large (typically d90 or d95) or small (d5 or dlO) particles. Even the highest energy-milling techniques will not produce a specific particle size, but instead will produce material with a distribution of particle sizes. In many cases, this distribution will be log normal, but it is not uncommon to produce multimodal distributions... 

The use of the characteristic functions facilitated the extension of the stochastic theory to two-site [99] or to generic multiple-site heterogeneous surfaces [100] that are very complex to handle otherwise [5,101,102]. Thus, the stochastic theory is able to model the band profiles that are due to any unimodal or multimodal distribution of adsorption energies. The stochastic model was further extended to describe the effect of mobile phase dispersion and size exclusion effects as well [7,8,103]. 

A graded size distribution, where fine particles fill the interstices between coarse particles as shown schematically in Fig. 4, will minimize the amount of void space that must be filled by fluid, and so will reduce the quantity of liquid needed to produce a flowable slurry. The best size distributions for this purpose have proven to be multimodal distributions, made up of several fairly narrow size fractions. An example of such a size distribution is shown in Table 1. To achieve such a multimodal size distribution, coal slurry production facilities are designed to generate several coal streams, which are each ground and sized to the desired particle sizes and then combined to give the proper size distribution. 

As a result, the multimodal distribution, occurring for the apparent thickness 2.4 nm, transforms into the unimodal one. This is a proof of conversion of the deposition character at a certain stage of the coating growth. The non-stationary deposition phase, observed during the very onset of the coating formation, is attributed... 

Figure 6.15 Pore size distribution in plasma-sprayed CaO-TiOj-ZrOj Oj coatings, (a) Unimodal distribution, mean porosity P = 10.8%. (b) Multimodal distribution, mean porosity P > 15.5% (Heimann, 2006b).

The set of data in Fig. 23B indicate the effect of [ED] (expressed by the [TMPC1]/[TEA] ratio) on Mn and MWD under the same conditions. In the absence of ED (conventional polymerization) or at low [ED] ([TMPC1]/ [TEA] = 1 0.042), polymerizations are rapid and conversions are complete in much less than 10 min, and the GPC traces show broad multimodal distributions suggesting a multiplidty of propagating spedes. With increasing [TEA] the rate of polymerization is reduced due to the decrease in the concentration of cationic species and a reduction in kp. At [TMPC1]/[TEA] = 1 0.5 most of the... 

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