Unimodal and Bimodal Distributions

Figure 11.1. Demonstrates a unimodal and bimodal distribution of drug metabolism. The theoretical unimodal (single gaussian distribution) on the left is typical of a nonpolymorphic metabolism, and the bimodal distribution (mixture of two Gaussian distributions) on the right represents the results of phenotyping for a polymorphic substrate, debrisoquine. The intersection of the two bimodal distributions is the antimode, or the estimated cutoff point between the two phenotypes used to predict the PM phenotype. Adapted, in part, from Henthorn T. K., Benitez, J., Avram,M. J., Martinez,C., Llerena, A., Cobaleda, J., Krejcie,T., and Gibbons, R. D. Clin. Pharmacol. Ther. 45, 328-333,1989 and Kiipfer, A., and Presage, R. Ear. J. Clin. Pharmacol. 26 753-759,1984.

Fig 1 Frequency and Cumulative Distributions of Size Fig 2 Comparison of Unimodal and Bimodal Distributions... 

Description The Hostalen process is a slurry polymerization method with two reactors parallel or in series. Switching from a single reaction to a reaction in cascade enables producing top quality unimodal and bimodal polyethylene (PE) from narrow to broad molecular weight distribution (MWD) with the same catalyst. 

The Hoechst slurry process was improved over the years and has evolved into what is now called the Hostalen process. Hostalen is a slurry-cascade process that is capable of producing a wide range of molecular weight distributions of HOPE. The modern Hostalen process employs 2 continuous stirred tank reactors that can be run in series or in parallel to produce unimodal and bimodal HOPE (11). 

Unimodal or bimodal distribution of r and s forms (mono- or polygenic change). 

Both processes can produce unimodal and bimodal molecular weight distributions. Currently, bimodal MWDs may have to be produced in dual reactor systems. They are energy intensive and require more capital and increase the control complexity. Some licensors claim now to achieve similar product quality with a single reactor by using a dual site catalyst with bimodal capability. 

In recent years, with the development of model networks it has been possible to prepare networks of controlled and junction functionality These are prepared by endlinking functionalized prepolymers with cross-linking agents of known functionality. Therefore, by choosing the appropriate molecular weight distribution of the prepolymers it is possible to prepare unimodal and bimodal networks. Mark and coworkers (5-11) have performed extensive studies on model networks to test the various theories of rubber elasticity. In the case of unimodal networks they find that the macroscopic properties such as stress or swelling ratios can be described reasonably well by the Flory-Erman theory (12,13). 

For the q range studied, the values for the bimodal networks are found to be similar to unimodal networks. The fact that the correlation size in a swollen network is much lager than in the corresponding semi-dilute solution is usually taken to be an indication of inhomogeneities in the network structure. However, the similarity of the results from the unimodal and bimodal networks seems to suggest that in the absence of non-random crosslinking, the distribution of crosslink densities is not the dominant factor in determining spatial correlations in a swollen network at the level of the network mesh size. 

Vicente et al. [30] used the heat of reaction and the open-loop observers developed in Section 7.2.5.3 to determine the concentration of monomer and CTA and hence to infer the instantaneous number-average molar masses during emulsion homo- and copolymerization reactions. In addition, the authors used the inferred values for online control of the molar mass distributions of copolymers with predefined distributions. They demonstrated that polymer latexes with unimodal MMD with the minimum achievable polydispersity index in free-radical polymerization (PI = 2) and bimodal distributions could be easily produced in linear polymer systems [15, 30]. 

The CONTIN method uses a regularization technique to seek smooth solutions, no matter whether the G(r) distribution is unimodal, multimodal, or broad. So the CONTIN method is appropriate for photocount correlation profile analysis without an a priori assumption on the form of the G(r) distribution. We used the CONTIN method, which was kindly provided by Dr. S.W. Provencher (European Molecular Biology Laboratory), mainly for correlation function profile analysis of unimodal and bimodal G(r) distributions. 

Novel general expressions were developed for the description of the behaviour of the height equivalent of a theoretical plate in various chromatographic columns such as unpacked (open capillary), packed with spherical nonporous particles and packed with spherical porous adsorbent particles. Particles may have unimodal or bimodal pore size distribution. The expression describing the mass balance in open capillaries is... 

The dimensions of the fluid voxels correspond to 100 pm depth and to about 30 nm width, (a) Mixing chamber the initial bimodal distribution becomes unimodal upon application of the AC field (b) exit stream a similar behavior is observed, leaving, however, a slight bimodality [25]... 

For strong scatterers an average diameter can be obtained within +/- 5 every 10 or 20 seconds. To get this precision with the poly-dispersity takes 2 or 3 minutes. To distinguish between a broad unimodal and a bimodal distribution may take several minutes. Finally, as already mentioned, it may take hours to obtain representative widths. 

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