Model B

Figure Bl.19.8. (a) STM image (5.7 mu x 5.7 mu) of 10-alkylcyanobiphenyl on graphite (b) model showing the packing of the molecules. The shaded and unshaded segments represent the alkyl tails and the cyanobiphenyl head groups, respectively. (Taken from [38], figure 2.)...

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

In this section the discretization of upper-convected Maxwell and Oldroyd-B models by a modified version of the Luo and Tanner scheme is outlined. This scheme uses the subdivision of elements suggested by Marchal and Crochet (1987) to generate smooth stress fields (Swarbrick and Nassehi, 1992a). 

Fig. 2. (a) Chain conformation of isotactic polypropylene, and (b) model of a polypropylene spheruHte. 

Demonstrations (a) Atomix (to show grain boundaries), (b) Model of dispersion strengthening. Take piece of PMMA sheet == 2.5 mm thick and = 7 cm square. Glue four PMMA strips of section =7X7 mm on top of the sheet to form a tray = 7 mm deep. Cut six = 7-mm lengths of an = 6-mm-diameter PMMA rod. Glue the ends of these to... 

Figure 10 Models of complexes between BLBP and two different fatty acids. The fatty acid ligand IS shown in the CPK representation. The small spheres in the ligand-bmdmg cavity are water molecules, (a) Model of the BLBP-oleic acid complex, in which the cavity is not filled, (b) Model of the BLBP-docosahexaenoic acid complex, m which the cavity is filled. The figure was prepared using the program MOLSCRIPT [236].

Protein toxins acting intracellularly are often composed of two subunits (A/B model). One subunit is catalytic (A-subunit) and the other is responsible for binding and cell entry (B-subunit). Following binding to an extracellular membrane receptor, the toxins are endocytosed. From the endosomes, the A-subunit is directly (pH dqDendent) transferred into the cytosol (e.g., diphtheria toxin and anthrax toxin) or the toxin is transported in a retrograde manner via the golgi to the ER (e.g., cholera toxin), where translocation into the cytosol occurs [1]. 

Fig. 5 a, b. Models of the crystallization of flexible-chain polymers with the formation of a folded-chain crystals and b extended-chain crystals b... 

Organocobalt B models axial ligand effects on the structure and coordination chemistry of coba-loximes. N. Bresciani-Pahor, M. Forcohin, L. G. Marzilli, L. Randaccio, M. F. Summers and P. J. Toscano, Coord. Chem. Rev., 1985, 63,1 (263). 

Figure 10. Theoretical model for the electrical double layer at an electrode with a polycrystalline surface, (a) Model of independent diffuse layers [Eq. (S3)], and (b) model of common diffuse layer [Eq. (54)).

More recently, two-state E/B models have been proposed by Chujo and Doi (9.10) for the analysis of polypropylene. Similar E/B models were proposed by Cheng(11) and Asakura, et al(12) for polybutylene. For copolymers, two-state B/B models have been proposed for ethylene-propylene copolymers,(11,13-15) and propylene-butylene copolymers.(11,13) Recently, Cheng(11) generalized these multi-state models and developed computer methodology for the general analysis of such systems. A number of polymer systems were treated. 

As an example, polybutylene is a commercially inq ortant polymer made with Ziegler-Natta catalysts. Such catalysts frequently produce more than one catalytic site, and the resulting polymer is a blend of several homopolymers differing only in propagation statistics. Previously, the two-site E/B model has been used to analyze the tacticity of this polymer.(11,12) Reasonably good fits with experimental data were observed. 

The theoretical expressions for the three-state E/E/B model are shown in Table II. The two-site E/B model can be... 

The pentad intensity data can be analyzed with the MIXCO program using the three-state E/E/B model. The fractions are taken two at a time. For each analysis 16 pentad intensities ere entered into the computer (8 from each fraction). The equations used are ... 

Similar analysis can be carried out for Samples II and III in Icenogle and Klingensmith s paper.(18) The results are tabulated in Table IV. It appears that Sample II (made with the same conventional catalyst as Sample I but without a selectivity control agent (18) also follows the three-site E/E/B model very well. Perhaps surprisingly the reaction probabilities for the two E-sites are virtually the same in Samples I and II (P l = 0.994, P 2 = 0.80). The B-site is indeed different. 

Copolymer sequence analysis follows the same procedure. A computer program (HIXCO.TRIAD) was previously written for the two-state B/B model-fitting of triad sequence distributions and applied to (unfractionated) propylene-butylene copolymers and... 

As an example of the use of MIXCO.TRIAD, an analysis of comonomer triad distribution of several ethylene-propylene copolymer samples will be delineated. The theoretical triad Intensities corresponding to the 2-state B/B and 3-state B/B/B mixture models are given In Table VI. Abls, et al (19) had earlier published the HMR triad data on ethylene-propylene samples made through continuous polymerization with heterogeneous titanium catalysts. The data can be readily fitted to the two-state B/B model. The results for samples 2 and 5 are shown In Table VII. The mean deviation (R) between the observed and the calculated Intensities Is less than 1% absolute, and certainly less than the experimental error In the HMR Intensity determination. 

Hote that we have fitted the HMR triad data to the 2-slte B/B model only. In the 2-site B/B model, we have 6 intensities and 3 unlcnowns (Pp,i> Pp,2> 1 Mathematically the... 

In order to test higher-order or multi-site models, it is preferable to study HMR data of polymer fractions. In this work, we shall use the pairwise HMR/fractions data to fit to multi-site models. This Is carried out in the same way as in the pairwise fraction on the tactlcity data. We can either use the two-state B/B model, or even three-state B/B/B model If applicable. The comonomer sequence intensity data for the two fractions are entered Into the computer. A total of 12 entries are Involved (6 for each fraction). The equations for the three-state copolymer... 

The HMR/fractionatlon approach gives very good results When applied to ethylene-propylene copolymer fractions reported by Abls, et. al. (19) These authors extracted sample 5 (In Table VII) with hexane to get soluble and Insoluble fractions (5a and 5b), and with ether to get soluble and insoluble fractions (5c and 5d). The hexane set (5a and 5b) and the ether set (5c and 5d) can be separately analyzed by the HIXCO.TRIADX program. The results are shown In Table VIII. In the 2-state (B/B) model, we have 4 parameters and 12 values to fit to HMR data of pairwise fractions. In the 3-state (B/B/B) model, we have 7 parameters and 12 values to fit. Thus, the use of pairwise fractions Is absolutely essential for 3-state analysis. 

Table VIII. Mixture Analysis of Ethylene-Propylene Copolymer Fractions Data by 2-Site (B/B>. and 3-Site (B/B/B) Model ...

The results for the 2-slte (B/B) model are satisfactory. The weighted sums (Table IX) indicate that there are two sites of roughly equal weights (41% w j for hexane set, and 51% W ] for ether set). The average Bemoullian reaction probabilities for the two sites are = 0.58, Pp,2 = 0.20. These results... 

For completeness, the results of the 3-state (B/B/B) model are also included in Table VIII and Table IX. As expected, the data fitted three Bemoullian polymers very well with the following Bemoullian probabilities ... 

Fig. 2.2 Most favored helical structures proposed for two crystal forms of poly(a-n-butyl-/ -L-aspartate) (7, R=Bu) [28]. (A) Model of a (P)-3.25i4-helix. (B) Model of a (P)-4is-helix...

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