Statistical randomization

Homogeneous nucleation occurs when, as a result of statistically random segmental motion, a few segments have adopted the same conformation as they would have in a crystallite. At one time it was considered that the likelihood of the formation of such nuclei was greatest just above the transition temperature... 

When reaction is carried out homogeneously in solution this state of affairs more or less exists and it is possible to achieve a statistically random degree of substitution. (It is to be noted that the primary hydroxyl groups will be more reactive than the secondary hydroxyl groups.)... 

This apparent time dependent cell disruption is caused because of the statistically random distribution of the orientation of the cells within a flow field and the random changes in that distribution as a function of time, the latter is caused as the cells spin in the flow field in response to the forces that act on them. In the present discussion this is referred to as apparent time dependency in order to distinguish it from true time-dependent disruption arising from anelastic behaviour of the cell walls. Anelastic behaviour, or time-dependent elasticity, is thought to arise from a restructuring of the fabric of the cell wall material at a molecular level. Anelasticity is stress induced and requires energy which is dissipated as heat, and if it is excessive it can weaken the structure and cause its breakage. 

Copolymer types Statistical Random Block or graft Alternating... 

The copolymerization of lactones took place through enzyme catalysis [92]. The copolymerization of e-CL with d-VL catalyzed by lipase PF affords the corresponding copolymer having a molecular weight of several thousand. From 13C NMR analysis, the copolymer was found to be of random structure having both units, suggesting the frequent occurrence of transesterifications between the polyesters. In the copolymerization of 8-OL with e-CL or DDL, random copolyesters were also formed [84], whereas the copolymer from e-CL and PDL was not statistically random [88]. 

Comonomers, such as 3-hydroxyvalerate (3HV, ethylene R-unit (-CH2-CH3) in Fig. 1) and 4-hydroxybutyrate, have been incorporated in the PHB chains using specific additives in the growth medium of the bacteria [21-25]. It has been shown by nuclear magnetic resonance (NMR) studies that poly(3HB-co-3HV) has a statistically random distribution of the monomer units throughout a range of compositions varying from 0 to 90 mol % 3HV [23-26]. 

Production of OBCs by chain shuttling catalysis can result in a copolymer with a melting point more than 50 °C higher than that expected for a statistically random copolymer prepared at equivalent density. Figure 17 shows the typical relationship between density and melting point for random ethylene-LAO copolymers. The circled symbols are several OBCs prepared by chain shuttling catalysis [10]. While a typical random copolymer with density of 0.88 g cm-3 would melt at 60 °C, the OBCs made by chain shuttling do not melt until almost 120 °C. 

Fig. 22 Storage modulus vs. temperature for statistically random ethylene-octene and propylene-ethylene copolymers compared to an ethylene-octene OBC...

Animals are assigned to groups (test and control) by one or another form of statistical randomization. Prior to assignment, animals are evaluated for some period of time after being received in house (usually at least one week for rodents and two for nonrodents) to ensure that they are healthy and have no discernible abnormalities. The randomization is never pure it is always blocked in some form or another (by initial body weight, at least) so that each group is not (statistically) significantly different from the others in terms of the blocked parameters. 

When the molecules are free to rotate at a rate that is much faster than the deexcitation rate of the donor (isotropic dynamic averaging), the average value of k2 is 2/3. In a rigid medium, the square of the average of k is 0.476 for an ensemble of acceptors that are statistically randomly distributed about the donor with respect to both distance and orientation (this case is often called the static isotropic average). 

Rule 4.2.1 Statistical, random and unspecified copolymers are treated as irregular polymers. 

A number of different types of copolymers are possible with ATRP—statistical (random), gradient, block, and graft copolymers [Matyjaszewski, 2001]. Other polymer architectures are also possible—hyperbranched, star, and brush polymers, and functionalized polymers. Statistical and gradient copolymers are discussed in Chap. 6. Functionalized polymers are discussed in Sec. 3-16b. 

Data used to describe variation are ideally representative of some population of risk assessment interest. Representativeness was a focus of an earlier workshop on selection of distributions (USEPA 1998). The role of problem formulation is emphasized. In case of representativeness issues, some adjustment of the data may be possible, perhaps based on a mechanistic or statistical model. Statistical random-effects models may be useful in situations where the model includes distributions among as well as within populations. However, simple approaches may be adequate, depending on the assessment tier, such as an attempt to characterize quantitatively the consequences of assuming the data to be representative. 

These can combine in turn in a polymeric chain in a variety of ways, resulting in types such as unspecified, statistical, random, alternating, periodic, block and graft copolymers. 

Fig. 4.—Frequency of Occurrence of Homopolymeric Sequences of D-Mannosyluronic Residues in Alginate from Ascophyllum nodosum. [Key 9, predicted values, based on a statistically random distribution of glycosyluronic residues and O, values determined experimentally.]...

A large body of work has been developed by DeSimone and co-workers on the solubility of fluorinated polymers, especially polyfl,l-dihydroper-fluorooctylacrylate) fPFOA), in C02 (Hsiao et al., 1995 Luna-Barcenas et al., 1998). An excellent example of utilizing creative chemistry to design a C02-soluble polymer, PFOA is one of the very few fluoropolymers that dissolves in C02 at modest temperatures and pressures less than 300 bar. The characteristics needed to make a fluoropolymer soluble in C02 can be ascertained from Figure 7.2, which shows the difference in cloud-point curves for polyfvinylidene fluoride) (PDVF), a statistically random copoly-... 

Polypeptides and poly(a-amino acid)s have a quite unique position amongst synthetic polymers. The reason for this is that most common synthetic polymers have very little long range order in solution and their properties are the products of statistical random coil conformations. Polypeptides, in contrast, can adopt well defined, ordered structures typical of those existing in proteins, such as a-helix and P-struc-tures. Moreover, the ordered structures can undergo conformational changes to the random coil state as cooperative transitions, analogous to the denaturation of proteins. 

Atactic Polymers with a statistically random configuration at their tertiary carbon atoms (see Fig. 15). They have an amorphous structure. 

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