Variable cluster model

Hoffman, J. D. (1983) Regime III crystallization in melt-crystallized polymers the variable cluster model of chain folding, Polymer, 24, 3-26. 

Fig. 10.11 Illustration of the metastable polymer conformation in the crystalline regions. From left to right are the fringed-micelle model, the lamellar crystal with adjacent chain folding, the switchboard model and the variable-cluster model...

Hoffman JD, Lauritzen JI (1961) Crystallization of bulk polymers with chain folding theory of growth of lamellar spherulites. J Res Natl Bur Stand 65A 297-336 Hoffman JD, Guttman CM, DiMarzio EA (1979) On the problem of crystallization of polymers from the melt with chain folding. Faraday Discuss Chem Soc 68 177-197 Hoffman JD (1983) Regime in crystallization in melt-crystallized polymers The variable cluster model of chain folding. Polymer 24 3-26... 

According to this intramolecular-nucleation model, when the melt of long-chain macromolecules is quenched to low temperatures for fast crystallization, each macromolecule may perform multiple local intramolecular nucleation events and hence will be included in several lamellae or several positions of the same lamellae, with only little changes of their unperturbed coil-size scaling. At each position, intramolecular nucleation yields folded-chain clusters. This picture is quite consistent with Hoffman s proposition of a variable-cluster model for the conformation of macromolecules in the semi-crystalline state [50[. 

On the other hand, Hoffman (76) showed that the density of the amorphous phase is better accounted for by having at least about 2/3 adjacent reentries, which he calls the variable cluster model. An illustration of how a chain can crystallize with a few folds in one lamella, then move on through an amorphous region to another lamella, where it folds a few more times and so on, is illustrated in Figure 6.36 (124). Thus a regime III crystallization according to the variable cluster model will substantially retain its melt value of Rg. 

Figure 6.36 The variable cluster model, showing how a chain can crystallize from the melt with some folding and some amorphous portions and retain, substantially, its original dimensions and its melt radius of gyration (124).

Hikosaka M, Okada H, Toda A, Rastogi S, Keller A (1995) Dependence of the lamellar thickness of an extended-chain single crystal of polyethylene on the degree of supercooling and the pressure. J Chem Soc Faraday Trans 91(16) 2573-2579 Hoffman JD (1983) Regime III crystallization in melt-crystallized polymers the variable cluster model of chain folding. Polymer 24(l) 3-26... 

Figure 13.1 Illustration of the models of (a) fringed micelle [5], (b) adjacent chain folding [6], (c) switchboard model [7], and (d) the variable cluster model [8],...

Several alternative models were proposed that attempted to reconcile tight fold adjacent reentry with the observed scattering phenomena these included the central core model [58] and the variable cluster model [59], shown schematically in Figures 22 and 23, respectively. These two models are based upon the premise... 

The exp-6 model is not well suited to molecules with large dipole moments. To account for this, Ree9 used a temperature-dependent well depth e(T) in the exp-6 potential to model polar fluids and fluid phase separations. Fried and Howard have developed an effective cluster model for HF.33 The effective cluster model is valid for temperatures lower than the variable well-depth model, but it employs two more adjustable parameters than does the latter. Jones et al.34 have applied thermodynamic perturbation theory to... 

The large number of TIs, and the fact that many of them are highly correlated, confounds the development of predictive models. Therefore, we attempted to reduce the number of TIs to a smaller set of relatively independent variables. Variable clustering " was used to divide the TIs into disjoint subsets (clusters) that are essentially unidimensional. These clusters form new variables which are the first principal component derived from the members of the cluster. From each cluster of indexes, a single index was selected. The index chosen was the one most correlated with the cluster variable. In some cases, a member of a cluster showed poor group membership relative to the other members of the cluster, i.e., the correlation of an index with the cluster variable was much lower than the other members. Any variable showing poor cluster membership was selected for further studies as well. A correlation of a TI with the cluster variable less than 0.7 was used as the definition of poor cluster membership. 

Variable clustering of the remaining 89 TIs resulted in ten clusters. These clusters explained 89.7% of the total variation. In Table 7, we present the indexes selected from each cluster for subsequent use in modeling the BP of hydrocarbons. 0, ICo, IQ, IC2, SlCo, and SICi were selected because of their poor relationship with their clusters (r<0.7). 

It is also worth to mention that exothermic heats of formation for similar O3 and O4 type moieties were observed for the reactions of the MeOxMe (X = 1—2) species with molecular oxygen (Table 11.9) [58, 59]. Close similarities are revealed between the O4 geometries obtained in our slab and the ones noted in the cluster models and periodic models of zeolites despite of more variable geometries in the last case. The closeness between bond lengths ( 0.02 A) with the similar O4 structures optimized at the PW91/PAW level for the Ca204 species in CaMOR zeolite is even surprising. 

Fig. 4.3 Illustration pictures on the models of (a) Switchboard (Flory 1962) and (b) variable clusters (Hoffman 1983)...

The next step was development of model describing the impact of regional characteristics in Germany on FATALR values in these regions. For this purpose, separated base of regional data was created the attempts were made to develop a model of impact of respective variables on modelled dependent variable. Cluster analysis allowed specification of classes of correlated variables. In individual models the impact of respective classes have been taken into account through selection of their representatives. 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

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