Random distribution, Flory

Solution of long-chain molecules When two liquids mix to form a mixture, the entropy change is similar to that of the volume expansion, as long as the solute molecules have the same size as the solvent molecules and are randomly distributed. But when the solute forms long-chain molecules, the correct method of calculating the entropy was given by Flory. First consider a lattice model where the solvent and the solute molecules have the same volume. Let i and 2 be the number of solvent and... 

Three common methods of measuring crosslinking (swelling, elastic modulus, and gel point measurements) have recently been critically appraised by Dole (14). A fourth method using a plot of sol + sol against the reciprocal dose has also been used extensively. However, Lyons (23) has pointed out that this relation, even for polyethylenes of closely random distribution, does not have the rectilinear form required by the statistical theory of crosslinking. Flory (19) pointed out many years ago that the extensibility of a crosslinked elastomer should vary as the square root of the distance between crosslinks. More recently Case (4, 5) has calculated that the extensibility of an elastomer is given by ... 

The Flory distribution is a random distribution useful in several modes of polymerization. This distribution results from addition polymerization reactions when the only significant processes that interrupt macromolecular growth are either or both of chain transfer (to any species but the polymer) or termination by disproportionation. Likewise, this molecular weight distribution describes linear condensation polymerization when equal reactivity is assumed for all ends only when the reaction involves an equilibrium between polymerization and depolymerization. The model describes the distribution with one parameter which is the number average molecular weight. The distribution equation is ... 

Homogeneous sorption occurs in cases where the interaction between the polymer and water is uniform. Flory (L7) described this case as the polymer and water being randomly distributed. The distribution can be described by ... 

Equation 3.6, together with Equation 3.4, describes a random distribution of molecular sizes this distribution is also known as the Flory-Schulz distribution or the most probable distribution [5]. Recently, Wutz and Kricheldorf [6] proposed a model describing the frequency distribution (/ ) and formulated the weight distribution (w,) of linear chains in step-growth polymerizations considering the cyclation reaction, which is one of the most important side reactions in step-growth polymerization. 

The simple Flory-Huggins theory discussed above is based on a series of questionable assumptions lattice sites of equal size for solvent segments and polymer monomeric units, uniform distribution of the monomeric units in the lattice, random distribution of the molecules, and the use of volume fractions instead of surface-area fractions in deriving the enthalpy of mixing. Proposed improvements, however, have led to more complicated equations or to worse agreement between theory and experiment. Obviously, various simplifications in the Flory-Huggins theory are self-compensating in character. 

The derivation of equation 3.72 assumes no interaction between polymers. Although the use of a three dimensional lattice to represent a polymer solution is admittedly artificial (particularly regarding the assumption that polymer segments and solvent molecules have the same dimensions), equation 3.72 has proven useful in correlating many experimental results. A major drawback, as mentioned previously, is the assumption that the polymer molecules are randomly distributed on the lattice. This assumption becomes untenable in dilute solutions in which polymer molecules exist as isolated islands surrounded by a sea of solvent. Flory [11], Flory and Krigbaum [12] and Ishihara and Guth [13] have derived expressions for the dilute solution case. 

Situation (iv) This situation involves those cases in which the homopolymer is rather unsoluble in the phases formed by the BCP. The low solubility of the homopoiymer induces the formation of macrophase-separated domains rich in diblock copolymer (in which additional microphase separation may be observed) randomly distributed in a homopolymer host. Micro- and macrophase separation is observed in blends either with diblock copolymers having large incompatibility between the blocks (given by the Flory-Huggins parameter, x) or in blends in which the molar masses of homopoiymer and diblock copolymer differ to a large extent [5,39,40]. 

For the isotactic polymer used in this study, containing 2% of randomly distributed r dyads, the results of the Flory calculations indicate that if co = 0, the characteristic ratio should be ca. 30. We measured the characteristic ratio for the polymer employed in this work using the molecular weight obtained from light scattering measurements (M 300,000) and the unperturbed end-to-end distance o obtained from viscosity measurements in diphenyl ether at 145°, which corresponds to 9 conditions. We find ... 

One further point should be emphasized. When an epoxy reacts with a carboxyl group, an hydroxyl group is formed, see reaction (4), Thus it could be argued that in this esterification reaction, the vernonia oil has a functionality of six (or 4.8 if it is assumed that the epoxy groups are randomly distributed). These values are also shown in Table I. Of these calculations, the Flory-Stockmayer trifunctional value fits best. Use of equation (7) suggests an actual functionality of 3.61. 

So, at high temperatures, the ELP is a highly flexible chain, which shows a random distribution of the end-to-end distance in agreement with the idea of Hoeve and Flory that the elasticity of elastin is rubber like [662, 663]. Irregular (or even random) location of the ordered... 

Observing that in solutions containing polar components, segment-segment interactions can be favored and perturb a random distribution of macromolecular chains, Flory proposed to take into account the existence of such interactions... 

The original Flory-Huggins Theory assumed random distribution of contiguous segments of polymer chains. Modem polymer solution theories have diverged significantly from this original concept and contain many correction terms for non-ideal behaviour. 

The Flory-Rehner equation in its original form does not take account of network imperfections due to the random distribution of lengths of effective chains or cilia consisting of chains that are bound to the network by only one end. To account for such imperfections in the network, Flory proposed the following modification [91], which was subsequently confirmed [92] ... 

Flory, P. J., Random reorganization of molecular weight distribution in linear condensation polymers, J. Am. Chem. Soc., 64, 2205-2212 (1942). 

From Eq. (4) it can be seen that as the conversion is driven towards completion, i.e.,p is close to unity, the molecular weight distribution increases dramatically. Theoretically, polycondensation of A2B monomers should form an infinite molecule at extremely high conversions, though in practice this is seldom observed. Flory concluded that condensation of A B monomers would give randomly branched molecules without network formation [1]. However, the occurrence of unwanted reactions (an A group reacts with an A group, for instance) will eventually give rise to an infinite network. Therefore, side-reactions have to... 

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