Stockmayer chain

That assumption simplifies the analysis of primitive path rearrangement. Local path jumps now correspond to random flips in the Orwoll-Stockmayer chain model, and we can apply these results directly. For our case the local jump distance is the path step length a, and the average time between jumps is 2r , where r is the mean waiting time for release of a constraint which allows a length preserving jump. The average number of such suitably situated constraints per cell is z(z < zo), and we assume for simplicity that all cells have z such constraints. 

The theoretical background of the confinement effect in (artificial) tubes was recently examined in detail with the aid of an analytical theory as well as with Monte Carlo simulations [70]. The analytical treatment referred to a polymer chain confined to a harmonic radial tube potential. The computer simulation mimicked the dynamics of a modified Stockmayer chain in a tube with hard pore walls. In both treatments, the characteristic laws of the tube/reptation model were reproduced. Moreover, the crossover from reptation (tube diameter equal to a few Kuhn segment lengths) to Rouse dy-... 

The ring-opening polymerization of is controUed by entropy, because thermodynamically all bonds in the monomer and polymer are approximately the same (21). The molar cycHzation equihbrium constants of dimethyl siloxane rings have been predicted by the Jacobson-Stockmayer theory (85). The ring—chain equihbrium for siloxane polymers has been studied in detail and is the subject of several reviews (82,83,86—89). The equihbrium constant of the formation of each cycHc is approximately equal to the equihbrium concentration of this cycHc, [(SiR O) Thus the total... 

At the end of the 1930s, the only generally available method for determining mean MWs of polymers was by chemical analysis of the concentration of chain end-groups this was not very accurate and not applicable to all polymers. The difficulty of applying well tried physical chemical methods to this problem has been well put in a reminiscence of early days in polymer science by Stockmayer and Zimm (1984). The determination of MWs of a solute in dilute solution depends on the ideal, Raoult s Law term (which diminishes as the reciprocal of the MW), but to eliminate the non-ideal terms which can be substantial for polymers and which are independent of MW, one has to go to ever lower concentrations, and eventually one runs out of measurement accuracy . The methods which were introduced in the 1940s and 1950s are analysed in Chapter 11 of Morawetz s book. 

The structure formation in an ER fluid was simulated [99]. The characteristic parameter is the ratio of the Brownian force to the dipolar force. Over a wide range of this ratio there is rapid chain formation followed by aggregation of chains into thick columns with a body-centered tetragonal structure observed. Above a threshold of the intensity of an external ahgn-ing field, condensation of the particles happens [100]. This effect has also been studied for MR fluids [101]. The rheological behavior of ER fluids [102] depends on the structure formed chainlike, shear-string, or liquid. Coexistence in dipolar fluids in a field [103], for a Stockmayer fluid in an applied field [104], and the structure of soft-sphere dipolar fluids were investigated [105], and ferroelectric phases were found [106]. An island of vapor-liquid coexistence was found for dipolar hard spherocylinders [107]. It exists between a phase where the particles form chains of dipoles in a nose-to-tail... 

Kurata, M. and Stockmayer, W. H. Intrinsic Viscosities and Unperturbed Dimensions of Long Chain Molecules. Vol. 3, pp. 196—312. 

The sedimentation coefficient for wormlike chains was first worked out by Hearst and Stockmayer [123], later improved by Yamakawa and Fujii [124] to give this expression for s ... 

Zimm, B.H. and Stockmayer, W.H.J. The dimensions of chain molecules containing branches and rings, Chem. Phys., 17, 1301, 1949. 

For long linear chains the second condition is supported by the Stockmayer bivariate distribution (8,9) which shows the bivariate distribution of chain length and composition is the product of both distributions, and the compositional distribution is given by the normal distribution whose variance is inversely proportional to chain length. 

In an analogous fashion, the weight average long chain branching parameters per molecule (B ) and per 1000 repeat units (Xy) can be calculated. First Bu(V) can be determined from the Zimm-Stockmayer equation (20),... 

M Kurata, WH Stockmayer. Intrinsic viscosities and unperturbed dimensions of long chain molecules. Fortschr Hochpolym-Forsch 3 196-312, 1963. 

An important source of experimental and theoretical studies of equilibria in ring formation is represented by the field of so-called macrocyclisation equilibria (Flory, 1969). Interest in this field appears to have been restricted so far to chemists conventionally labelled as polymer chemists. Experimental evidence of cyclic oligomer populations of ring-chain equilibrates such as those obtained in polysiloxanes (Brown and Slusarczuk, 1965) may be delated to the statistical conformation of the corresponding open-chain molecules (Jacobson and Stockmayer, 1950 Flory, 1969). In these studies experimental results are expressed in terms of molar cyclisation equilibrium constants Kx (14) related to the x-meric cyclic species Mx in equilibrium with the... 

A satisfactory theory of macrocyclisation equilibria for chains obeying Gaussian statistics was presented by Jacobson and Stockmayer (1950) long before the availability of suitable experimental data for the proper testing of the theory. According to this theory, the macrocyclisation equilibrium constant Kx (see p. 10) is related to the density W(0) of the distribution of the end-to-end vector r in the region r = 0 through relation (57), where NA is... 

There are several reasons why the behaviour of the shorter chains deviate from the original formulation of the Jacobson and Stockmayer theory (Flory, 1969). First, if the ring size is small enough to induce strain, the enthalpy change for cyclisation (16) will differ from that for the intermolecu-lar process (17). In terms of the 0 operator (39), 0AH° will differ from zero and, presumably, be positive. Secondly, (57) is based on the implicit assumption that the relative orientation of the reacting bonds, when they come in close proximity in the cylisation reaction, is random. This independence of orientation and proximity, which leads to the absence of any factor referring to orientation in (57), must fail for short chains. Thirdly, short chains may not follow Gaussian statistics. When this occurs, an appropriate expression for the density of end-to-end vectors is required. 

Flory s viscosity theory also furnishes confirmation of the w temperature as that in which a=V.2, and it permits the determination of the unperturbed dimensions of the Polymer chain. Even if a Q solvent is not available, several extrapolation techniques can be used for the estimating the unperturbed dimensions from viscosity data in good solvents. The simplest of these techniques seems to be that of Stockmayer. 

The introduction of branching in the Kirkwood formula and the KR calculations can be accomplished in a relatively easy way if Gaussian statistics corresponding to ideal chains are maintained. This description cannot, however, be very accurate in molecules with centers of high functionality because of the presence of cores with a high density of polymer units, which profoundly perturbs the internal distribution of distances. Stockmayer and Fixman [81 ] employed the Kirwood formula and Gaussian statistics to calculate h in the case of uniform stars, obtaining an analytical formula. They also performed a KR evaluation of the viscosity and proposed that g could be evaluated from the approximation... 

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