Influence of Polydispersity

The overall polymer content of the coexisting phases is given by the intersection points of the horizontal line, at the considered temperature, with these branches. 

The coexistence curves are usually no closed curves but rather divided into two branches beginning at corresponding points of the cloud-point curve and of the shadow curve. Only if the composition of the initial homogeneous phase equals 

Owing to this polydispersity, characterization of polymers usually does not provide the number of the individual molecules or their mole fraction, mass fraction, and so on, but requires the use of continuous distribution functions or their averages. Continuous thermodynamics, developed by Ratzsch and Kehlen [39] and Cotterman et ai [41] can direr dy be applied for the calculation of thermodynamic properties including phase equilibria, because this theoretical framework is based completely on continuous distribution functions, which include all information about the polydispersity and allow a mathematically exact Leatment of all related thermodynamic properties. Within this approach, a g -model or an EOS can be used. 

An alternative to the application of continuous thermodynamics is the careful selection of pseudo-components [2]. One option for a suitable selection of the pseudocomponents is the Gauss-Hermitian quadrature. 

For the calculation of the stability of a multicomponent polymer solution (spinodal line and critical solution points), the stability theory can be applied [42]. One possible consequence of the poiydispersity, especially if the distribution function is bimodal, is the appearance of tri-critical solution points [2]. Suggestions for the phase equilibrium calculations of such systems can be found in the literature [2, 39, 43). 

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Hamau, L., Winkler, R. G., and Reineker, P., Influence of polydispersity on the dynamic structure factor of macromolecules in dilute solution, Macromolecules, 32, 5956, 1999. 

If the statistical model of a paracrystalline stack is assumed, it turns out that the renormalization attenuates the influence of polydispersity on the position of the first zero. In general, the first-zero method is more reliable than the valley-depth method, although it is not perfect. Even the first-zero method is overestimating the value of V . The deviation is smaller than 0.05, if the found crystallinity is smaller than 0.35. If bigger crystallinities are found, the significance of the determination is... 

Another uncertainty arises from the influence of polydispersity. Intrinsic viscosity data were mostly obtained from fractions but the second virial coefficient data were chosen from unfractionated samples. The resulting error is probably not large since A2 depends only slightly on the width of the distribution [183, 184]. 

From the point of view of Section 3.3 the influence of polydispersity is treated in the most satisfactory way by calculating the influence on the relaxation times. From the spectrum of these times, all rheological properties can then be derived. This has been tried by Peticolas (75). For this purpose, eq. (3.37) is rewritten as follows ... 

The facts just mentioned indicate that the influence of polydispersity is by far the most important effect when compared with those discussed in the previous sections (branching, excluded volume, theoretical refinements). 

Clewlow, A.C., Rowe, A.J., and Tombs, M.P. (1995). Pectin-gelatin phase separation the influence of polydispersity. In S.E. Harding, S.E. Hill, and J.R. Mitchell (Eds.), Biopolymer Mixtures, Nottingham University Press, Nottingham, UK, pp. 173-191. 

Two types of rheological phenomena can be used for the detection of blend s miscibility (1) influence of polydispersity on the rheological functions, and (2) the inherent nature of the two-phase flow. The first type draws conclusions about miscibility from, e.g., coordinates of the relaxation spectmm maximum cross-point coordinates (G, CO ) [Zeichner and Patel, 1981] free volume gradient of viscosity a = d(lnT]) / df the initial slope of the stress growth function S = d(lnr +g)/dlnt the power-law exponent n = d(lnOj2)/dlny = S, etc. The second type involves evaluation of the extrudate swell parameter, B = D/D, strain (or form) recovery, apparent yield stress, etc. 

The simple thermodynamic model derived in Sect. 2.1 has been useful to get a qualitative insight into the phase separation process. When one intends to apply it to an actual system, the significant influence of polydispersity is clearly evidenced. For example, Fig. 13 shows the experimental cloud-point curve for a DGEBA-CTBN binary mixture, together with binodal and spinodal curves calculated by assuming monodisperse components [66] (curves are arbitrarily fitted to the critical point). The shape of the CPC and precipitation threshold temperature (maximum of the CPC) appearing at low modifier concentrations are a clear manifestation of the rubber polydispersity [77]. 

Mclntyre,D., Wims,A.M., 0 Mara,J.H. Tlic influence of polydispersity on pol)mer-solvent coexistence curves. Polymer Preprints ACS 6, 1037 (1965). 

Presently, the influences of polydispersity and chain branching on the dynamics of polymers have been analyzed theoretically. The disentanglement along the reptation of a long chain is accelerated by the surrounding short chains due to the... 

Nowadays, many polymerization reactions lead to products having very narrow molar-mass distributions, but such product samples still differ in average molar-mass values. Also, fractionation [2] can narrow the distribution function, but never produces a monodisperse polymer. As a hypothetical case, we consider a monodisperse polymer dissolved in one solvent as quasi-binary. In this chapter, first the thermodynamic models (g -models and equations of state (EOS)) for a quasi-binary system, consisting of a monodisperse polymer and a solvent, will be introduced, and later the influence of polydispersity on the phase behavior... 

Fig. 19. Influence of polydispersity on the relative variation of Rj (= , with Y from Numericm calculation for a block copolymer with a Schulz-Ziirun distribution. X.,...

The influence of polydispersity is exemplarily shown in Fig. 4.24 for DLCA aggregates. Variation in primary particle size (monodisperse or log-normal distributed with = 0.4) as well as in aggregation number (log-normal distributed and a self-preserving distribution function, Eq. (4.10)) were considered. As in the previous section, a reciprocal correlation between N and jCp v was assumed for the calculation. The normalised diffusion coefficients are plotted versus a -axis that is scaled with the effective radius of gyration Rg etc (cf- Elq- (4.59)) ... 

Every synthetic polymer has a molar mass distribution which may have a profound influence on the phase diagram. The evaluation of thermodynamic properties is certainly complicated by polydispersity. To exemplify the influence of polydispersity, spinodal data for three polydisperse PS samples having comparable mass average molar masses summarised in Table 1. In the computations, the polymer samples were presented by r-equivalent distributions [15] with the molar mass averages and... 

The hole theory offers an excellent basis to evaluate the phase behavior of polymer systems. The description of the spinodal conditions are almost quantitative without the introduction of empirical parameters. The cell free volume is very important for this quantitative success. The influence of polydispersity on the spinodal conditions in the Simha-Somcynsky theory is not restricted to the mass average molar mass. 

An imderstanding of the influence of polydispersity on chain dynamics in the melt was achieved by Baschnagel and co-workers (177) in a simulation using BFM. These dynamic Monte Carlo simulations showed that long chains move more rapidly in the presence of short chains, and the short ones move more slowly in the presence of the long ones. The net effect is that the dsniamics of a polydis-perse melt is close to Rouse theory predictions, ie, the chains act as if they are not entangled An indirect approach to reptation dynamics was described by Byutner... 

The influence of polydispersity on the analysis of quasi-elastic lightscattering data is considerable. For non-interacting particles in the Stokes-Einstein regime, the effective diffusion coefficient is ... 

Influence of Polydispersity on the Liquid + Liquid Equilibrium of a Polymer Solution... 

Polymers in coexisting phases have different molar-mass distributions which are also different from that of the initial homogeneous system. Obviously, the influence of polydispersity on the LLE is not only of a quantitative nature but of a qualitative nature as well. This demixing behaviour is important for some practical problems, for example, in the high-pressure synthesis of low-density polyethene [polyethylene] or of poly(but-3-enoic acid ethene) [poly(ethylene-co-vinylacetate)]. The polyethene is obtained as a solute in supercritical ethene... 

The effect of molecular weight polydispersity is shown in Fig. 10 for blends of one PE with two different series of polystyrenes with constant (18,100 and 107,200) and different polydispersities. The interfacial tension decreased with increasing polydispersity in both cases, and the influence of polydispersity was higher for lower PS molecular weights. The decrease in interfacial tension could be... 

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