Polymers exponents

The reason the critical exponents in the large b limit do not tend to the Euclidean value may be understood as a crossover effect. For large b, the space looks Euclidean at length scales smaller than b, and the effective polymer exponents for e > 1/6 " would be near the Euclidean values. However, for e 1/6 ", the polymer has to go through the constrictions, and the asymptotic value of exponents for large polymers can, and do, take different values. 

Figure A2.5.27. The effective coexistence curve exponent P jj = d In v/d In i for a simple mixture N= 1) as a fimction of the temperature parameter i = t / (1 - t) calculated from crossover theory and compared with the corresponding curve from mean-field theory (i.e. from figure A2.5.15). Reproduced from [30], Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 358, figure 3, by pennission of Elsevier Science.

The viscosity average molecular weight is not an absolute value, but a relative molecular weight based on prior calibration with known molecular weights for the same polymer-solvent-temperature conditions. The parameter a depends on all three of these it is called the Mark-Houwink exponent, and tables of experimental values are available for different systems. 

P. G. De Gennes. Exponents for the excluded volume problem as derived by the Wilson method. Phys Lett 38A 339, 1972 J. des Cloiseaux. The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys 26 281-291, 1975. 

B. Duplantier, H. Saleur. Exact tricritical exponents for polymers at the 0-point in two dimensions. Phys Rev Lett 59 539-542, 1987. 

MHS exponents than their coil analogs), but it is a reasonable rule of thumb for many polymers. MHS exponents are conveniently found in references such as Du et al. (2) or Grulke (3). 

A criterion for selecting a right pore size to separate a given polydisperse polymer is provided here. To quantify how much the MW distribution narrows for the initial fraction, an exponent a is introduced (2). The exponent is defined by [PDI(0)] = PDI(l), where PDI(O) and PDI(l) are the polydispersity indices of the original sample and the initial fraction, respectively. A smaller a denotes a better resolution. If a = 0, the separation would produce a perfectly monodisperse fraction. Figure 23.7 shows a plot of a as a function of 2RJd (2). Results... 

FIGURE 23.7 Exponent a plotted as a function of the ratio of the polymer dimension to the pore diameter, 2Rg/d. (Reprinted from Polymer, 39, 891, Copyright 1998, with permission from Elsevier Science.)... 

Organic Polymers, Natural and Synthetic 610 Appendix 1 Units, Constants, and Reference Data 635 Appendix 2 Properties of the Elements 641 Appendix 3 Exponents and Logarithms 643 Appendix 4 Nomenclature of Complex Ions 648 Appendix 5 Molecular Orbitals 650... 

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

Concentration of right moving steps at position x Concentration of left moving steps at position x Lauritzen number = L2i/4g Concentration of polymer in solution Concentration exponent... 

The heat of dissociation in hexane solution of lithium polyisoprene, erroneously assumed to be dimeric, was reported in a 1984 review 71) to be 154.7 KJ/mole. This value, taken from the paperl05> published in 1964 by one of its authors, was based on a viscometric study. The reported viscometric data were shown i06) to yield greatly divergent AH values, depending on what value of a, the exponent relating the viscosity p of a concentrated polymer solution to DPW of the polymer (q DP ), is used in calculation. As shown by a recent compilation 1071 the experimental a values vary from 3.3 to 3.5, and another recent paper 108) reports its variation from 3.14 to 4. Even a minute variation of oe results in an enormous change of the computed AH, namely from 104.5 KJ/mole for oe = 3.38 to 209 KJ/mole for oe = 3.42. Hence, the AH = 154.7 KJ/mole, computed for a = 3.40, is meaningless. For the same reasons the value of 99.5 KJ/mole for the dissociation of the dimeric lithium polystyrene reported in the same review and obtained by the viscometric procedure is without foundation. 

In a number of experimental studies of polymer diffusion, molar mass exponents close to 2 have been found, though always with some deviations. For example, using radio-labelled molecules, the diffusion coefficient of polystyrene in dibutyl phthalate was found to follow the relationship... 

Experimental values of the molar mass exponent close to 2 have been obtained. For example, for poly(methyl methacrylate), a value of 2.45 has found (see P. Prentice, Polymer, 1983, 24, 344—350). As with values of selfdiffusion coefficient, this has been regarded as close enough to 2 for reptation to be considered a good model of the molecular motion occurring at the crack tip. 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

Relationships between dilute solution viscosity and MW have been determined for many hyperbranched systems and the Mark-Houwink constant typically varies between 0.5 and 0.2, depending on the DB. In contrast, the exponent is typically in the region of 0.6-0.8 for linear homopolymers in a good solvent with a random coil conformation. The contraction factors [84], g=< g >branched/ <-Rg >iinear. =[ l]branched/[ l]iinear. are another Way of cxprcssing the compact structure of branched polymers. Experimentally, g is computed from the intrinsic viscosity ratio at constant MW. The contraction factor can be expressed as the averaged value over the MWD or as a continuous fraction of MW. 

Another investigation involved the SC VP of a macroinimer 8 via ATRP [46]. GPC/viscosity measurements indicated that the intrinsic viscosity of the branched polymer is less than 40% of that of the linear one at highest MW area (Fig. 6). A significantly lower value for the Mark-Houwink exponent (a=0.47 compared to a=0.80 for linear Pf-BuA) was also observed, indicating the compact nature of the branched macromolecules. 

The MW dependences of the normalized chain relaxation times in melts of linear and branched samples are compared in Fig. 12. Both can be represented by scaling power laws, but with remarkably different scaling exponents. For the melts of linear chains, the exponent 3.39 is observed close to the typical value of 3.4 for such systems. In contrast, for the fractions of the branched polymer, the exponent is considerably lower (2.61). It is interesting to note that the value of the normalized chain relaxation time for the feed polymer with the broad M WD fits nicely into the data for the fractions with narrow MWDs. This seems to indicate that conclusions can also be drawn from a series of hyperbranched polymers with broad MWDs. 

Some information concerning the intramolecular relaxation of the hyperbranched polymers can be obtained from an analysis of the viscoelastic characteristics within the range between the segmental and the terminal relaxation times. In contrast to the behavior of melts with linear chains, in the case of hyperbranched polymers, the range between the distinguished local and terminal relaxations can be characterized by the values of G and G" changing nearly in parallel and by the viscosity variation having a frequency with a considerably different exponent 0. This can be considered as an indication of the extremely broad spectrum of internal relaxations in these macromolecules. To illustrate this effect, the frequency dependences of the complex viscosities for both linear... 

Values obtained for and a for a number of polymer-solvent pairs are given in Table XXX. It will be observed that the exponent a varies with both the polymer and the solvent. It does not fall below 0.50 in any case and seldom exceeds about 0.80. Once K and a have been established for a given polymer series in a given solvent at a specified temperature, molecular weights may be computed from intrinsic viscosities of subsequent samples without recourse to a more laborious absolute method. 

The viscosity average molecular weight depends on the nature of the intrinsic viscosity-molecular weight relationship in each particular case, as represented by the exponent a of the empirical relationship (52), or (55). However, it is not very sensitive to the value of a over the range of concern. For polymers having the most probable distribution to be discussed in the next chapter, it may be shown, for example, that... 

The exponent a is a function of solvent power usually a > 1/2, but for an idealized random-flight polymer, a = 1/2. 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

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