Monodisperse Linear Polymers

The occurrence of a secondary phase separation inside dispersed phase particles, associated with the low conversion level of the p-phase when compared to the overall conversion, explains the experimental observation that phase separation is still going on in the system even after gelation or vitrification of the a-phase [26-31]. A similar thermodynamic analysis was performed by Clarke et al. [105], who analyzed the phase behaviour of a linear monodisperse polymer with a branched polydisperse polymer, within the framework of the Flory-Huggins lattice model. The polydispersity of the branched polymer was treated with a power law statistics, cut off at some upper degree of polymerization dependent on conversion and functionality of the starting monomer. Cloud-point and coexistence curves were calculated numerically for various conversions. Spinodal curves were calculated analytically up to the gel point. It was shown that secondary phase separation was not only possible but highly probable, as previously discussed. 

Because it is evaluated in the limit of infinite dilution, the intrinsic viscosity provides information about the average size of molecules in a solution in which there is no interaction between molecules. In practice, for a linear, monodisperse polymer, the relationship used to calculate the molecular weight from the intrinsic viscosity is the one proposed by Mark [40], Houwink [41] and Sakurada [42] and given here as Eq. 2.79. 

Figure 4.6 is a sketch of the typical shape of the creep compliance curve using linear scales. The limiting long-time slope and its extrapolation to t = 0 to obtain the steady-state compliance are shown. Also shown is the creep curve of a crosslinked elastomer. The creep compliance for an entangled, linear, monodisperse polymer sample is shown in Fig. 4.7, this time using logarithmic scales for both axes. 

We note that the plateau modulus, which is a feature of the relaxation modulus and storage modulus of linear, monodisperse polymers, should vary with temperature in the same way as bT-,i.e. ... 

The steady-state compliance /° of a linear, monodisperse polymer increases linearly with M when M is less than a critical value M q but becomes independent of M when M > M q, i.e., when entanglements become important. Curiously, M q is much larger than Mq, the critical value of M for the effect of entanglements on 7o, often by a factor of four or five. This behavior is sketched in Fig. 5.12. 

For more non-uniform samples the effect is much more dramatic. For example, in a blend of two compatible, linear, monodisperse polymers /° can be several times larger than the values of 7° of either of the two components. This is illustrated in Fig. 5.13, showing the steady-state compliances of two monodisperse samples as well as those of various binary blends of... 

Where Xq(M) is the reptation time of a linear, monodisperse polymer having a molecular weight M. Leonardi et al. [50] assembled a complete model of G(t) for polydisperse systems by combining the ideas of BenaUal et al. [44] and Montfort etal. [48]. The final model for G( t) is similar to one published by Carrot and GuiUet [51], except that the latter do not account for tube length fluctuations and use a different modeling of the Rouse processes. 

Tube models have been used to predict this material function for linear, monodisperse polymers, and a so-called standard molecular theory [159] gives the prediction shovm in Fig. 10.17. This theory takes into account reptation, chain-end fluctuations, and thermal constraint release, which contribute to linear viscoelasticity, as well as the three sources of nonlinearity, namely orientation, retraction after chain stretch and convective constraint release, which is not very important in extensional flows. At strain rates less than the reciprocal of the disengagement (or reptation) time, molecules have time to maintain their equilibrium state, and the Trouton ratio is one, i.e., % = 3 7o (zone I in Fig. 10.17). For rates larger than this, but smaller than the reciprocal of the Rouse time, the tubes reach their maximum orientation, but there is no stretch, and CCR has little effect, with the result that the stress is predicted to be constant so that the viscosity decreases with the inverse of the strain rate, as shown in zone II of Fig. 10.17. When the strain rate becomes comparable to the inverse of the Rouse time, chain stretch occurs, leading to an increase in the viscosity until maximum stretch is obtained, and the viscosity becomes constant again. Deviations from this prediction are described in Section 10.10.1, and possible reasons for them are presented in Chapter 11. 

Tube models predict that in a linear, monodisperse polymer, the extensional viscosity should decrease with strain rate at moderate rates, pass through a minimum, and then rise as chain stretch begins to occur. At a sufficiently high rate, the chain reaches its maximum stretch, and the extensional viscosity should approach a final plateau. However, strain rates capable of generating chain stretch in linear molecules are very high and out of the range of most experimental methods. Some aspects of this behavior, however, have been observed in experimental data. 

Linear monodisperse tt-conjugated oligopyrroles and oligothiophenes as model compounds for polymers 99AG(E)1350. 

For example, the parameters g = 0.77, h = 0.94, p = 1.4, and C = 0.158 measured for a polymer sample and compared with the plots in Figures 7.11 through 7.13 were most consistent with athree-arm star monodisperse polymer a poly disperse three-arm star would have g= 1.12,/ = 1.05,p= 1.6, and C close to 0.2. °° The second example was poly(vinyl acetate) (PVAc) prepared by emulsion polymerization. Since no data for linear equivalent were available, g and h were not calculated. At lower conversion/MW p= 1.84 was found, only slightly higher than the theoretically expected p = 1.73 for a randomly branched architecture, p slightly decreased with increasing M, indicating... 

R. E. Martin, F. Diederich, Linear Monodisperse ji-Conjugated Oligomers Model Compounds for Polymers and More , Angew. Chem. Int. Ed. Engl. 1999, 38, 1350-1377. 

Martin RE, Diederich F Angew Chem Int Ed (1999) 38 1350-1377 Linear monodisperse n-conjugated oligomers model compounds for polymers and more... 

The spreading factor C is the variance of the chromatograms of the monodisperse polymer species, i.e. of the instrumental spreading fimction G(V,Vr), If O g varies linearly vd.th the retention volume of the monodisperse polymer, then<0 > is numerically equal to the interpolated value 0 (v) of the function (T (Vr) for the polydisperse sample at its mean elution volume. 

We have synthesized and tested an example boron-chelating polymer based on a commercially available dendrimeric poly(amido amine) (PAMAM). Dendrimeric chelants offer several advantages over polymers typically used in PAUF. Foremost among these is the reduced viscosity of dendrimer solutions as compared to solutions of linear polymers[6]. This allows the use of higher polymer concentrations than previously feasible (though in the present study we worked at polymer concentrations of less than 5% due to the expense of the starting dendrimer). In this study, the dendrimeric chelant also serves as a convenient, monodisperse polymer with which to test the mathematical model for boron speciation which is derived from the work of Wise and Weber[l],... 

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

NMRP is not a true living polymerization but it has some attributes of a living polymerization, e.g. rather narrow polydispersity polymers can be produced, polymer molecular weight increases linearly with monomer conversion, and sequential addition of monomers leads to block copolymers. However, no one has yet produced truly monodisperse polymer using NMRP. Therefore, there are likely side reactions going on during chain growth that lead to adventitious termination. 

In practice, SEC columns are often calibrated using linear monodisperse polystyrene standards, generating a calibration curve like Fig. 1.26. Then any linear polymer that is soluble in the same solvent, for which a Mark-Houwink equation is known, can have its molar mass determined by this calibrated SEC experiment. The elution volume of the polymer determines [q M from the calibration curve and the Mark-Houwink... 

Doi and Edwards (1978, 1979, 1986). They started with the Rouse-segmented chain model for a polymer molecule. Because of the presence of neighboring molecules, there are many places along the chain where lateral motion is restricted, as shown in Fig. 21. To simplify the representation of these restrictions, Doi and Edwards assume that they are equivalent to placing the molecule of interest in the tube as shown in Fig. 22. This tube has a diameter d and length L. The mean field is represented by a three-dimensional cage. The primitive chain can move randomly forward or backward only along itself. For a monodisperse polymer, the linear viscoelasticity is characterized by... 

Synthesis and Properties of Conjugated Poly(aryleneethynylene)s Linear Monodisperse jr-Conjugated Oligomers Model Compounds for Polymers and More Molecular Rods. 1. Simple Axial Rods... 

Dendrimers are monodisperse polymers with precisely controlled macromo-lecular architecture. Although dendrimer synthesis is complex, the lack of chain entanglements results in a lower viscosity with respect to linear polymers of the same MM. ... 

The second virial coefficient A2 of a monodisperse polymer solution is defined by eq 1-2.13. Historically, it has been one of the central subjects in polymer solution studies. Yet, the theories available at present are not self-contained for typical experimental data. In this section, we give an account highli ting the gaps remaining between theory and experiment on the A2 of solutions of linear flexible polymers. 

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