Homopolymer structure factors

Fig. 24 NSE spectra from the PEP homopolymer (above) and the triblock (below) at 492 K. The solid lines are the result of a fit with the Ronca model [49] the dashed line presents the expected dynamic structure factor for Rouse relaxation corresponding to the highest Q-value. ( Q = 0.058 A"1 V Q = 0.068 A"1 Q = 0.078 A"1 A Q = 0.097 A"1 o Q = 0.116 A"1 Q = 0.116A 1). The arrows mark the crossover time xe. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

The denominator in Eq. (B.25) is recognized as the static scattering function, and the numerator is called the dynamic structure factor S (q, t) which for a homopolymer is given as... 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

We assume, now, that the three component blend considered in the previous section consists of a copolymer A/B (could be a diblock, triblock, etc, or an alternating copolymer) and a homopolymer C [11-15]. The notation and formalism of the previous section hold but now XAB(Q) + 0 (note that Xab(Q) shows a peak in the scattering function). The partial structure factors become ... 

Among the structural factors that should be controlled in polymer syntheses (Fig. 1, Section I), perhaps the least exploited is the sequence of constitutional repeat units along a polymer main chain. We have already discussed the syntheses of block copolymers, where two or more homopolymer segments are connected, such as AAAAA-BBBBB- -, which is among the most primitive examples of sequence control in synthetic polymers. 

Fig 13a-c. Time evolution of structure factor during the phase separation for a a blend without block copolymer b blend with block copolymers of attractive interaction (A05f5) c blend with block copolymers of repulsive interaction (Rllf5) between the blocks and the homopolymers [70]. The total chain length of block copolymers is ATblock = 12... 

The special situation of a homogeneous saddle point, corresponding to a homogeneous disordered phase, is particularly interesting. In that case, ex-phdt analytical relations between the single chain partition function and the fields can be obtained, and one recovers the well-known random phase approximation (RPA). To illustrate this approach, we shall now derive the RPA structure factor for the case of a symmetric binary homopolymer blend. 

A symmetric binary blend is defined as consisting of two types of homopolymer chains, A and B, which are structurally identical in every respect. Hence, the corresponding single chain structure factors obey the identity (OaW = cobW = )(r). The pair potentials between sites are ... 

The same molecular closures proposed for homopolymer blends [68-70] apply to copolymers but the intramolecular structure factor matrix is now non-diagonal. Equation (8.8) becomes [86]... 

The structurally simplest polymer, and one of the most commercially important, is polyethylene. It consists of a linear chain of CH2 units, which we model as single spherical sites in the single-site homopolymer spirit. There exist well-developed ideal rotational isomeric state chain models where the bond rotational degrees of freedom are represented as discrete trans and gauche isomers. Numerical calculation of the required single-chain structure factor can be achieved via Monte Carlo simulation or using the recently developed computationally convenient approximate methods of McCoy and co-workers. ... 

The principal mechanical properties of PP homopolymers are good rigidity and high thermal resistance, with limited impact resistance at low temperature. The main structural factors affecting these properties are isotacticity, molecular weight, and MW distribution, mostly through their infiuence on crystallinity. 

Here we have the situation that one of the (A/B) monomers of the homopolymer and of the copolymer was always deuterated with the same relative amount of deuterium. Under such conditions the structure factor S(Q) measures thermal composition fluctuations with respect to the total monomer fraction, which corresponds to a scalar order parameter represented by the local concentration =

basic thermodynamic features of those systems near their binodal line are well described by the common Landau expansion of the free energy according to... 

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. 

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