Structure factors of a polymer

The defining property of a structural glass transition is an increase of the structural relaxation time by more than 14 orders in magnitude without the development of any long-range ordered structure.1 Both the static structure and the relaxation behavior of the static structure can be accessed by scattering experiments and they can be calculated from simulations. The collective structure factor of a polymer melt, where one sums over all scattering centers M in the system... 

Fig. 6.12 Temperature dependence of the static structure factor of a polymer 5p( ) and comparison of Sp q) with the Debye formula (eq. [6.17]). The diamonds (O) and the crosses (+) correspond to the simulation results at T = 0.05 and T = 1.8, respectively, whereas the Debye formula at the two temperatures T = 0.05 and T = 1.8 is represented by the dashed and the dotted lines. The model is the same as in Fig. 6.7.

According to Benoit and Hadziioannou [148] the partial structure factors of a f functional star polymer with diblock arms of A- B-type in the Gaussian approximation are given by... 

For K 7 0, a part of Aqo cancels 1 jft exactly in the non-free-draining limit and the remainder is dependent on the structure factor of the polymer and the size exponent v. For large values of KRg, p becomes... 

The calculation of the X-ray structure factor of distorted polymer structures should be based on a strategy of where to place approximations. As shown by the literature this is a difficult task. 

Fig. 5. Structure factor of (a) noncross-linked versus cross-linked polymer (b) an imprinted network in presence of 250 imprinting agents of different sizes.

A systematic SANS study of the PS-PA block copolymer has been achieved. The PS-PA co-polymers were obtained through thermal annealing of the diblock precursor PS-PPVS (polyphenylsulfoxide), a detailed description of the synthesis and preparation of the PA sequence free from defect can be found in references [54-55], To get a perfectly dispersed micelle structure in solution, the PA must be short and so the study was devoted to the diblock PS64-PA1 [56-58], The contrast factor of the PS unit is about 95 x 10 cm , while the contrast of the PA chain is negligible. As given in Section 1.2.2 the structure factor of a particle varies as a function of its geometrical shape. For instance, a... 

It is well known that the morphological and molecular structures of polymers play an important role in their wear behavior. It seems that the degree of crystallinity is also a structural factor of semicrystalline polymers important to their wear. Lontz et al. ( ) reported that the wear of poly(tetrafluoroethylene),(PTFE) decreased with the increase in crystallinity. Tanaka et al. (2 ) studied the wear of heat-treated PTFE specimens and concluded that the wear rate was affected by the width of the band in the fine structure rather than crystallinity. Recently, Hu et al. ( 3) have studied the effect of crystallinity on wear of PTFE using various heat-treated specimens. They have shown that the wear decreases with the increase in crystallinity, when molecular weight is constant. Eiss et al. ( ) reported that poly(chlorotrifluoroethylene) of a crystallinity of 65% exhib-ted higher wear than that of 45%. The results obtained by the authors mentioned above indicate that the effect of crystallinity on the wear of polymers is somewhat complicated and further investigation is needed to clarify the effect of crystallinity on polymer wear. 

The calculation of M depends on structural factors of a given polymer, and the result depends on the correct... 

The structure factor (or scattering function) only depends on the form factor of an isolated, noninteracting polymer chain, Soiq), the second-virial coefficient, V2, the fraction / of charged monomers, and the interaction between monomers, which in the present case is taken to be the DH potential i>dh( ) The structure factor of a noninteracting semiflexible polymer is characterized, in addition to the monomer... 

An equation fully equivalent to Eq. (A.80) can be applied in studies of polymer solutions. The limiting value for g —> 0 of the structure factor of a solution is given by... 

The surface structure factor of the polymer layer, Sppiq) can be measured directly in a contrast matching experiment by choosing rig - rig. In principle, Spp(q) depends both on the average concentration profile 4>(2) and on the concentration fluctuations within the layer.In the simple case, where the adsorbed layer can be assumed homogeneous, Spp(q) is related to the square of the Fourier transform of the profile 4>(2). 

The neutron experiments have been made at Laboratoire Leon Brillouin on the spectrometer PACE. The range of scattering vector, 10 A < q < 10 A, is chosen to probe the inner structure of the layers, ctftg > > 1. For both systems, the partial structure factors of the polymer are extracted from the difference between the signal of the polymer-coated solid and that of the bare solid, observed at three different contrasts. 

The theoretical expression (12) of the structure factor of the polymer layer, Spp(q), suggests a plot of the product Spp(q) as a function of q. This is done in Fig. (5). We observe a linear variation ... 

As for the first system, we plot the structure factor of the polymer layer Spp(q) in the representation Sppiq) as a function of (Fig. 7). Here also we obtain a straight line ... 

The SANS observations also show the interest of performing full contrast variation experiments. Indeed, the information given by the two partial structure factors of the polymer are not equivalent because of the very singular nature of the profile at close distances from the wall. The experimental data yield values of the surface excess F. They correspond roughly to one full monolayer of dense polymer. It must be stressed, however, that the chains are not adsorbed in a flat configuration on the solid surface, as demonstrated as well by the EWIF method. 

Pdh(Q) cross-term in structure factor of a partly deuterated polymer... 

Fifth, there are hints in measurements of the dynamic structure factor of a transition for polymer concentrations near 800 g/1. Konak and Brown(8) examined the slow mode of polystyrene toluene at volume fractions 0.9 and 0.8. The slow mode scaled as (diffusive) or (structural), depending on concentration and temperature, q being found at larger c and smaller T, but with a transition to diffusive behavior (melting) at larger T, especially at the smaller concentration. Koch, et al.(9), measuring VV and VH spectra of 700 and 810 g/1 polystyrene in dioxane, similarly found that (tvv) depends on q at higher temperature but not at lower temperature. 

This is an important result. It says that the problem of calculating the structure factor of a mixture of polymer chains is equivalent to the problem of calculating response functions. De Gennes was the first to use this equivalence in pol uner theories. The procedure is usually addressed as random phase approximation, abbreviated RPA, adopting a name firstly chosen by Bohm, Pines and Nozieres in a work on electron properties in metals. Equation (A.91) can be applied in two directions. If response coefficients are known. 

Enantioseparation is typically achieved as a result of the differences in interaction energies A(AG) between each enantiomer and a selector. This difference does not need to be very large, a modest A(AG) = 0.24 kcal/mol is sufficient to achieve a separation factor a of 1.5. Another mechanism of discrimination of enantiomers involves the preferential inclusion of one into a cavity or within the helical structure of a polymer. The selectivity of a selector is most often expressed in terms of retention of both enantiomers using the separation factor a that is defined as ... 

Chain length is another factor closely related to the structural characterization of conducting polymers. The importance of this parameter lies in its considerable influence on the electric as well as the electrochemical properties of conducting polymers. However, the molecular weight techniques normally used in polymer chemistry cannot be employed on account of the extreme insolubility of the materials. A comparison between spectroscopic findings (XPS, UPS, EES) for PPy and model calculations has led some researchers to conclude that 10 is the minimum number of monomeric units in a PPy chain, with the maximum within one order of magnitude n9- 27,i28) mechanical qualities of the electropolymerized films,... 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

In contrast to -conditions a large number of NSE results have been published for polymers in dilute good solvents [16,110,115-120]. For this case the theoretical coherent dynamic structure factor of the Zimm model is not available. However, the experimental spectra are quite well described by that derived for -conditions. For example, see Fig. 42a and 42b, where the spectra S(Q, t)/S(Q,0) for the system PS/d-toluene at 373 K are shown as a function of time t and of the scaling variable (Oz(Q)t)2/3. As in Fig. 40a, the solid lines in Fig. 42a result from a common fit with a single adjustable parameter. No contribution of Rouse dynamics, leading to a dynamic structure factor of combined Rouse-Zimm relaxation (see Table 1), can be detected in the spectra. Obviously, the line shape of the spectra is not influenced by the quality of the solvent. As before, the characteristic frequencies 2(Q) follow the Q3-power law, which is... 

Many of the properties of a polymer depend upon the presence or absence of crystallites. The factors that determine whether crystallinity occurs are known (see Chapter 2) and depend on the chemical structure of the polymer chain, e.g., chain mobility, tacticity, regularity and side-chain volume. Although polymers may satisfy the above requirements, other factors determine the morphology and size of crystallites. These include the rate of cooling from the melt to solid, stress and orientation applied during processing, impurities (catalyst and solvent residues), latent crystallites which have not melted (this is called self-nucleation). 

Authors concluded, that although the swelling degree is not directly connected with molecular characteristics of absorbed liquids, however determining factor is their parameter of solubility in spite of the fact that at detailed consideration of the dependencies Q =f(3) (or f(32)) there are a number of deviations (as same as in the work [5]) from the ideal curve for many solvents. It is necessary to notify that although it is hard to estimate the verisimilitude of determined in such a way molecular weights of structural links of a coal between the points of cross bonds, however, in a case of synthetic polymers in a same way determined masses of links visible don t agree with the values obtained in accordance with others methods. 

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