Dynamic structure factors solutions

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. 

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]. 

For non-interacting, incompressible polymer systems the dynamic structure factors of Eq. (3) may be significantly simplified. The sums, which in Eq. (3) have to be carried out over all atoms or in the small Q limit over all monomers and solvent molecules in the sample, may be restricted to only one average chain yielding so-called form factors. With the exception of semi-dilute solutions in the following, we will always use this restriction. Thus, S(Q, t) and Sinc(Q, t) will be understood as dynamic structure factors of single chains. Under these circumstances the normalized, so-called macroscopic coherent cross section (scattering per unit volume) follows as... 

Fig. 40a, b. NSE spectra of a dilute solution under 0-conditions (PDMS/ d-bromobenzene, T = = 357 K). a S(Q,t)/S(Q,0) vs time t b S(Q,t)/S(Q,0) as a function of the Zimm scaling variable ( t(Q)t)2/3. The solid lines result from fitting the dynamic structure factor of the Zimm model (s. Tablet) simultaneously to all experimental data using T/r s as adjustable parameter. 

The crossover from 0- to good solvent conditions in the internal relaxation of dilute solutions was investigated by NSE on PS/d-cyclohexane (0 = 311 K) [115] and on PDMS/d-bromobenzene(0 = 357K) [110]. In Fig. 45 the characteristic frequencies Qred(Q,x) (113) are shown as a function of t = (T — 0)/0. The QZ(Q, t) were determined by fitting the theoretical dynamic structure factor S(Q, t)/S(Q,0) of the Zimm model (see Table 1) to the experimental data. This procedure is justified since the line shape of the calculated coherent dynamic structure factor provides a good description of the measured NSE-spectra under 0- as well as under good solvent conditions. 

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. 

Figure 61 presents the Q(Q)/Q2 relaxation rates, obtained from a fit with the dynamic structure factor of the Zimm model, as a function of Q. For both dilute solutions (c = 0.02 and c = 0.05) Q(Q) Q3 is found in the whole Q-range of the experiment. With increasing concentrations a transition from Q3 to... 

Now, the decay rate of the incoherent dynamic structure factor is proportional to fc(i+2v)/v xherefore, the -dependence of the decay rate for salt-free solutions is independent of whether the hydrodynamic interaction is present or not. 

Fig. 4.20 Temperature dependence of the average relaxation times of PIB results from rheological measurements [34] dashed-dotted line), the structural relaxation as measured by NSE at Qmax (empty circle [125] and empty square), the collective time at 0.4 A empty triangle), the time corresponding to the self-motion at Q ax empty diamond),NMR dotted line [136]), and the application of the Allegra and Ganazzoli model to the single chain dynamic structure factor in the bulk (filled triangle) and in solution (filled diamond) [186]. Solid lines show Arrhenius fitting curves. Dashed line is the extrapolation of the Arrhenius-like dependence of the -relaxation as observed by dielectric spectroscopy [125]. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

Recently a very detailed study on the single chain dynamic structure factor of short chain PIB (M =3870) melts was undertaken with the aim to identify the leading effects limiting the applicability of the Rouse model toward short length scales [217]. This study was later followed by experiments on PDMS (M =6460), a polymer that has very low rotational barriers [219]. Finally, in order to access directly the intrachain relaxation mechanism experiments comparing PDMS and PIB in solution were also carried out [186]. The structural parameters for both chains were virtually identical, Rg=19.2 (21.3 A). Also their characteristic ratios C =6.73 (6.19) are very similar, i.e. the polymers have nearly equal contour length L and identical persistence lengths, thus their conformation are the same. The rotational barriers on the other hand are 3-3.5 kcal/mol for PIB and about 0.1 kcal/mol for PDMS. We first describe in some detail the study on the PIB melt compared with the PDMS melt and then discuss the results. 

Fig. 5.8 Chain dynamic structure factor of PDMS empty symbols) and PIB full symbols) in toluene solution at 300 K (a) and 378 K (b). The corresponding Q-values [A ] are indicated. Lines through the points represent the single exponential fits, which describe the data well. (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

The dynamic structure factor of block copolymer liquids (melts and solutions) has been accounted for using dynamical mean-field theory by Benmouna et al. (1987a,6). For a block copolymer melt, the dynamic structure factor can be written (Stepanek and Lodge 1996)... 

The solute dynamic variables required to calculate the density and current contribution are the self-dynamic structure factor, Fs(q, t) and inertial part of the self-intermediate structure factor, Fs0(q, t). Fs0(q, f) is given by... 

The long-time tail in the solvent dynamic structure factor is responsible for the large value of the viscosity in supercooled liquid. The solute dynamics for smaller solutes are faster than that of the solvent. Thus, the solute dynamics is decoupled from this long-time tail of the solvent dynamic structure factor. 

The other dynamic variables required to calculate Rpp(t) and Rrr(t) are the dynamic structure factor of the solvent, F(q, t), the inertial part of the dynamic structure factor, Fo(q, t), the transverse current autocorrelation function of the solvent, C (q,t), the inertial part of the same, Ctf0(q, t), the self-dynamic structure factor of the solute, Fs(q, t), and the inertial part of the self-dynamic structure factor of the solute, Fs0(q,t). The expressions for all the above-mentioned dynamic quantities are similar to that given in Section IX but in two dimensions. 

The self-dynamic structure factor of the solute at all time is calculated from the mean square displacement (MSD) using the following definition ... 

The aqueous LiCl solution with the composition of LiC1.6.0H2O was prepared in the same way as described for the X-ray samples. In the aqueous solution the proton ( H) has a very large incoherent scattering cross section the observed differential scattering cross section can be approximated to the incoherent dynamic structure factor through... 

T. Maeda, Matrix representation of the dynamic structure factor of a solution of rodlike molecules in the isotropic phase, Macromolecules, 23 (1990) 1464-1474. 

Figure 12. Half-peak time ti,2 of coherent dynamic structure factor for atactic polystyrene plotted vs. Q. Left, CgD solution right, CS2 solution continuous lines, calculated results at T = 30 °C dashed line, at F = 70 °C. Experimental points from ref. 14. [Model assumptions and parameters same as in Figures 10 and 11, tB = 0.012.] (Reprinted with permission from ref. 14, Copyright 1984, American Chemical Society.)...

Dynamic modes of semidilute solutions can also be studied using dynamic scattering. The dynamic structure factors S(q, t) of these more complicated systems still have a simple diffusive behaviour at low values of... 

Relatively few theoretical studies have been devoted to the conformational characteristics of nonlinear block copolymers in different solvent environments. Burchard and coworkers [284] studied theoretically the behavior of the static and dynamic structure factors for regular star-block copolymers in dilute solutions. They considered different cases where the refractive index (n)s of the solvent takes certain values with respect to the corresponding refractive indices of the inner and outer blocks. A different dependence of the ratios... 

On the other hand, Doi and Onuki (Doi and Onuki 1992) proposed both a dynamic structure factor S(q,t) and a time-dependent modulus G(t), considering the dynamic coupling between stress and composition in polymer solutions and blends ... 

Dynamic Structure Factor and Mean Square Displacement Here we learn how gi(T) obtained in DLS gives an estimate of the diffusion coefficient. We are concerned with dilute solutions here. Hence, gi(T) = Si(k, t). 

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