The dynamic structure factor

Light scattering studies of two-component polymer solvent systems naturally partition into several categories of measurement  

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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]. 

The dynamic structure factor is S(q, t) = (nq(r) q(0)), where nq(t) = Sam e q r is the Fourier transform of the total density of the polymer beads. The Zimm model predicts that this function should scale as S(q, t) = S(q, 0)J-(qat), where IF is a scaling function. The data in Fig. 12b confirm that this scaling form is satisfied. These results show that hydrodynamic effects for polymeric systems can be investigated using MPC dynamics. 

In the ideal case being performed at X-ray energy transfers much higher than the characteristic energies of the scattering system, the impulse approximation [14] is applicable. In this case, the dynamical structure factor is directly connected with the electron momentum density p(p) ... 

Taking the photon scattering vector q in z-direction, the dynamical structure factor is related to the Compton profile J(pz) by... 

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... 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

For different momentum transfers the dynamic structure factors are predicted to collapse to one master curve, if they are represented as a function of the Rouse variable. This property is a consequence of the fact that the Rouse model does not contain any particular length scale. In addition, it should be mentioned that Z2/ or the equivalent quantity W/4 is the only adjustable parameter when Rouse dynamics are studied by NSE. 

Though the functional form of the dynamic structure factor is more complicated than that for the self-correlation function, the data again collapse on a common master curve which is described very well by Eq. (28). Obviously, this structure factor originally calculated by de Gennes, describes the neutron data well (the only parameter fit is W/4 = 3kBT/2/C) [41, 44],... 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Fig. 7. Characteristic relaxation rate for the Rouse relaxation in polyisoprene as a function of momentum transfer. The insert shows the scaling behavior of the dynamic structure factor as a function of the Rouse variable. The different symbols correspond to different Q-values. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

Under these conditions, Eq. (32) indicates the maximum extent to which a particular mode p can reduce S(Q,t) as a function of the momentum transfer Q. Figure 10 presents the Q-dependence of the mode contributions for PE of molecular weights Mw = 2000 and Mw = 4800 used in the experiments to be described later. Vertical lines mark the experimentally examined momentum transfers. Let us begin with the short chain. For the smaller Q the internal modes do not influence the dynamic structure factor. There, only the translational diffusion is observed. With increasing Q, the first mode begins to play a role. If Q is further increased, higher relaxation modes also begin to influence the... 

Fig. 10a, b. Contributions of the different modes to the relaxation of the dynamic structure factor S(Q,t)/S(Q,0) (see text) for PE of molecular masses, a Mw = 2.0 x 103 g/mol and b Mw = 4.8 x 103 g/mol. The experimental Q-values are indicated by vertical lines curves correspond to mode numbers increasing from bottom to top. (Reprinted with permission from [52]. Copyright 1993 The American Physical Society, Maryland)... 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

Following the mode analysis approach described in Section 3.2.1, the spectra at different molecular masses were fitted with Eqs. (32) and (33). Figure 13 demonstrates the contribution of different modes to the dynamic structure factor for the specimen with molecular mass 3600. Based on the parameters obtained in a common fit using Eq. (32), S(Q,t) was calculated according to an increasing number of mode contributions. 

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

Regime of creep In the time range t > tr, the chain creeps out of the tube. Doi and Edwards give a simple argument for the shape of the dynamic structure factor for the range QRG 1 [5]. For the parts of chain still in the original tube S(Q, t) = 12/Q V2 and for those parts of the chain which have already crept out of the tube all correlations have subsided and S(Q, t) = 0. It follows... 

Fig. 21. Comparison of the dynamic structure factors from long PTHF chains in a matrix of long chains (x) with that in a matrix of short chains ( ). The Q-value of the experiment was Q = 0.09 A-1, the temperature T = 418 K (Reprinted with permission from [57]. Copyright 1985 Royal Society of Chemistry, Cambridge, UK)...

Figure 31 compares the dynamic structure factors obtained from the crosslinks and the chain ends for two different Q-values. Without any analysis a strong reduction of the cross-link mobility compared to that of the chain end is obvious. A closer inspection also shows that the line-shape of both curves differs. While S(Q,t)/S(Q, 0) from the chain end decays continuously, S(Q,t) from the cross-links appears to decay faster at shorter than at longer times. This difference in line shape is quantified via the line shape parameter p. For the end-labelled chains, p is in close agreement with the p = 1/2 prediction of the... 

Comparing Eqs. (83), (84) and Eqs. (21), (22) it follows immediately that Rouse and Zimm relaxation result in completely different incoherent quasielastic scattering. These differences are revealed in the line shape of the dynamic structure factor or in the (3-parameter if Eq. (23) is applied, as well as in the structure and Q-dependence of the characteristic frequency. In the case of dominant hydrodynamic interaction, Q(Q) depends on the viscosity of the pure solvent, but on no molecular parameters and varies with the third power of Q, whereas with failing hydrodynamic interaction it is determined by the inverse of the friction per mean square segment length and varies with the fourth power of Q. 

From Fig. 35, where the normalized coherent scattering laws S(Q, t)/S(Q,0) are plotted as a function of 2 (Q)t for Zimm as well as for Rouse relaxation, one sees that hydrodynamic interaction results in a much faster decay of the dynamic structure factor. 

Since the random forces f(Q, t) at time t are not correlated with the densities p(Q, t ) at t tthe time evolution of the dynamic structure factor (90) follows immediately from (92) 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. 

Obviously, in the case of PS these discrepancies are more and more reduced if the probed dimensions, characterized by 2ti/Q, are enlarged from microscopic to macroscopic scales. Using extremely high molecular masses the internal modes can also be studied by photon correlation spectroscopy [111,112], Corresponding measurements show that - at two orders of magnitude smaller Q-values than those tested with NSE - the line shape of the spectra is also well described by the dynamic structure factor of the Zimm model (see Table 1). The characteristic frequencies QZ(Q) also vary with Q3. Flowever, their absolute values are only 10-15% below the prediction. 

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... 

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