Zimm function

The form factor term, P(q), contains information on the distribution of segments within a single dendrimer. Models can be used to fit the scattering from various types of particles, common ones being a Zimm function which describes scattering from a collection of units with a Gaussian distribution (equation (3a)), a... 

Tanaka and Sole used both Schulz-Zimm and logarithmic-normaF distributions to calculate the effects of polydispersity on second virial coefficients for the Fade approximant expression. For the Schulz-Zimm function, A 2 /A increases from unity with increasing dispersion, as in the case of hard sphere interactions, but it approaches a finite limit and the ratio of. 4 2 to A2 for the monodisperse species matching M passes through a maximum with increasing dispersion and may then become less than unity. The trends exhibited in these results suggest that careful conventional fractionation procedures applied to typical noncrystalline polymers should diminish the effects of polydispersity on the second virial coefficient to the extent that a solution of a polymer fraction can safely be treated as a binary system. 

The inverse scattering function of dilute polymer solutions for small scattering wavenumber qR- 1) obeys tire so-called Zimm relation [22] ... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

To generate the necessary distribution functions, the ratio of is used to approximate the true molecular weight distribution by a Schulz-Zimm distribution. It is also assumed that the reactive functional groups are distributed randomly on the polymer chain. The Schulz-Zimm parameters used to calculate distribution functions and probability generating functions (see below) are defined as follows ... 

Equations 22 and 23 can be solved numerically using the method described in Ref. 5. For oligomers, the probability generating functions are calculated by the appropriate sums. For random copolymers analytical expressions for and t can be written for a polymer or crosslinker using the appropriate Schulz-Zimm parameters (5) ... 

These parameters are used calculate the site and mass distribution functions assuming a Schulz-Zimm molecular weight distribution. The Schulz-Zimm parameters are calculated in lines 930-950. The weight fraction of diluent (as a fraction of the amount of polymer) is then sought. If there is no diluent enter 0. If there is a diluent, the functionality and molecular weight of the diluent is requested (line 1040). The necessary expectation values are computed (lines 1060-1150). 

The potential energy function prohibits double occupancy of any site on the 2nnd lattice. In the initial formulation, which was designed for the simulation of infinitely dilute chains in a structureless medium that behaves as a solvent, the remaining part of the potential energy function contains a finite repulsion for sites that are one lattice unit apart, and a finite attraction for sites that are two lattice units apart [153]. The finite interaction energies for these two types of sites were obtained by generalizing the lattice formulation of the second virial coefficient, B2, described by Post and Zimm as [159] ... 

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. 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

The coherent structure factor of the Zimm model can be calculated [34] following the lines outlined in detail in connection with the Rouse model. As the incoherent structure factor (83), it is also a function of the scaling variable ( fz(Q)t)2/3 and has the form... 

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. 

On the basis of scaling arguments, general functional dependencies can also be derived. For example, dimensional analysis shows that the center of mass diffusion coefficient DG for Zimm relaxation has the form... 

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. 

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... 

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. 

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... 

Fig. 64. Single-chain behavior in semi-dilute PDMS/d-chlorbenzene solutions. Line-shape parameter (3 as a function of Q at the concentration c = 0.18 and c = 0.45, indicating the occurance of two crossover effects, as predicted by the concept of incompletely screened hydrodynamic interactions. (----), (---) asymptotic Zimm and Rouse behavior, respectively. (Reprinted with per-...

In Figure 2 the scattering function l(K) is plotted against the wave vector K for PS dispersions containing PS(D) blocks in n-heptane. These data may be analysed by the Zimm treatment (7.) according to Equation U... 

The complex viscosity, i.e., the viscosity observed in the presence of an oscillatory shear rate, is a dynamic property that can be straightforwardly obtained from the Rouse, or Rouse-Zimm theory as the Fourier transform of the stress time-correlation function. Thus, these theories give [15]... 

Fig. 18. Stress time-correlation function of EV linear chains and stars with different functionalities. Comparison of Brownian dynamics (crosses) and generalized Zimm calculations from MC averages (solid lines). Reprinted with permission from [89]. Copyright (1996) American Institute of Physics...

Equation (29) was previously derived by Zimm and Stockmayer [49] who used another technique. Figure 12 shows a plot of the theoretical increase of the radii of the stars as a function of the number of arms. 

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