Zimm polymer dynamics

The exponent is fixed to 2/3 similar to the Zimm polymer dynamics. Within the approximations used in [75] to arrive at a closed analytical expression, the slope and the elastic constant k are connected by... 

These models are designed to reproduce the random movement of flexible polymer chains in a solvent or melt in a more or less realistic way. Simulational results which reproduce in simple cases the so-called Rouse [49] or Zimm [50] dynamics, depending on whether hydrodynamic interactions in the system are neglected or not, appear appropriate for studying diffusion, relaxation, and transport properties in general. In all dynamic models the monomers perform small displacements per unit time while the connectivity of the chains is preserved during the simulation. 

Zimm, BH, Dynamics of Polymer Molecules in Dilute Solution Viscoelasticity, Flow Birefringence and Dielectric Loss, Journal of Chemical Physics 24, 269, 1956. 

In dilute solutions, hydrodynamic interactions between the monomers in the polymer chain are strong. These hydrodynamic interactions also are strong between the monomers and the solvent within the pervaded volume of the chain. When the polymer moves, it effectively drags the solvent within its pervaded volume with it. For this reason, the best model of polymer dynamics in a dilute solution is the Zimm model, which effectively treats the pervaded volume of the chain as a solid object moving through the surrounding solvent. 

The Zimm limit applies to dilute solutions, where the solvent within the pervaded volume of the polymer is hydrodynamically coupled to the polymer. Polymer dynamics are described by-the Zimm model in dilute-... 

All in all, the Rouse model provides a reasonable description of polymer dynamics when the hydrodynamic interactions, excluded volume effects and entanglement effects can be neglected a classical example of its applicability is short-chain polymer melts. Since the Rouse model is exactly solvable for polymer chains, it represents a basic reference frame for comparison with more involved models of polymer dynamics. In particular, the decouphng of the dynamics of the Rouse chain into a set of independently relaxing normal modes is fundamental and plays an important role in other cases, such as more complex objects of study, or in other models, such as the Zimm model. 

Just as the Gaussian chain is the basic paradigm of the statistics of polymer solutions, so is its extension to the bead-spring model still basic to current work in the held of polymer dynamics. The two limiting cases of free draining (no hydrodynamic interaction between beads, characterized by the draining parameter A = 0) and non-free draining (dominant hydrodynamic interaction, A= CO, due to Rouse and Zimm, respectively, are sufficiently familiar that the approach is often known as the Rouse-Zimm model. ... 

The general expectations embodied in Equations 7.12, 7.16, and 7.19 are borne out to be valid as shown by experiments in dilute solutions of uncharged polymers. Depending on the experimental conditions, the value of the size exponent changes and this change is directly manifest in D, rj, and t in terms of their dependencies on the molecular weight of the polymer and solvent conditions. In order to obtain the numerical prefactors for the above scaling laws and to understand the internal dynamics of the polymer molecules, it is necessary to build polymer models that explicitly account for the chain connectivity. The two basic models of polymer dynamics are the Rouse and Zimm models (Rouse 1953, Kirkwood and Riseman 1948, Zimm 1956), which are discussed next. 

As the hydrodynamic interaction is screened in semidilute solutions, the molecular weight dependencies of the diffusion coefficient, the longest relaxation time, and the viscosity change in the semidilute solutions are exactly the same as in the Rouse model. However, since the Rouse model was originally designed for an isolated chain, the concentration dependencies of these quantities are not captured by the Rouse model. Nevertheless, we shall refer to the correct description of polymer dynamics in semidilute solutions as the Rouse regime. A summary of the main results for the Zimm model in dilute solutions... 

The hydrodynamic scaling model is an extension of the Kirkwood-Riseman model for polymer dynamics(l). The original model considered a single polymer molecule. It effectively treats a polymer coil as a bag of beads. For their collective coordinates, the beads have three center-of-mass translations, three rotations around the center of mass, and unspecified other coordinates. The use of rotation coordinates causes the Kirkwood-Riseman model to differ from the Rouse and Zimm models(2,3). The other collective coordinates of the Kirkwood-Riseman model are lumped as internal coordinates whose fluctuations are in first approximation ignored. The beads are linked end-to-end, the links serving to estabhsh and maintain the coil s bead density and radius of gyration. However, the spring constant of the finks only affects the time evolution of the internal coordinates it has no effect on translation or rotation of the coil as a whole. 

The dynamical behavior of macromolecules in solution is strongly affected or even dominated by hydrodynamic interactions [6,104,105]. Erom a theoretical point of view, scaling relations predicted by the Zimm model for, e.g., the dependencies of dynamical quantities on the length of the polymer are, in general, accepted and confirmed [106]. Recent advances in experimental single-molecule techniques provide insight into the dynamics of individual polymers, and raise the need for a quantitative theoretical description in order to determine molecular parameters such as diffusion coefficients and relaxation times. Mesoscale hydrodynamic simulations can be used to verify the validity of theoretical models. Even more, such simulations are especially valuable when analytical methods fail, as for more complicated molecules such as polymer brushes, stars, ultrasoft colloids, or semidilute solutions, where hydrodynamic interactions are screened to a certain degree. Here, mesoscale simulations still provide a full characterization of the polymer dynamics. 

B. Zimm. Dynamics of polymer molecules in dilute solutions viscoelasticity, low birefringence and dielectric loss. J Chem Phys 24 269-278, 1956. 

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. 

Zimm theory, multiparticle collision dynamics, polymers, 123-124... 

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 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. 

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

For small chains in solution the translational diffusion significantly contributes to the overall decay of Schain(Q>0- Therefore precise knowledge of the centre of mass diffusion is essential. Combing dynamic light scattering (DLS) and NSE revealed effective collective diffusion coefficients. Measurements at different concentrations showed that up to a polymer volume fraction of 10% no concentration dependence could be detected. All data are well below the overlap volume fraction of (p =0.23. Since no -dependence was seen, the data may be directly compared with the Zimm prediction [6] for dilute solutions ... 

Lodge, A. S., Wu, Y.-J. Exact relaxation times and dynamic functions for dilute polymer solutions from the bead/spring model of Rouse and Zimm, Report 16. Rheology Research Center, University of Wisconsin (July, 1972). 

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