Renormalized Rouse Models

The preaveraging approximation given in Eq. 107 is again used. The memory matrix then becomes a scalar relative to space rotations, and does not depend on the momentary configuration of the tagged chain  

The general equation of motion (Eq. 109) becomes a linear function of the tagged chain variables, and the system is considered to be isotropic. 

Intermolecular interactions determining the force /(r) with which a segment from a chain acts on a segment from another one are approximated by a hard sphere interaction with a certain diameter d comparable with the Kuhn segment length h. The entanglement effects or the topological interactions are the consequence of chain connectivity and excluded-volume intermolecular interactions. 

The matrix density correlation function (0,r )5p (r,r)) for projected dynamics is approximated as a fc-space integral (for a detailed argument see Ref. [98]), 

The intrachain dynamical structure factor for projected dynamics is defined by 

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Keywords NMR Relaxation Diffusion Rouse model Reptation Renormalized Rouse model... 

Renormalization in this context means an attempt to find a physically plausible ansatz for the unknown memory matrix given by Eq. 106. In principle one could postulate a power law with an exponent being a fitting parameter to experimental data after having derived expressions of observables on this basis. However, in the frame of the renormalized Rouse models (RRM) a somewhat less formal and less phenomenological approach is possible. 

The essence of the first and second renormalization ansatzes to be described in the following refers to heuristic replacements of the mean squared segment displacement for projected dynamics, (R (())q. In the (once) renormalized Rouse model this unknown function is replaced by the result of the ordinary Rouse model, given in Eq. 58 [98]. In the... 

Table 2. Theoretical dependences on time (f), angular frequency (

Below it will be shown that field-cycling NMR relaxometry studies unambiguously reveal a crossover between high-frequency and low-frequency dispersion regimes that can be identified with the high-mode-number and low-mode-number limits of the renormalized Rouse models. Moreover, the variation of the power law exponents closely corresponds to that predicted by the renormalized Rouse models. These dynamic regimes cannot be ex-... 

The appearance of dispersion regions I and II in experiments is an exultant confirmation of the high- and low-mode-number, short-time limits predicted by the twice renormalized Rouse models (Tables 2 and 3). The exponents of the power laws suggested by the experimental data even match the theoretical predictions almost perfectly [47, 49]. Nevertheless, the good coincidence of the numerical values of these exponents is not considered to be the decisive finding backing up the renormalized Rouse models. The problem is that the theoretical exponents are slightly affected by the renormalization... 

The time dependence of the mean squared segment displacement in the laboratory frame was derived on the relevant time scale as low-mode-number, short-time limits of the renormalized and twice renormalized Rouse models as and (R ,(f)) oc (limit (II)trr)> respectively... 

Based on the generalized Langevin equation, the renormalized Rouse models suggest dynamic high- and low-mode-number limits as an implicit structural feature of this equation of motion. This is a stand-alone prediction of paramount importance independent of any absolute values of power law exponents that arise and are measured in the formalism and in experiment, respectively. The two limits manifesting themselves as power law spin-lattice relaxation dispersions were clearly identified in bulk melts of entangled polymers of diverse chemical species. 

However, the inclusion of the long-range correlations places the Zimm model into a different dynamic universality class than the Rouse model. Although there exist sophisticated renormalization- oup... 

Apart from the introductory section, the article is subdivided into four major sections NMR Methods Modeling of Chain Dynamics and Predictions for NMR Measurands Experimental Studies of Bulk Melts, Networks, and Concentrated Solutions and Chain Dynamics in Pores. First, the NMR techniques of interest in this context will be described. Second, the three fundamental polymer dynamics theories, namely the Rouse model, the tube/reptation model, and the renormalized Rouse theories are considered. The immense experimental NMR data available in the literature will be classified and described in the next section, where reference will be made to the model theories wherever possible. Finally, recent experiments, analytical treatments, and Monte Carlo simulations of polymer chains confined in pores mimicking the basic premiss of the tube/reptation model are discussed. 

Three basic model theories have been considered the Rouse model, the tube/reptation concept, and the renormalized Rouse formalism. Depending on the sample system and the experimental conduct, characteristic features of all these theories have been shown. Some spectacular predictions of theories were verified, others were ruled out. 

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