Collective Dynamics

In our example S(Q) (see Eq. 6.21) would be the Leibler structure factor (Eq. 6.12) and Aj(Q) describes the collective dynamics of the diblock copolymer melt. For small momentum transfers Aj(Q) is given by  

At large Q quantitative agreement between experiment and RPA prediction is found. Here the first cumulant is proportional to - we are in the Rouse 

In the low Q-regime RPA describes well the static structure factor for the short chain melt, where the ODT is sufficiently far away (kN 7). In the dynamics we would expect the diblock breathing mode to take over around QRg 2 (Q=0.04 A ). Instead, deviations from Rouse dynamics are already observed at Q values as high as QR =5. At QJ g=3 a crossover to a virtually Q-independent relaxation rate about four to five times faster than the predicted breathing mode is found. This phenomenon is only visible under h-d labelUng. Under single chain contrast (see below) these deviations from RPA are not seen. Thus, the observed fast relaxation mode must be associated with the block contrast. 

Similarly, the results on dPEP-hPEE already exhibit deviations from the RPA-Rouse regime around QRg=7 (T=533 K). Thereby, the experimentally observed rates differ from the predicted rates by up to more than one order of magnitude (see Fig. 6.11). At the phase transition temperature (Todt=473 K), however, the effect is greatly reduced and a general slowing down of the fluctuations is found, though some deviations persist. 

The single chain dynamics of one given block or of one chain in a diblock copolymer melt is observed if a matched deuterated diblock is mixed with a small amount of labelled diblocks, where the label could be a protonated a or b block or a protonated chain. In terms of the dynamic RPA such a system is a four-component polymer mixture. It is characterized by four different relaxation modes A1-A4 which - depending on the contrast conditions - appear with 

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The fundamental concept of AL is emergence, or the appearance of higher-level properties and behaviors of a system that - while obviously originating from the collective dynamics of that system s components - are neither to be found in nor are directly deducible from the lower-level properties of that system. Emergent properties are properties of the whole that are not possessed by any of the individual parts making up that whole an air molecule is not a tornado and a neuron is not conscious. 

Yakubov, G. E., Loppinet, B., Zhang, H., Ruhe, J., Sigel, R. and Fytas, G. (2004) Collective dynamics of an end-grafted polymer brush in solvents of varying quality. Phys. Rev. Lett., 92, 115501. 

Since both the temperature dependence of the characteristic ratio and that of the density are known, the prediction of the scaling model for the temperature dependence of the tube diameter can be calculated using Eq. (53) the exponent a = 2.2 is known from the measurement of the -dependence. The solid line in Fig. 30 represents this prediction. The predicted temperature coefficient 0.67 + 0.1 x 10-3 K-1 differs from the measured value of 1.2 + 0.1 x 10-3 K-1. The discrepancy between the two values appears to be beyond the error bounds. Apparently, the scaling model, which covers only geometrical relations, is not in a position to simultaneously describe the dependences of the entanglement distance on the volume fraction or the flexibility. This may suggest that collective dynamic processes could also be responsible for the formation of the localization tube in addition to the purely geometric interactions. 

NSE Results from Semi-Dilute Solutions of Linear Homopolymers Transition from Single Chain to Collective Dynamics... 

Using and S, the translational friction coefficient and the electrophoretic mobility can be calculated by simply replacing G by in the respective formulas. Before we proceed to do this, we present the collective dynamics of monomer density and counterions. 

Apart from the further refinements of the BO approach, there has been a continuing interest in theoretically describing molecular systems with a method that treats the motions of both nuclei and electrons equivalently. This type of methodology has to entirely depart from the PES concept. It is particularly interesting how this type of approach describes the conventional notions of the molecular and electronic stmctures. In particular, the concept of chemical bonding, which at the BO level is an electronic phenomenon, has to be described in an approach departing from the BO approximation, as an effect derived from collective dynamical behavior of both electrons and nuclei. 

Then we address the dynamics of diblock copolymer melts. There we discuss the single chain dynamics, the collective dynamics as well as the dynamics of the interfaces in microphase separated systems. The next degree of complication is reached when we discuss the dynamic of gels (Chap. 6.3) and that of polymer aggregates like micelles or polymers with complex architecture such as stars and dendrimers. Chapter 6.5 addresses the first measurements on a rubbery electrolyte. Some new results on polymer solutions are discussed in Chap. 6.6 with particular emphasis on theta solvents and hydrodynamic screening. Chapter 6.7 finally addresses experiments that have been performed on biological macromolecules. 

Fig. 5.25 Result of applying shift factors corresponding to an activation energy of 0.43 eV to the relaxation times observed for the collective dynamics (empty symbol) and the self-correlation (full symbol) of PIB 335 K (circle), 365 K (square), and 390 K (triangles) (reference temperature 365 K). The dotted line through the self-correlation data shows the dependence implying Gaussian behaviour (Reprinted with permission from [147]. Copyright 2002 The American Physical Society)...

The relation between the dynamics of a diblock copolymer and that of the homopolymers composing the diblock chains. Is it possible to understand the single chain and collective dynamics of A-B diblock copolymer chains from the dynamics of the homopolymers A and B ... 

A detailed test of dynamic RPA requires a combination of static and dynamic neutron experiments involving investigations of the respective component dynamics. All three issues addressed in the introduction were investigated. We will start with the collective dynamics of a diblock copolymer and then address the single component dynamics. Finally, we will discuss aspects of the interface dynamics. The experiments were performed on diblock copolymers of the PE-PEE and PE-PEP type. In order to access the different dynamics a series of materials with different h-d labelUng was employed (see Table 6.1). 

The combination of careful chemical synthesis with NSE and SANS experiments sheds some light on the fast relaxation processes observed in the collective dynamics of block copolymers melts. The results reveal the existence of an important driving force acting on the junction points at and even well above the ODT. Modelling the surface forces by an expression for the surface tension, it was possible to describe the NSE spectra consistently. The experimental surface tension agrees reasonably well with the Helfand predictions, which are strictly valid only in the strong-segregation hmit. Beyond that, these data are a first example for NSE experiments on the interface dynamics in a bulk polymer system. 

Note that the MCT treatment presented above is quite general and can be extended to describe relaxation in many different systems, such as orientational relaxation in dipolar liquids [54]. This approach can also be extended to multicomponent systems, in particular to describe transport properties of electrolyte solutions [55]. The usefulness and the simplicity of the expressions lies in the separation between the single particle and collective dynamics (as in Eq. 98). Actually this sepration allows one to make connections with hydrodynamic (or continuum frameowrk) models where only the collective dynamics is included but the single particle motion is ignored. However, the same separation is also the reason for the failure... 

S(q, co). As stated earlier, such a slow tail in F(q, t) can significantly increase the friction. In calculating the friction, the effect of the inhomogeneity on the static and dynamic correlations is to be considered only for intermediate and long wavenumbers. The small wavevector limit probes the collective dynamics of the solvent, and hence it is the average friction which contributes in this region. It is fair to assume that qa 1 separates small from intermediate wavenumber regime. 

Watts D.)., Strogatz S.H. Collective dynamics of small-world networks. Nature, 1998, 393,440-442. 

Levesque et al. [132] also arrived at the conclusion that the AT potential increases the pressure. In addition, they shown that the local structure is not reinforced by the AT potential and that the collective dynamical properties are affected only in a minor way. 

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