Rouse friction

When the slip velocity is further increased, the Rouse friction [148] finally becomes dominant, for Vs>V ocN l. A linear friction regime is then recovered, with a constant extrapolation length, b much larger than b0 and comparable to what would be observed on an ideal surface without anchored chains [139]. 

Prove that the average Rouse friction of the constraint release process at time t due to short A-mers with N [Pg.418]

The explanation proposed by Ngai and Plazek [1990], was based on the postulate that the number of couplings between the macromolecules varies with concentration and temperature of the blend. The number of couplings, n, can be calculated from the shift factor, a. = [ (T)/ (T )] < ">, where ( g(T) is the Rouse friction coefficient. Thus, in miscible, single phase systems, as either the concentration or temperature changes, the chain mobility changes and relaxation spectra of polymeric components in the blends show different temperature dependence, i.e., the t-T principle cannot be obeyed. Similar conclusions were reached from a postulate that the deviation originates from different temperature dependence of the relaxation functions of the blend components [Booij and Palmen, 1992]. 

Now consider the case of flexible polyelectrolytes such as DNA. There are two dominant hydrodynamic models for polymer motion, the Rouse model and the Zimm model [4]. In the Rouse model, the different parts of the chain are assumed to be hydrodynamically independent, so that the total friction of the chain is simply the sum of the friction of each segment The Rouse model is also referred to as freely draining, since the polymer chain appears to be hoUow to the fluid. The Zimm model includes both the Rouse friction of the individual segments with the fluid and the hydrodynamic interactions between different segments. The Zimm model is not freely draining the polymer appears as a solid object to the surrounding fluid. 

Relaxation or residence time, and characteristic relaxation time Angular frequency Rouse friction coefficient... 

Let us now consider an interface A/ B between two molten polymers, with 1 > X K X is large, there will be no entanglements between A and B, and slippage may occur. This problem was first considered by Furukawa ( 8) -but without a full appreciation of the role of entanglements versus Rouse friction. A slightly improved (qualitative) discussion is given in ref. (29) and ill be summarized here. 

Then the parameters of the FENE potential of the bonded monomers and the parameters and Va are chosen in a way that a melt of such chains had the same (R ), R ), pressure and Rouse friction as the LJ melt. For p = 0.S5cr the parameters are = 2.22e,= l.Ocr and Ro = 1.75cr (in the FENE Potential). The amplitude of the FENE was also reduced by a factor of 0.175. Monomers can cross each other with a penalty of a few ksT. Since the macroscopic properties are not altered, these two potentials can be used to identify the contributions from the noncrossability of the chains. The general monomer packing constraints are only weakly affected, as the pressure and the temperature remain the same. 

As only a small proportion of the material is in contact with the roUs and friction on the rollers is low, hard materials can be processed with tittle wear. The high pressure action creates a slab of ultrafine particles which usually requires a low speed impact milting system to disagglomerate. Used in closed circuit with such a disagglomerator and an air classifier, such machines can reduce the energy requirement for fine grinding many minerals. 

Vanes may be used to improve velocity distribution and reduce frictional loss in bends, when the ratio of bend turning radius to pipe diameter is less than 1.0. For a miter bend with low-velocity flows, simple circular arcs (Fig. 6-37) can be used, and with high-velocity flows, vanes of special airfoil shapes are required. For additional details and references, see Ower and Pankhurst The Mea.surement of Air Flow, Pergamon, New York, 1977, p. 102) Pankhurst and Holder Wind-Tunnel Technique, Pitman, London, 1952, pp. 92-93) Rouse Engineering Hydraulics, Wiley, New York, 1950, pp. 399 01) and Joreensen Fan Engineerinp, 7th ed., Buffalo Forge Co., Buffalo, 1970, pp. Ill, 117, 118). 

Rouse motion has been best documented for PDMS [38-44], a polymer with little entanglement constraints, high flexibility and low monomeric friction. For this polymer NSE experiments were carried out at T = 100 °C to study both the self- and pair-correlation function. 

Rouse behavior observed on PI homopolymer melts has to be modified if the labelled (protonated) PI species are replaced by diblock copolymers of proto-nated PI and deuterated polystyrene (PS) [46]. The characteristic frequency Q(Q) is slowed down considerably due to the presence of the non-vanishing X-parameter. Thus, the reduction is stronger at smaller Q-values or at larger length scales than in the opposite case. In addition, as a minor effect, Q(Q) becomes dependent on both friction coefficients per mean square monomer length, //2, valid for PI and for PS. 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

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. 

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

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. 

To guarantee the transformational invariance of DG = kBT/(NQ (see Table 4) in the case of Rouse relaxation, the replacement of N by N/A, requires the simultaneous replacement of the segmental friction coefficient by X which is natural, since friction is proportional to the number of segments involved. 

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