Center-of-mass diffusion

It appears that all those data perfectly corroborate the validity of limit (IV)de of the tube/reptation model. However, there are four reasons why the molecular weight dependence observed in experiments may be modified by factors other than the tube constraints  

Free volume connected with chain-end blocks influences the molecular weight dependence of chain dynamics in the vicinity of Me- An apparently steeper molecular weight dependence results on these grounds particularly below Me- The failure of NMR techniques to directly render the molecular 

At very high molecular weights, when the diffusion coefficient due to flip-flop spin diffusion of the order 10 m /s is getting comparable to the ordinary Brownian self-diffusion coefficient, a flatter chain length dependence of the effective diffusion coefficient is expected [10]. 

In view of the contour-length fluctuation mechanism suggested by Doi as an explanation of the fractional exponent of the power law for the zero-shear viscosity [142], a molecular weight dependence somewhat stronger 

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Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

In Fig. 20 we show the MSQ of a system of GM [66] with different mean chain lengths (depending on 7, cf. Eq. (12)) for three values of LO=l, 0.1, 0. 01. Since the individual chains have only transient identity, it is meaningless to discuss their center of mass diffusion. It is evident from Fig. 20 that the MSQ of the segments, g t) = ([x( ) - x(O)j ), follows an intermediate sub-diffusive regime, g(t) oc which is later replaced by conventional diffusion at some characteristic crossover time which grows... 

Most in depth studies of termination deal only with the low conversion regime. Logic dictates that simple center of mass diffusion and overall chain movement by reptation or many other mechanisms will be chain length dependent. At any instant, the overall rate coefficient for termination can be expressed as a weighted average of individual chain length dependent rate coefficients (eq. 20) 39... 

Mahabadi and O Driscolm considered that segmental motion and center of mass diffusion should be the dominant mechanisms at low conversion. They analyzed data for various polymerizations and proposed that k, J should be dependent on chain length such that the overall rale constant obeys the expression ... 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

For small angles (QRe 1), the second and third terms in (24) are negligibly small and S(Q,t) describes the center-of-mass diffusion of the coil... 

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. 

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... 

Within the framework of the Rouse model the characteristic frequency for the center of mass diffusion follows the equation... 

Particular attention was placed on the crossover from segmental diffusion to the center of mass diffusion at Q 1/Rg and to the monomer diffusion at Q /i, respectively, by Higgins and coworkers [119,120]. While the transition at small Q is very sharp (see Fig. 43, right side), a broader transition range is observed in the regime of larger Q, where the details of the monomer structure become important (see Fig. 44). The experimental data clearly show that only in the case of PDMS does the range 2(Q) Q3 exceed Q = 0.1 A-1, whereas in the case of PS and polytetrahydrofurane (PTHF) it ends at about Q = 0.06-0.07 A-1. Thus, the experimental Q-window to study the internal dynamics of these polymers by NSE is rather limited. 

Chain and ring macromolecules are topologically distinct. Thus it is not surprising that many differences in their microscopic properties are observed [127], Besides many other experimental techniques, which were applied to specify these differences, NSE was used to compare the center of mass diffusion and the internal relaxation of linear and cyclic PDMS systems in dilute solutions under good solvent conditions [120,128,129]. An important parameter for these investigations was the molecular mass, which was varied between 800 and 15400 g/mol and which was almost identical for the corresponding linear (L) and ring (R) systems. 

For simple center-of-mass diffusive motion, the mean decay rate F can be shown to be D(f, where D is the intensity-weighted average D). The use of D in the Stokes-Einstein equation, Eq. (10), yields an effective radius R. The second moment /t2 is given by... 

Luedtke K, Jordan R, Furr N, Garg S, Forsythe K, Naumann CA (2008) Two-dimensional center-of-mass diffusion of lipid-tethered poly(2-methyl-2-oxazoline) at the air-water interface studied at the single molecule level. Langmuir 24 5580-5584... 

Dsim Modified (center-of-mass) diffusivity in a single-file system... 

Table 1 Results of the best fit Dsim and N and the center-of-mass diffusivity Dgff and site number N in the final time domain of tracer exchange (controlled by center-of-mass diffusion) [57]...

Termination dominated by center-of-mass diffusion of shorter radicals... 

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