Center-of-mass self-diffusion coefficient

Figure 9 Chain center of mass self-diffusion coefficient for the bead-spring model as a function of temperature (open circles). The full line is a fit with the Vogel-Fulcher law in Eq. [3]. The dashed and dotted lines are two fits with a power-law divergence at the mode-coupling critical temperature. 

Calculation and results. For each temperature we calculated the HMX molecular center-of-mass self-diffusion coefficient determined as ... 

The Rouse model describes the dynamical properties of melts of macromolecules of a relatively small number of Kuhn segments, Ncritical number Nc is the number of Kuhn segments for the critical molecular mass Me- Flexible polymers have critical Kuhn segment numbers typically in the range Mc=40- 60 [1, 42-44, 52]. On the other hand, chain dynamics in concentrated systems of polymers with N Nc is much slower than expected on the basis of the Rouse model. Alluding to chain entanglements that are considered to become relevant in this case, one speaks of entangled dynamics. For example, experimental terminal relaxation times and center-of-mass self-diffusion coefficients scale as and... 

The zero mode is the self-diffusion of the center of mass whose diffusion coefficient is given by the Stokes-Einstein relation D = k TIN. The time Tj will be proportional to the time required for a chain to diffuse an end-to-end distance, that is, R )/D = t N b lk T. This means that for time scales longer than Tj the motion of the chain will be purely diffusive. On timescales shorter than Tj, it will exhibit viscoelastic modes. However, the dynamics of a single chain in a dilute solution is more complex due to long-range forces hydrodynamic interactions between distant monomers through the solvent are present and, in good solvents, excluded volume interactions also have to be taken into account. The correction of the Rouse model for hydrodynamic interaction was done by Zimm [79]. Erom a mathematical point of view, the problem becomes harder and requires approximations to arrive at some useful results. In this case, the translational diffusion coefficient obtained is... 

Figure 27. Dynamical scaling of DNA confined to the surface of a supported lipid membrane, (a) Time sequence (At = 30 s) of a DNA molecule diffusing on a cationic lipid membrane. The image on the right depicts an overlay of 16 images time average yields a smeared fluorescence distribution, (b) Scaling behavior of the self-diffusion coefficient of the center of mass D with the number of base pairs, (c) Scaling behavior of the rotational relaxation time r, with the number of base pairs.140...

Among the pure solvents we have treated so far, other available data regard the calculation of the self diffusion coefficient (D) for liquid ethanol at different temperatures." The D parameter was obtained from the long-time slope of the mean-square displacements of the center of mass experimental changes of D over the 285-320K range of temperatures were acceptably reproduced by molecular dynamics simulations. 

In case of the self-diffusion coefficient, A(t) is the position vector of a given molecule at some time t and A(t) is its center of mass velocity vector. In this way. 

Polymer molecules are in constant motion in solution. The long-time trajectory of the center of mass is governed by the self-diffusion coefficient, Dj. The molecular theory of the self-diffusion coefficient of a polymer molecule will be presented. The concept of the molecular friction coefficient will be developed. The process of mutual diffusion will then be presented, and an expression for the mutual-diffusion coefficient, D , will be derived. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

The molecular dynamics calculations were used to calculate the self-diffusion coefficient (D) of the guest molecule using the mean square displacements (MSD) of the center of mass with respect to time (Figure 6). Using the Einstein relation... 

Figure 6. Mean-square center-of-mass displacement vs time plot. The slope/6 is equal to the self-diffusion-coefficient.

Self-diffusion coefficient V, center-of-mass velocity of tagged molecule ... 

Various transport coefficients can also be related to time-correlation functions. For instance, as was shown in Section 5.9, the translational self-diffusion coefficient is proportional to the area under the time-correlation function of the velocity of the center of mass of the particle. 

Bulk polymer properties such as viscosity and elasticity are concerned with averaged responses of an assembly of polymer chains to external stimuli. On the other hand, the self-diffusion coefficient has something to do with the average speed of translation of the centers-of-mass of individual chains. Thus its study should give us a clue to the clarification of the modes of Brownian motion of a single chain on long timescales. This expectation must have been in the mind of polymer workers for many years, but, except in dilute solutions, few measurements of Ds were undertaken until recently, probably on the one hand because of experimental difficulties and on the other because of the lack of an adequate guiding theory. 

There is a distinct difference between a different pair (m + n) and the same pair (m = n self-diffusion of each bead). For the same pair, ([r (0] = 6Dq(N + l)r = 6 k T/C)t. It means that each bead moves freely with its own friction coefficient as if the other beads were absent or not connected. For a pair of different beads, the short-time mean square displacement increases as 6Dot, the same as the center of mass diffusion. Different beads are uncorrelated. 

Measurement of the Center-of-Mass Diffusion Coefficient The center-of-mass diffusion coefficient Dq we obtained here is the self-diffusion coefficient Dj. DLS cannot measure the self-diffusion coefficient. It is necessary to use more specialized techniques such as FRS, FRAP, and PFG-NMR, described in Section 3.2.11. Figures 4.39 and 4.40 show examples of FRS studies of D for dye-labeled polystyrene in benzene. ... 

Figure 3 shows the mean square displacement of the chain center of mass, Remit) — i cm(0)) ), in time for the longer-chain systems, Cise, C200, and C250- From the linear part of these curves the self-diffusion coefficient D can be obtained using the Einstein relation. 

The self-diffusion coefficient is used to describe the center-of-mass motion for a simple liquid. It can also be used in connection with the Rouse-Zimm model to describe the behavior of a long chain. The determination of diffusion coefficient D... 

---