Translational diffusion coefficient calculation

We have also reported on the ordinate axis of Fig.3 the values of the translational diffusion coefficient calculated from radius values measured by transient electric birefringence, using ... 

FIGURE 2.2 (Upper panel) Simulated MSDs for the overall systems as a function of time at four representative compositions. (Middle panel) Simulated normalized VACF as a function of time at the same four representative compositions. (Lower panel) Overall translational diffusion coefficient (calculated from the VACF and MSD plot) as a function of the mole... 

A number of bulk simulations have attempted to study the dynamic properties of liquid crystal phases. The simplest property to calculate is the translational diffusion coefficient D, that can be found through the Einstein relation, which applies at long times t ... 

Table 7. Molecular masses, functionalities, calculated and measured radii of gyration, and translational diffusion coefficients of the investigated polyisoprene stars...

With the help of the Stokes-Einstein relation, the translational diffusion coefficient may be calculated according to... 

The translational diffusion coefficient of micelles loaded with a fluorophore can be determined from the autocorrelation function by means of Eqs (11.8) or (11.9). The hydrodynamic radius can then be calculated using the Stokes-Einstein relation (see Chapter 8, Section 8.1) ... 

The failure of the model to reproduce satisfactorily the dynamics of PeMe (Figure 3B) can be attributed to its slower dynamics. Diffusional processes then become more relevant and our rough estimation of kr, by means of the Stokes-Einstein expression, is probably not good enough. Much better agreement can be obtained when the translational diffusion coefficient is calculated with the semiempirical expression of Spemol and Wirtz [5]. 

The error concerned an explicit formula for the translational diffusion coefficient. Kirkwood calculated the diffusion tensor as the projection onto chain space of the inverse of the complete friction tensor he should have projected the friction tensor first, and then taken the inverse. This was pointed out by Y. Ikeda, Kobayashi Rigaku Kenkyushu Hokoku, 6, 44 (1956) and also by J. J. Erpenbeck and J. G. Kirkwood, J. Chem. Phys., 38, 1023 (1963). An example of the effects of the error was given by R. Zwanzig, J. Chem. Phys., 45, 1858 (1966). In the present article this question does not come up because we use the complete configuration space. 

Fig. 14. Apparent weight distributions calculated from the translational diffusion coefficient distributions corresponding to low-mass (O) and high-mass ( ) copolymer segmented polyethylene terephthalate-co-caprolactone) (PET-PCL) containing 13% PET in tetrahydrofuran (THF) at 25 °C...

Using the Laplace inversion program, CONTIN, in the correlator, we were able to calculate G(T) from G(2>(t, q) on the basis of Eqs. 3 and 4. For a pure diffusive relaxation, r is related to the translational diffusion coefficient D as [47]... 

Recently, Stechemesser and Eimer published hydrodynamic radii obtained from translational diffusion coefficients measured by means of recovery after photobleaching experiments on selected generations of PAMAM dendrimers [76]. The values of Rh obtained in methanol are compared with Rg data on the same dendrimers in Fig. 4. It can be calculated that the ratio Rh/R 1.4 for all but the tenth generation dendrimer. This ratio is somewhat higher than expect-... 

A measurement of physical parameters in solution for isolated macromolecules provides a manner by which the shape of a macromolecule can be determined. The approximate dimensions and axial ratio or radius can be calculated by applying Equations (4.3) through (4.17). As shown in Figure 4.10, the particle scattering factor for collagen molecules depicted in Figure 4.9 is more sensitive to bends than is the translational diffusion coefficient. 

The r-average translational diffusion coefficient l> is calculated from the equation Dj = V/q2. For a collection of identical spheres undergoing ordinary Brownian motion in solution. 

The calculated ratio of the two translational diffusion coefficients is always slightly lower than the experimental one. The systematic deviation can be rationalized considering the approximation made by assuming the same... 

Calculation with the variation of translational diffusion coefficients showed that the nonlinear effects lessened effects of translational diffusion (Fig. 2). In the linearized equation, solvent relaxed quickly for a large translational diffusion coefficient. That is, a value of S(t) at t=(2DR) decreased with the increase in the translational diffusion coefficient for the linearized equation. For calculation with the nonlinear Smoluchowski-Vlasov equation (1), however, a value of S(t) at t=(2DR) increased slightly with the translational diffusion coefficient. Thus, the ratio of a nonlinear value to a linear value increased with the translational diffusion coefficient. 

Fig. 14. Translational diffusion coefficients (D) of protein 4 and the corresponding calculated effective hydrodynamic radius (Re) as a function of temperature. From Ahmed et al. (1975), reproduced with permission.

B(t) is the scattering amplitude of a particle, which depends on the particle polarizability at given orientation. B(t) changes with time due to reorientation of the particle. If the scatterers are spherical, B(t) is constant and Cg(x) = 1. Note that Cg(x) does not depend on the scattering angle and can be calculated if the polarizability tensor and the rotational diffusion tensor of the particles are known. The calculation of CJi q,x) requires averaging of the translational diffusion tensor of the particle over all possible orientations to obtain the averaged translational diffusion coefficient. 

Using Eq. (2.58), from the mean relaxation rate T the average apparent translational diffusion coefficient D can be calculated. The measured apparent diffusion coefficient D depends on the concentration [C] of the scattering particles. When [C] is not too large (4> < 0.1), one has... 

As stated above, this theory assumes that translational diffusion is isotropic that is, in a molecule-fixed frame, the diffusion constant parallel to the long molecular axis is the same as that perpendicular to it. For highly anisotropic large molecules this is probably not a good assumption. Maeda and Saito (1969) have calculated the spectrum taking into account the anisotropy of the translational diffusion constant. Their resulting expressions are rather complex and will not be given here. Their results are expressed as a power series in the translational diffusion coefficient anisotropy,... 

The translational diffusion coefficient of a chain molecule is measurable using quasi-elastic light scattering (23, 24). It can be calculated from an appropriate chain model using the Kirkwood (25) equation... 

Table 16.2 Comparison of the experimental translational diffusion coefficient (Dq p) and the experimental rate sedimentation coefficient (Sq p) with the respective calculated values and of these parameters for rhodexman in aqueous solution ...

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