The Translational Diffusion Coefficient

While the mean-square radius of gyration (S2)z can be obtained from the procedure of differentiating the particle-scattering factor Pz(q2) with respect of q2, the translational diffusion coefficient is obtained by integration of Pz(q2) over the whole q2 region94  

At low monomer conversion, the polymerization leads to fairly small molecules, and the branching process can proceed largely unimpeded by the finite volume. Thus, the common behavior of randomly branched polymers is observed. At larger conversion of monomer, the polymer has grown in size, such that the largest species have already reached dimensions of the latex sphere. On further branching, the molecular dimensions 

Information from the Combination of Static and Dynamic Light Scattering1 2) 

In the last two chapters some major properties of the static and dynamic scattering functions have been discussed separately. This chapter deals with the combination of both techniques and with the question of whether such a combination can produce additional information. 

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It is possible to get molecular weight from the sedimentation coefficient if we assiune a conformation or if we combine with other measurements, namely the translational diffusion coefficient via the Svedberg equation [50]... 

The translational diffusion coefficient in Eq. 11 can in principle be measured from boimdary spreading as manifested for example in the width of the g (s) profiles although for monodisperse proteins this works well, for polysaccharides interpretation is seriously complicated by broadening through polydispersity. Instead special cells can be used which allow for the formation of an artificial boundary whose diffusion can be recorded with time at low speed ( 3000 rev/min). This procedure has been successfully employed for example in a recent study on heparin fractions [5]. Dynamic fight scattering has been used as a popular alternative, and a good demonstra-... 

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 ... 

We have applied FCS to the measurement of local temperature in a small area in solution under laser trapping conditions. The translational diffusion coefficient of a solute molecule is dependent on the temperature of the solution. The diffusion coefficient determined by FCS can provide the temperature in the small area. This method needs no contact of the solution and the extremely dilute concentration of dye does not disturb the sample. In addition, the FCS optical set-up allows spatial resolution less than 400 nm in a plane orthogonal to the optical axis. In the following, we will present the experimental set-up, principle of the measurement, and one of the applications of this method to the quantitative evaluation of temperature elevation accompanying optical tweezers. 

Under the condition that the Stokes-Einstein model holds, the translational diffusion coefficient, D, can be represented by Eq. (8.3). the diffusion time, Xd, obtained through the analysis is given by Eq. (8.4). 

In contrast to normal diffusion, Ar2n does not grow linearly but with the square root of time. This may be considered the result of superimposing two random walks. The segment executes a random walk on the random walk given by the chain conformation. For the translational diffusion coefficient DR = kBT/ is obtained DR is inversely proportional to the number of friction-performing segments. 

A and B are instrumental factors and g(T) the electric field correlation function which is related to the translational diffusion coefficient D by... 

Measurement of the translational diffusion coefficient, D0, provides another measure of the hydrodynamic radius. According to the Stokes-Einstein relation... 

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

The translational diffusion coefficient on the molecular scale is then... 

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) ... 

Figure Bll.2.1 shows the normalized autocorrelation functions of various micelles loaded with octadecyl rhodamine B chloride (ODRB) at pH 7 (PBS buffer)3 . The differences in size of the micelles are clearly reflected by the differences in diffusion times td- The translational diffusion coefficients are reported in Table Bll.2.1, together with the hydrodynamic radii and the aggregation numbers.

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

This is a surprising result in view of the following argument [51], due to Hiickel. Defining Q and D to be respectively the net charge and the translational diffusion coefficient of the polyelectrolyte, the balance of frictional force ksT/D) p, and electrical force gE gives... 

When a chain has lost the memory of its initial state, rubbery flow sets in. The associated characteristic relaxation time is displayed in Fig. 1.3 in terms of the normal mode (polyisoprene displays an electric dipole moment in the direction of the chain) and thus dielectric spectroscopy is able to measure the relaxation of the end-to-end vector of a given chain. The rubbery flow passes over to liquid flow, which is characterized by the translational diffusion coefficient of the chain. Depending on the molecular weight, the characteristic length scales from the motion of a single bond to the overall chain diffusion may cover about three orders of magnitude, while the associated time scales easily may be stretched over ten or more orders. 

Figure 5.8 presents typical spectra taken on both polymer solutions at 300 K (a) and 378 K (b). The PDMS data are represented by open symbols, while the PIB data are shown by full symbols. Let us first look at the data at 378 K. At Q=0.04 A"i (QR =0.S) we are in the regime of translational diffusion, where the contributions of the intrachain modes amount to only 1%. There the spectra from both polymers are identical. Since both polymers are characterized by equal chain dimensions, the equality of the translational diffusion coefficients implies that the draining properties are also equal. In going to larger Q-values, gradually the spectra from the PlB-solutions commences to decay at later times. This effect increases with increasing Q and is maximal at Q=0.4 A" (see Fig. 5.8a). 

Specific diffusion control The rate constants depend on the size and topology of the molecule the group is bound to i.e., they depend on the translation diffusion coefficient of the species. 

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 ... 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

Fig. 25. Time evolution of the translational diffusion coefficient distribution G(D) for blend STVPh-9/PEMA after mixing, where the total polymer concentration is 1.0x10" g/ml, the weight ratio of STVPh-9/PEMA is 100 8, and the scattering angle is 15° [152]...

When the diffusion time is short enough, the translation on the spherical surface is approximately identical with that on the tangent plane to the spherical surface. The latter is the two-dimensional diffusion process treated by the Green function method in Appendix C, and we can use Eq. (C17) again. Since the rotational diffusion coefficient Dr is related to the translational diffusion coefficient D<2) in Eq. (Cl7) by Dr = D(2)/(Le/2)2, we have... 

Fig. 16a, b. Computer simulation results for rodlike polymers in solution a the translational diffusion coefficients [122,123] b the rotational diffusion coefficient [119,122,123]... 

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]. 

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

The translational diffusion coefficient D is (2d) l times the mean square displacement per unit time of the center of mass of the molecule. When bead i flips from its position Rj, as measured with respect to an arbitrary origin, to a new position R(, the square of the resulting displacement of the center of mass of the chain is (R( — R,-)2/(W + l)2. Since bead i flips with a frequency w, we have... 

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. 

Knowing these functions, the mean-square radius of gyration (S2)z and the translational diffusion coefficient Dz can easily be derived eventually by application of the Stokes-Einstein relationship an effective hydrodynamic radius may be evaluated. These five... 

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