Translational diffusion coefficients determination

Figure 7. Translational diffusion coefficients determined by PCS as a function of concentration for dilute sols S3 (a), S4 (b), and S2 (c).

The hydrodynamic radius reflects the effect of coil size on polymer transport properties and can be determined from the sedimentation or diffusion coefficients at infinite dilution from the relation Rh = kBT/6itri5D (D = translational diffusion coefficient extrapolated to zero concentration, kB = Boltzmann constant, T = absolute temperature and r s = solvent viscosity). 

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. 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

In fluorescence correlation spectroscopy (FCS), the temporal fluctuations of the fluorescence intensity are recorded and analyzed in order to determine physical or chemical parameters such as translational diffusion coefficients, flow rates, chemical kinetic rate constants, rotational diffusion coefficients, molecular weights and aggregation. The principles of FCS for the determination of translational and rotational diffusion and chemical reactions were first described in the early 1970s. But it is only in the early 1990s that progress in instrumentation (confocal excitation, photon detection and correlation) generated renewed interest in FCS. 

Translational diffusion coefficients of fluorophores like rhodamine 6G have been determined by FCS and reasonable values of 3 x 10-6 cm2 s 1 were found (Figure 11.12). Tests with latex beads showed good agreement with known values. 

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 z-averag translational diffusion coefficient aj infinite dilution, D, could be determined by extrapolating r/K to zero scattering angle and zero concentration as shown typically in Figs. 4 and 5. D is related to the effective hydrodynamic radius, by the Stokes-Einstein relation ... 

Determination of translational diffusion rates of proteins requires measurements at longer timescales, one-tenth of a second to several minutes. Eosin derivatives are also commonly used to measure translational diffusion coefficients using the Fluorescence Recovery After Photobleaching technique [138-141],... 

We thus come to the conclusion that the correct value of the coefficient C for structure determination can be gained only if the measurements were made at different delay times and if the resultant coefficients C (t0) are extrapolated to to = 0. On the other hand, we can also state that the correct translational diffusion coefficient is always obtained if T (t0)/q2 is extrapolated to zero q2. This is because the coefficient C (to) loses its influence at low q2, as may be recognized from Eq. (D.40), and from Fig. 54. 

The question about the difference between the macroscopic and microscopic values of the quantities characterizing the translational mobility (viscosity tj, diffusion coefficient D, etc.) has often been discussed in the literature. Numerous data on the kinetics of spin exchange testify to the fact that, with the comparable sizes of various molecules of which the liquid is composed, the microscopic translational mobility of these molecules is satisfactorily described by the simple Einstein-Stokes diffusion model with the diffusion coefficient determined by the formula... 

It has already been pointed out that the power spectrum of this function at zero frequency determines the translational diffusion coefficient, D. The full-time dependence of this function can be obtained indirectly from inelastic slow neutron experiments.57 Unfortunately, these experiments are not yet precise enough to say anything quantitatively about this function. /(t) s memory function, K t), is defined by... 

K represents the following constant parameters n is the index of refraction of the liquid, X is the laser wavelength in air, and 0 is the angle at which the scattering intensity is measured. For polydisperse samples, the autocorrelation function plot is the sum of exponentials for each size range. Once the average translational diffusion coefficient of the sample is determined, the equivalent spherical diameter can be determined by using the Stokes-Einstein... 

It is important to note that the translational diffusion coefficient can be determined this way for macromolecules that are less than 2 pm in their largest dimension at scattering angles less than 4°. Macromolecules larger than this or measurements made at higher scattering angles need to be corrected to get accurate values of Dt. 

Figure 4.7. Determination of translational diffusion coefficient. Plots of ln[(G(nAf))/B) - 1] versus time for collagen a chains (top) and mixtures of a chains and P components (bottom). The translational diffusion coefficient is obtained by dividing the slope of each line by -2Q2, where Q is the scattering vector. B is the baseline of the autocorrelation function. Note for single molecular species, the slope is constant, and for mixtures with different molecular weights, the slope varies (molecular weight for a chains is 95,000 and for y components is 285,000). (reproduced from Silver, 1987).

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. 

Microemulsions consist of oil, water and an oil-water interfacial Him. DLS and SLS have been used to determine the translational diffusion coefficient and the interaction potential of microemulsions [45—47). The thickness of the inter-facial film and its curvature were measured by the contrast variation method in neutron scattering [48,491. This method is based on changing the scattering strength by changing the relative amount of light and heavy water in the microemulsion. 

Tanford [13] and Perrin [14] determined the translational diffusion coefficient for ellipsoidal particles ... 

Here M is the molecular weight and V the partial specific volume of the solute, N the Avogadro number, k the Boltzmann constant, and T the absolute temperature s and D are the sedimentation and translational diffusion coefficients (after extrapolation to infinite dilution). The translational frictional coefficients from both measurements are regarded as identical, i.e., f, = fd. The rotary frictional coefficient, designated as f, can be determined from either flow birefringence or non-Newtonian viscosity measurements. 

Fig. 12. Translational diffusion coefficients of protein 4 determined at different scattering angles (6) are plotted against their corresponding K2 values where K = (4irn/ ) sin (0/2). This anomalous behavior is due to sample heterogeneity. From Ahmed et al. (1975), reproduced with permission.

Translational Diffusion Coefficients (D) of Glycoproteins 4 and 8 Determined at Temperatures Close to Freezing in the Presence and in the Absence of Ice Crystals ... 

Therefore, the power spectrum is a sum of two Lorentzians which are shifted in frequency, but their half-width remains determined by the translational diffusion coefficient. The autocorrelahon function in this case is... 

Figure 2.6 Measured intermediate scattering functions of a w/o-droplet microemulsion for the system D2 0/n-octane-d- 8/CioE4. The four curves were obtained at four different q values close to the minimum of the droplet form factor. The solid lines are double exponential fits with only two adjustable parameters. The translational diffusion coefficient was determined using PCS (see Fig. 2.5) and used as input for the analysis of the NSE data. (Figure redrawn with data taken from Ref. [67].)...

Figure 2.7 Relaxation times t for the deformation modes as obtained from the analysis of the NSE intermediate scattering functions using DLS data for the determination of the translational diffusion coefficient. The lines indicate the average relaxation time for the different samples.

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