Determination of translational diffusion coefficients

For a single fluorescent species undergoing Brownian motion with a translational diffusion coefficient Dt (see Chapter 8, Section 8.1), the autocorrelation function, in the case of Gaussian intensity distribution in the x, y plane and infinite dimension in the z-direction, is given by 

For a finite volume element with a Gaussian intensity distribution in three dimensions, the autocorrelation can be written as 

5) Translational diffusion can also be studied by fluorescence recovery after photobleaching (FRAP). This technique v ill not be described 

Applications to fluorescent or fluorescently labeled proteins and nucleic acids, and to fluorescent lipid probes in phospholipid bilayers, have been reported. In the latter case, the diffusion coefficients measured above the chain melting temperature were found to be 10 cm s which is in agreement with values obtained by other techniques. 

Translational diffusion times of micelles can be measured by PCS, which allows calculation of the aggregation number (see Box 11.2). 

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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).

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],... 

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... 

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. 

Selected Examples. - Pulsed magnetic field gradient (PFG) NMR is today a routine method for the determination of self-diffusion coefficients. However, a remaining goal is the improvement of the precision of the method. The best procedure for the determination of accurate diffusion coefficients by PFG NMR is a calibration with a sample of precisely known D value. Thus Holz et al presented temperature-dependent self-diffusion coefficients of water and six selected molecular liquids. The gained accurate self-diffusion data are suited for an elaborate check of theoretical approaches in the physics of molecular liquids. Price et al examined the translational diffusion... 

This should be compared to the result 0 = ksT/STitja3 for the sticky sphere of radius a. Perrin also determined the translational diffusion coefficients for ellipsoidal molecules.19 His result is... 

Figure 14 illustrates the viability of the PFS method, but here it is worth taking stock of the assumptions that underpin PCS and PFS. A principal virtue of size estimates derived from PCS are that they are independent of the optical properties of the particle. They are however implicitly dependent on the particle shape, for the shape is required to determine the translational diffusion coefficient in the suspending medium. The particles are usually assumed to be spherical. The aspect ratios displayed in Figure 13 were reconstracted assuming the size of the particles and their refractive index was known, and... 

In a quasielastic light scattering study of macromolecular solutions, it is essential to establish a clear relationship between the experimental conditions and the type of relaxation phenomena manifest in the light scattering data. To accomplish this, one must perform experiments over a range of scattering angles and solution concentrations. One of our objectives is to determine the translational diffusion coefficient (D ) for xanthan, and to relate this parameter to a hydrodynamic size. 

Figure 2. Concentration dependence of translational diffusion coefficient for PSM in O.IM NaCl as determined by the method of cumulants.

Two general methods exist for determining the translational diffusion coefficient, D, in polymer melts (a) by measuring the broadening of concentration... 

The study of processes of translational diffusion pamits determining the translational diffusion coefficient D based on the dependence of the dispersion... 

Brownian motion of particles in solution gives rise to a spectral distribution in the scattered light. From measurements on the line width of scattered laser light by photon correlation spectroscopy (PCS) the translational diffusion coefficient may be determined. Unlike the traditional method of investigating translational diffusion, PCS does not require a macroscopic concentration gradient and therefore can more readily be applied to investigate association processes. The method has been used to determine the translational diffusion coefficients, of micelles formed by a number of block copolymers in selectively bad solvents. ... 

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) ... 

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. 

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. 

Table 4.3. Determination of translational and rotational diffusion coefficients for connective tissue macromolecules from quasi-elastic light scattering 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. 

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