Translational diffusion coefficient figure

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. 

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.

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). 

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]. 

Figure 4.5. Autocorrelation function of scattered light. Schematic diagram showing the decay of the autocorrelation function to a baseline. The autocorrelation function is related to the product of two intensities separated by a time interval. As the time interval increases, the function decays to a baseline. The rate of decay is proportional to the translational diffusion coefficient. 

Figure 4.6. Normalized autocorrelation function. Autocorrelation function for collagen single molecules. The autocorrelation function G(nAt) is normalized by dividing all points by the first experimental point G(l). The autocorrelation function decays to a value of the average squared intensity of scattered light divided by G(l). The average squared intensity is proportional to the weight average molecular weight, whereas the rate of decay is related to the translational diffusion coefficient. Reproduced from Silver, 1987.

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. 

Figure 4.10. Theoretical dependence of translational diffusion coefficient of rods on location of bends. Theoretical relationships shown between translational diffusion coefficient (a), particle scattering factor at a scattering angle of 175.5° (b), and bend angle obtained using bead model with bends at ID and 2D from the end. Points above horizontal lines shown for D o,w and P(175.5°) are significantly different from those values for a straight rigid rod. The total rod length is 4.41). (reproduced from Silver, 1987).

In addition, by measuring the intensity fluctuations at low scattering angles changes in the translational diffusion coefficient can be measured during the early phases of self-assembly as illustrated by Figure 5.5. The translational diffusion coefficient decreases as the ratio of the length to width increases as linear assembly occurs. 

Figure 5.5. Measurement of physical properties during initiation of collagen self-assembly. Translation diffusion coefficient (D20-w) (top) and intensity of scattered light at 90° (bottom) versus time for type I collagen. Note translational diffusion constant decreases, whereas intensity of scattered light remains initially unchanged.

Figure 2. Values of S(t) at t=(2DR)" for some translational diffusion coefficients. Circles show values by the nonlinear equation for z=0- l, diamonds show values by the linearized equation, triangles show ratios between them.

Figure 3.27 gives three MD snapshots at different molecular cross-sections (indicated). At 0.25 nm the layer is disordered, the more strongly compressed layers are ordered. The change In tilt is pronounced. In the simulated xlOj) diagram a small double knee-bend is observed (not shown) between 0.21 and 0.23 nm. Translational diffusion coefficients D can be obtained from the root mean square displacement... 

Figure 2.5 Relaxation rates F of the intensity correlation functions as a function of q2 obtained via a photon correlation spectroscopy experiment. The sample was a w/o-droplet microemulsion made of D2 0/n-octane-di8/CioE4. On the oil-continuous side of the phase diagram the scattered light intensity is usually low leading to rather large errors of the individual data points. Nevertheless, from the slope of the linear fit the translational diffusion coefficient is obtained. (Figure redrawn with data from Ref. [67].)...

The translational diffusion coefficient from the PCS measurements is then used in Eq. (2.8) to reduce the number of adjustable parameters. In Fig. 2.6, the experimentally obtained intermediate scattering functions for four scattering angles are shown. The solid lines in this figure are fits using Eq. (2.8). The relaxation t2 for the deformation mode with... 

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.

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

Figure 8. The variation of translational diffusion coefficients, Dr, measured by PCS with different sol concentrations for dilute samples of the dialyzed (+) and undialyzed (%) silica sol SI.

Figure 9. The effect of sol concentration on translational diffusion coefficients, Dr, determined by PCS for silica sols S2 (a) and S4 (b).

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... 

Figure Effective translational diffusion coefficient Dgff = r/q of polydisp e PS s in cyclohexane at 37°C, (A) = sample withM = lOOfiOO, = 250,-)ith 3 n = I...

Figure 4. Flory-Mandelkern correlation of intrinsic viscosity, molecular weight, and translational diffusion coefficient for a variety of polymer solvent systems, demonstrating the insensitivity of these data to the structure of the macromolecule ( X PS/tetrahydrofuran (37) (O), protein random coils in 6M guanidine hydrochloride-0,IM mercaptoethanol (26) (X), tobacco mosaic virus in aqueous solution (11) ( ), bovine serum albumen in aqueous solution (37).

Figure 5. Effective translational diffusion coefficient D ff = T/q of mixtures of poly-ia-lysine HBr and chondroitin 6-sulfate in aqueous 0.1 M NaCl at 25°C Lysine to disaccharide ratio is 1 1. Inset is the variation in relative scattered intensity per unit scattering volume with the square of the wave vector q for...

Figure 3. Concentration dependence of translational diffusion coefficient D, of PCS Al-Dl fraction treated with CNBr in 0./5M NaCl, pH 7.0 at a 40° scattering...

Figure 3.7. Translational diffusion coefficient as a fiinetion of density. Note the maximum at intermediate density. The figure is reprodueed from the thesis of Dr. Pradeep Kmnar. http //polymer.bu.edu/ hes/water/thesis-kumar.pdf...

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

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