Measurements of Translational Diffusion Coefficient

Caroline and co-workers have recently reported measurements of translational diffusion coefficients in solutions of PS in two mixed-solvent systems at or near theta conditions. In the solvent CCb-methanol (85), they observed the diffusion theta state, defined when the coefficient y of Equation 41 equals 0.5, to occur at 25°C and a volume fraction of CCI4, (fyCCU = 0.8025. In this system there is strong preferential adsorption of the polymer for CCI4, and it is not possible to define a true theta state such that y = a = V2 and A2 = 0 simultaneously. Under diffusion theta conditions, the concentration dependence of Dt apparently is closely described by the Pyun-Fixman hard-sphere model. In the mixed solvent benzene—2 propanol, polystyrene exhibits a true theta condition at T = 25.5°C and (benzene) = 0.04. Frost and Caroline confirmed that y = 0.5 within experimental error in this system (86) and report that values of the parameter fcf are scattered between the extreme values corresponding to the predictions of Yamakawa (and Imai) and the soft-sphere model of Pyun-Fixman (or the Freed theory). 

It should also be noted that a similar treatment is possible for the translational hydrodynamic radius, Rhj, obtained from measurements of translational diffusion coefficients or sedimentation coefficients of branched polymers. One may define a parameter gn = Rh,fb/Rhji - the ratio of the hydrodynamic radius of the branched polymer relative to that of a linear polymer of the same molecular weight. Again, it is expected that gH < 1. For star polymers with uniform subchain lengths having... 

The high frequency relaxation is attributed in part to the modulation of intermolecular dipolar interactions by the translational diffusion. The cutoff frequency (60 MHz at 55°C) corresponds to the local diffusive jump frequency that is estimated from measurements of the diffusion coefficient (D 10"6 cm2/sec at 55°) (19, 21). This cutoff frequency also varies in temperature with the same activation energy (Eact 0.25 eV) as the diffusion frequency. 

Pulse field gradient (PFG) NMR spectroscopy is now generally regarded as the method of choice for measuring the translational diffusion coefficients of molecules of virtually any type under many conditions (48). H, H, F, and P variants of this method have been used successfully to study lateral diffusion of cholesterol, phospholipids, and water in model membranes (49,50). This technique introduces two identical gradient pulses of the external magnetic field into the standard spin-echo NMR... 

Quasi-elastic laser light scattering (also called intensity fluctuation spectroscopy, light-beating spectroscopy or photon correlation spectroscopy) is an accurate method to measure the translational diffusion coefficients of macromolecules. The diffusion coefficient is a parameter, that depends on the size and shape of the macromolecules and on the thermodynamic and hydrodynamic interaction between the macromolecules. 

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.

Thus, if ti is in a time range accessible to autocorrelation (roughly 1—10—7 sec), fluorescence fluctuations may be used to measure macromolecular translational diffusion coefficients. The presence of a fluorescent label enables this method to measure the translational diffusion coefficient of a molecule in a complex mixture. Such a measurement would be very difficult in an ordinary light-scattering experiment because all components of the mixture contribute.17 The advantage of fluorescent probes is that they allow particular species to be labeled and thereby separately studied. For or of the order of 10 4 cm and for a particle with a diffusion coefficient of the order 10 5 cm2/sec, tt = 10 3 sec, well within our ability to measure. This leads to the interesting possibility of measuring diffusion coefficients of labeled molecules in membranes, and in cells in vivo. 

The fundamental rate expression to be considered is the Smoluchowski relation k = 4n iVDAB AB (Equation (2.1)). The derived expression ART/r] (Equation (2.3a)), is a useful approximation, but deviations from it are observed, because the Stokes-Einstein equation which is involved is derived by hydrodynamic theory for spherical particles moving in a continuous fluid, and does not accurately represent the measured values of translational diffusion coefficients in real systems. Although the proportionality Da 1 /rj is indeed a reasonable approximation for many solutes in common solvents, the numeral coefficient 1 /4 is subject to uncertainty. In the first place, this theoretical value derives from the assumption that in translational motion there is no friction between a solute molecule and the first layer of solvent molecules surrounding it, i.e., that slip conditions hold. If, however, one assumes instead that there is no slipping ( stick conditions), so that momentum is... 

Nuclear magnetic resonance (NMR) provides a powerful method for the study of molecular motion. The techniques can distinguish molecular reorientation and translation and have proved particularly valuable for the study of self-diffusion in bulk liquids. The molecular motion of liquids in the confined geometry provided by their containment in porous materials has been of considerable interest for many years. It is of importance both as a fundamental scientific problem and because of its technological importance in such diverse systems as oil recovery from rocks and catalytic agents. The purpose of this paper is to question the reliability of many previous investigations and the validity of their interpretation. Potential sources of error are demonstrated by measurements on mobile liquids adsorbed into porous silicas with different geometrical characteristics. The principles illustrated are equally valid for other porous systems. Preliminary measurements of the diffusion coefficient of n-butane in silica as a fimction of temperature and the effect of pore dimensions are presented. 

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-... 

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. 

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

PVCL microgels prepared via covalent binding of PEO exhibit different temperature dependence (Fig. 19). In this case, a considerable increase in the diffusion coefficient takes place above the LCST of PVCL. The sudden increase may be attributed to the shrinking of the particle, which leads to an increase in the rate of its translational diffusion and, consequently, also in the rate of diffusion of the grafts bound to the particle surface. The values of the diffusion coefficients above the LCST should be taken as apparent ones, as the measurements were complicated by the heterogeneity of the collapsed samples. 

In addition to Ti and T2, which reflect the rotational motion of water, NMR can also be used to measure the translational motion of water. If an additional, relatively small (compared to B0), steady magnetic field gradient is incorporated into a pulsed NMR experimental setup, a translational diffusion coefficient (D, m2/s) can be measured (called pulsed field gradient NMR). 

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

We have identified three diffusion coefficients. These are the self-translational diffusion coefficient D, cooperative diffusion coefficient Dc, and the coupled diffussion coefficient fly. fl is the cooperative diffusion coefficient in the absence of any electrostatic coupling between polyelectrolyte and other ions in the system, fly is the cooperative diffusion coefficient accounting for the coupling between various ions. For neutral polymers, fly and Dc are identical. Furthermore, we identify fly as the fast diffusion coefficient as measured in dynamic light scattering experiments. The fourth diffusion coefficient is the slow diffusion coefficient fl discussed in the Introduction. A satisfactory theory of flj is not yet available. 

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. 

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 ... 

---