Single-particle diffusion

Further sections treat single-particle diffusion, light scattering spectroscopy, rotational diffusion, viscosity, and viscoelasticity. Colloid and polymer dynamics are found to have many striking similarities. A final section brings the findings together. 

There is considerable interest in the concentration dependence of the colloidal self-diffusion coefficient, whose small-concentration limiting behavior may be written 

Pusey and van Megen compare their measurements with theoretical predictions. At lower concentrations, theory indicates = —1.73, consistent with their experiments. At elevated concentrations, their predicted Ds(c) lies well helow their measurements. They concluded that one must include three- and many-hody hydrodynamic interactions to calculate the concentration dependence of Ds at large / . 

Depolarized light scattering spectroscopy was applied by Degiorgio, et al. to solutions of a fiuorinated latex polymer(10). The orientations of pairs of spheres are uncorrelated, so as discussed in Section 10.4 the VH spectrum is determined entirely by the self- part of the dynamic structure factor appropriate analysis of the VH spectrum determines both Ds and the rotational diffusion coefficient Dr. Degiorgio, et al. found that Ds(4 ) and Dr(4 ) are both accurately described, for 

Kops-Werkhoven, et al. measured tracer diffusion of dilute 38 nm radius silica spheres diffusing through suspensions of 33 nm silica spheres, all in cyclo-heptane(12). The experimental method was dynamic light scattering particle motions were observed over distances large compared to their radius. They found ku =-2.7 0.3. 

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The case of protein synthesis, i.e., diffusion of many segments on the same one-dimensional lattice, is clearly recognized as a species of traffic problem. However, the existing traffic literature has been of no help to its solution. Indeed, only rather special types of solutions have presented themselves to date. On the other hand, methods for a full solution of the DNA synthesis problem, i.e., single-particle diffusion, are reasonably well known but not of much use, since they give solutions in terms of only slowly converging series. Nevertheless, the first two moments of the distribution of degrees of polymerization in the ensemble at each time are easily obtained. 

At low ionic strength (kR 1), other effects connected with the finite diffusivity of the small ions in the EDL surrounding the particle are present. The noninstantaneous diffusion of the small ions (with respect to the Brownian motion of the colloid particle) could lead to detectable reduction of the single particle diffusion coefficient, Dq, from the value predicted by the Stokes-Ein-stein relation. Equation 5.447. For spherical particles, the relative decrease in the value of Dq is largest at k/ 1 and could be around 10 to 15%. As shown in the normal-mode theory, the finite diffusivity of the small ions also affects the concentration dependence of the collective diffusion coefficient of the particles. Belloni et al. obtained an explicit expression for the contribution of the small ions in Ac)... 

Ala-Nissila, T., Ferrando, R., Ying, S.C. Collective and single particle diffusion on surfaces, Adv. Phys. 2002,51,949. 

If Dp depends significantly on Cp, extrapolation to Cp 0 must be performed. The initial slope of the dependence of Dp on probe concentration, and the slope s dependence on matrix concentration, have been measured in some systems and should be accessible to theoretical analysis. In this review If the probe and matrix polymers differ appreciably in molecular weight or chemical nature, the phrase probe diffusion coefficient is applied. If the probe and matrix polymers differ primarily in that the probes are labelled, the phrase self diffusion coefficient is applied. The tracer diffusion coefficient is a single-particle diffusion coefficient, including both the self cind probe diffusion coefficients as special cases. The interdiffusion and cooperative diffusion coefficients characterize the relaxation times in a ternary system in which neither m lcrocomponent is dilute. 

Jones [20] used a Smoluchowski approach to examine interacting spherical polymers. Jones predicted that, if one polymer species is dilute and labelled, the measured diffusion coefficient from QELSS is determined only by hydrodynamic interactions of the tagged polymers and their untagged matrix neighbors, and is the single-particle diffusion coefficient. The hydrodynamic approach culminated in analyses of Carter, et al. [21] and Phillies [22] of mutual and tracer diffusion coefficients, including hydrodynamic and direct interactions and reference frame issues. 

Elementary excitations also include single particle diffusive excitations beside quantized vibrations (i.e., molecular vibrations and vibrations of the crystal as a whole associated with phonons/magnons). Consider the incoherent dynamic structure factor 5snc(Q,(o), which is the Fourier transform pair of the time-dependent self-correlation function, compare... 

This chapter examines the diffusion of mesoscopic rigid probe particles through polymer solutions. These measurements form a valuable complement to studies of polymer self- and tracer diffusion, and to studies of self- and tracer diffusion in colloid suspensions. Any properties that are common to probe diffusion and polymer self-diffusion cannot arise from the flexibility of the polymer probes or from their ability to be interpenetrated by neighboring matrix chains. Any properties that are common to probe diffusion and to colloid diffusion cannot arise from the flexibility of the matrix polymers or from the ability of matrix chains to interpenetrate each other. Conversely, phenomena that require that the probe and matrix macromolecules be able to change shape or to interpenetrate each other will reveal themselves in the differences between probe diffusion, single-chain diffusion, and colloid single-particle diffusion. 

Phillies, etal. (77) re-examined results of Brown and Zhou(78) and Zhou and Brown (79) on probe diffusion by silica spheres and tracer diffusion of polyisobutylene chains through polyisobutylene chloroform solutions. These comparisons are the most precise available in the literature, in the sense that all measurements were made in the same laboratory using exactly the same matrix polymer samples, and were in part targeted at supporting the comparison made by Phillies, et al.(Jl). Comparisons were made between silica sphere probes and polymer chains having similar Dp and Dt in the absence of polyisobutylene. For each probe sphere and probe chain, the concentration dependence of the single-particle diffusion coefficient is accurately described by a stretched exponential in c. For large probes (160 nm silica spheres, 4.9 MDa polyisobutylene) in solutions of a small (610 kDa) polyisobutylene, Dp c)/Dt(c) remains very nearly independent of c as Dp c) falls 100-fold from its dilute solution limit. 

The list of experimentally accessible properties of colloid solutions is the same as the list of accessible properties of polymer solutions. There are measurements of single-particle diffusion, mutual diffusion and associated relaxation spectra, rotational diffusion (though determined by optical means, not dielectric relaxation), viscosity, and viscoelastic properties (though the number of viscoelastic studies of colloidal fluids is quite limited). One certainly could study sedimentation in or electrophoresis through nondilute colloidal fluids, but such measurements do not appear to have been made. Colloidal particles are rigid, so internal motions within a particle are not hkely to be significant the surface area of colloids, even in a concentrated suspension, is quite small relative to the surface area of an equal weight of dissolved random-coil chains, so it seems unlikely that colloidal particles have the major effect on solvent dynamics that is obtained by dissolved polymer molecules. 

Goyal, G. Freedman, K. J. Kim, M. J. Gold nanoparticle translocation dynamics and electrical detection of single particle diffusion using solid-state nanopores. Anal Chem 2013, 85, 8180-8187. 

H q)=D(q)S(q)IDo here S(q) is the static structure factor and Do is the single particle diffusion constant in the dilute limit. 

The kinetic contribution dominates for A 2> a, while the collisional contribution dominates in the opposite limit. Two other transport coefficients of interest are the thermal diffusivity, Dt, and the single particle diffusion coefficient, D. Both have the dimension square meter per second. As dimensional analysis would suggest, the kinetic and collisional contributions to Dj exhibit the same characteristic depen-... 

This section extends the outer radiation boundary condition in the previous model to include geminate reaction within the micelle. In this model a single particle diffuses with a sink at the centre and an outer radiation boundary. The required initial condition is S2(( = 0) = 1, inner boundary S2(r = a) = 0 and outer boundary condition — wS2(r = R). Solving the backward diffusion equation with these boundary conditions gives the analytical expression for the survival probability to be... 

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