Optical Probe Diffusion in Polymer Solutions

Optical Probe Diffusion in Polymer Solutions George D. J. Phillies∗ Department of Physics, Worcester Polytechnic Institute, Worcester, MA 01609 The experimental literature on the motion of mesoscopic probe particles through polymer solutions is systematically reviewed. The primary focus is the study of diffusive motion of small probe particles. Comparison is made with measurements of solution viscosities. A coherent description was obtained, namely that the probe diffusion coefficient generally depends on polymer concentration ν 0.8 as Dp = Dp0 exp(−αc ). One finds that α depends on polymer molecular weights as α ∼ M , and ν appears to have large-M and small-M values with a crossover linking them. The probe diffusion coefficient does not simply track the solution viscosity; instead, Dpη typically increases markedly with increasing polymer concentration and molecular weight. In some systems, e.g., hydroxypropylcellulose:water, the observed probe spectra are bi- or tri-modal. Extended analysis of the full probe phenomenology implies that hydroxypropylcellulose solutions are characterized by a single, concentration-independent, length scale that is approximately the size of a polymer coil. In a very few systems, one sees re-entrant or low-concentration-plateau behaviors of uncertain interpretation; from their rarity, these behaviors are reasonably interpreted as corresponding to specific chemical effects. True microrheological studies examining the motion of mesoscopic particles under the influence of a known external force are also examined. Viscosity from true microrheological measurements is in many cases substantially smaller than the viscosity measured with a macroscopic instrument. I. INTRODUCTION ticles are recorded. Historically, Brown used particle tracking to study This review treats probe diffusion and related methods the motion now called Brownian. Optical probe diffusion as a method for studying polymer solutions dates of investigating polymer dynamics. In a probe diffusion 1 experiment, a dilute dispersion of mesoscopic probe par- back to Turner and Hallett , who in 1976 examined ticles is mixed into a polymer solution. The motions or polystyrene spheres diffusing through dextran solutions. relative motions of the probe particles are then measured. The microrheology approach to interpreting probe dif- In most experiments discussed here, probe motions arise fusion stems from early work on Diffusing Wave Spec- troscopy (DWS), e.g., the 1993 chapter by Weitz and from diffusion; the small literature on driven motions of 2 mesoscopic probes through polymer solutions is also re- Pines . Particle tracking methods were early applied to observing the motion of labelled tags in cell membranes viewed here. In some systems, probe motions involve 3 multiple relaxations whose time dependences can be in- as reviewed, e.g., by Saxton and Jacobson . The ex- dependently determined. In others, a single relaxation perimental literatures in these three areas are less than entirely communicating. determined a probe diffusion coefficient Dp. Dp is sensitive to the probe radius R, polymer molecular weight M and concentration c, solution viscosity η, solvent viscosity ηs, and other variables. The dependence of Dp on A. Other Reviews these and other variables is used to infer how polymers move in solution. The optical probe diffusion literature has not recently The remainder of this Section presents a historical been reviewed systematically. In 1985, Phillies, et al.4 background. Section II remarks briefly on the theory un- re-examined extant optical probe studies of solutions of derlying major experimental methods for studying probe bovine serum albumin, polyethylene oxide, and poly- arXiv:0705.3222v1 [cond-mat.soft] 22 May 2007 diffusion. Section III presents the experimental phe- acrylic acid, primarily from their own laboratory. A nomenology. Section IV discusses the systematics of that uniform stretched-exponential concentration dependence ν 0.8±0.1 phenomenology. exp(−αc ) of Dp was found. A dependence α ∼ M The literature on probe diffusion studies of polymer was noted over a range of matrix species. α for probes solutions tends to be divided into three parts, namely in BSA solutions complies with this dependence if BSA (i) optical probe diffusion studies, largely with quasi- is assigned an effective M corresponding to its radius. elastic light scattering spectroscopy (QELSS), fluores- Phillies and Streletzky5 presented in 2001 a short re- cence recovery after photobleaching (FRAP), and Forced view (38 references) of the literature on optical probe Rayleigh Scattering (FRS), of the thermal motion of di- diffusion. They treat primarily systems studied with lute probe particles, (ii) microrheology studies in which QELSS. In some cases Dp follows the Stokes-Einstein an inferred mean-square particle displacement and a gen- equation evaluated using the solution viscosity. In other eralized Stokes-Einstein relation are used to compute dy- cases, diffusion is appreciably faster than would be ex- namic moduli of the solution, and (iii) particle-tracking pected from the solution viscosity. In a few cases one studies in which the detailed motions of individual par- finds re-entrant behavior in which the Stokes-Einstein 2 equation fails but only over a narrow band of concentra- ment of particle interactions with laser tweezer methods. tions. Finally (taking probes in hydroxypropyl-cellulose The focus is the application of these techniques to col- as an examplar), in some cases the spectral mode struc- loidal particles in solution. ture is too complicated to be characterized with a single diffusion coefficient. Phillies and Streletzky’s review does not contact the microrheology literature. II. THEORY UNDERLYING EXPERIMENTAL A series of reviews treat microrheology: METHODS Diffusing Wave Spectroscopy was reviewed (49 references) by Harden and Viasnoff6. The primary emphases Many methods have been applied to study the mo- are on two-cell light scattering measurements, in which tion of mesoscopic probe particles through polymer so- light scattered by the sample of interest is rescattered by lutions, including QELSS, FRS, FRAP, DWS, particle a second scattering cell before being collected, and on tracking, and interferometry. Optical probe methods re- CCD methods, which allow one to collect simultaneously fer to a special case of scattering from a general three- the light scattered into a substantial number of coherence component probe:matrix polymer:solvent system. In the areas. special case, one of the components, the probe, is dilute Solomon and Lu7 review (54 references) microrheology yet nonetheless dominates the scattering process, even studies of probe diffusion in complex fluids, in particular though the other solute component, the matrix, may be uses of the general Stokes-Einstein equation and its range far more concentrated than the probe. Most of these of validity, correlations in the motion of pairs of large methods, in particular QELSS, FRS, and FRAP, have particles, data analysis methods, and possible paths for also been used to study motion of dilute labelled polymer extending their methods to smaller probe particles. The chains in polymer solutions, leading to measurements of reference list has very limited contact with the optical polymer self- and tracer- diffusion coefficients. probe diffusion literature. Light scattering spectroscopy is sensitive to the dy- MacKintosh and Schmidt8 discuss (60 references) stud- namic structure factor S(q, τ) of the scattering parti- ies on the diffusion of microscopic probe particles through cles. The theoretical basis for applying QELSS, FRS, polymer solutions by means of particle tracking and dif- and FRAP to three-component solutions appears in our 12 fusing wave spectroscopy, as performed under the cog- prior review and need not be repeated here in detail. nomen microrheology, as well as studies of viscoelasticity For dilute probe particles diffusing in a non-scattering using atomic force microscopy and the driven motion of matrix, S(q, τ) reduces to mesoscopic particles. The excellent list of references lacks N contact with the QELSS/FRAP-based optical probe dif- S(q, τ)= h exp(iq · ∆ri(τ))i. (1) fusion literature. Xi=1 9 Mukhopadhay and Granick review (41 references) ex- Here i labels the N probe particles whose locations at perimental methods of driving and measuring the dis- time t are the ri(t). S(q, τ) is determined by the particle placement of mesoscopic particles, such as DWS and op- displacements ∆ri(τ)= ri(t + τ) − ri(t) during τ. tical tweezer techniques. They obtain the complex mod- The scattering vector q, with ulus of the fluid is via a generalized Stokes-Einstein equa- 4πn tion. The discussion and references do not contact the q = sin(θ/2), (2) optical probe diffusion literature. λ Among reviews of particle tracking methods, note: determines the distance a particle must diffuse to have a Saxton and Jacobson3 treat (105 references) experi- significant effect on S(q, τ). Here n is the solution index mental methods for tracking single particles as they move of refraction, λ is the illuminating wavelength in vacuo, in cell membranes. Motion of membrane proteins and and θ is the scattering angle. In observing a polymer coil other probes is complex, because the motion may be dif- with QELSS, complications arise if the polymer coil ra- fusive, obstructed, include nondiffusive convective mo- dius Rg is comparable with q, because coil internal modes tion, or involve trapping. Correlations between single enter the spectrum

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