arXiv:0705.3222v1 [cond-mat.soft] 22 May 2007 cosity ouin ed ob iie notreprs namely parts, three into divided be (i) to tends solutions phe- experimental that of the systematics the phenomenology. presents discusses IV III Section nomenology. Section probe studying un- for diffusion. theory methods the experimental on major briefly derlying remarks II Section background. polymers how infer to used solution. is in variables move other and these eednl eemnd nohr,asnl relaxation coefficient single diffusion a probe others, involve a in- In determined be motions can probe determined. dependences systems, dependently time whose some re- relaxations In multiple also is of here. solutions motions viewed polymer driven through on probes literature mesoscopic arise small motions the probe diffusion; here, or from discussed motions measured. experiments then The most are particles In solution. probe polymer the of a motions par- relative into probe mixed mesoscopic diffusion is of probe ticles dispersion a dilute In a experiment, dynamics. polymer investigating of alihSatrn FS,o h hra oino di- (ii) of motion particles, thermal probe the lute of (FRS), Forced fluores- Scattering and Rayleigh (FRAP), (QELSS), photobleaching after spectroscopy recovery cence scattering light elastic tde nwihtedtie oin fidvda par- individual of motions detailed (iii) the and which solution, in the studies of dy- compute moduli to namic used are gen- relation a Stokes-Einstein and eralized displacement particle mean-square inferred an iiet h rb radius probe the to sitive M h ieaueo rb iuinsuiso polymer of studies diffusion probe on literature The historical a presents Section this of remainder The hsrve raspoedffso n eae methods related and diffusion probe treats review This n concentration and pia rb diffusion probe optical η s n te aibe.Tedpnec of dependence The variables. other and , ssseaial eiwd h rmr ou stesuyo study solutio gene the coefficient of diffusion is probe as measurements the with that focus namely made obtained, primary was is The Comparison particles. reviewed. systematically is iuincecetde o ipytakteslto visco solution the track simply not does coefficient diffusion and esrmnsi nmn ae usatal mle hnth than smaller substantially examine instrument. cases also many are force in external is known measurements a rea of ex influence are studies the under behaviors microrheological low-conce True these or effects. re-entrant rarity, chemical sees specific their one systems, from that few interpretation; very scale hydroxypropyl a length that In concentration-independent, implies coil. single, phenomenology a probe by full molecu the spectr and of probe concentration observed polymer the hydroxypropylcellulose:water, increasing with markedly h xeietlltrtr ntemto fmssoi pro mesoscopic of motion the on literature experimental The D .INTRODUCTION I. ν p = per ohv large- have to appears eateto hsc,WretrPltcncIsiue W Institute, Polytechnic Worcester Physics, of Department D c ouinviscosity solution , p 0 microrheology exp( R pia rb iuini oye Solutions Polymer in Diffusion Probe Optical tde,lreywt quasi- with largely studies, oye oeua weight molecular polymer , − αc ν .Oefid that finds One ). tde nwhich in studies particle-tracking M D η p ovn vis- solvent , n small- and . D ereD .Phillies J. D. George p ssen- is D α p M on eed nplmrmlclrwihsas weights molecular polymer on depends auswt rsoe ikn hm h probe The them. linking crossover a with values h oinnwcalled now motion the recorded. are ticles rsoy(W) .. h 93catrb et and Weitz by chapter Spec- Wave 1993 Diffusing the Pines on e.g., work (DWS), dif- early troscopy probe from interpreting stems to fusion approach solutions. microrheology dextran The through diffusing spheres polystyrene eietlltrtrsi hs he ra r esthan less are areas Jacobson three communicating. and these entirely Saxton in membranes literatures by cell perimental in e.g., tags reviewed, labelled as of motion the observing cyi cd rmrl rmteronlbrtr.A laboratory. own exp( poly- their dependence concentration and from stretched-exponential uniform primarily oxide, polyethylene acid, of acrylic solutions albumin, al. of et serum studies Phillies, bovine probe 1985, optical In extant re-examined systematically. reviewed been ina ehdfrsuyn oye ouin dates Hallett solutions and polymer Turner studying for to method back a as sion a oe vrarneo arxspecies. matrix of range a over noted was iuin hytetpiaiyssessuidwith studied probe systems cases optical some primarily on In treat literature QELSS. They the of diffusion. references) (38 view etdfo h ouinvsoiy nafwcssone Stokes-Einstein cases the few which a in In ex- behavior viscosity. be re-entrant would solution finds than the faster from appreciably other pected is In viscosity. diffusion solution cases, the using evaluated equation nBAsltoscmle ihti eednei BSA if dependence effective this an with assigned is complies solutions BSA in itrcly rw sdpril rcigt study to tracking particle used Brown Historically, h pia rb iuinltrtr a o recently not has literature diffusion probe optical The hlisadStreletzky and Phillies − mnn h oino eocpcparticles mesoscopic of motion the amining r i rtimdl xeddanalysis Extended tri-modal. or bi- are a 2 oal nepee screpnigto corresponding as interpreted sonably αc atcetakn ehd eeeryapidto applied early were methods tracking Particle . sapoiaeytesz fapolymer a of size the approximately is icst esrdwt macroscopic a with measured viscosity e tainpaeubhvoso uncertain of behaviors ntration-plateau al eed nplmrconcentration polymer on depends rally .Vsoiyfo remicrorheological true from Viscosity d. ∗ ν eprilstruhplmrsolutions polymer through particles be iy instead, sity; icste.Achrn description coherent A viscosities. n of ) ells ouin r characterized are solutions cellulose a egt nsm ytm,e.g., systems, some In weight. lar iuiemto fsalprobe small of motion diffusive f D p retr A01609 MA orcester, a on.Adependence A found. was .OhrReviews Other A. D M p Brownian D η 5 p orsodn oisradius. its to corresponding yial increases typically rsne n20 hr re- short a 2001 in presented olw h Stokes-Einstein the follows 1 h n17 examined 1976 in who , pia rb diffu- probe Optical . α ∼ M 0 . 8 α , α 3 ∼ h ex- The . o probes for M 0 . 8 ± 0 . 4 1 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 refer- ences) 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 if qRg is not ≪ 1. However, the particle tracking and FRAP are considered, including the rigid probes considered here have no significant internal FRAP-determined nominal fraction of immobile parti- modes, so light scattering spectroscopy measures rigid- cles. probe center-of-mass motion regardless of qRg. Tseng, et al.10 review (52 references) particle tracking For dilute monodisperse rigid probes in a simple New- methods as applied to solutions of F-actin and other cy- tonian solvent, probe motion is described to good approx- imation by the Langevin equation. In this case, the dy- toskeletal proteins. While mention is made of measuring 13 mean-square displacements of diffusing particles, this is namic structure factor reduces to a simple exponential 2 in the context of a technique that actually does measure S(q, τ) = exp(−2Dpq τ) (3) particle positions and displacements. in which D is the probe diffusion coefficient. In one A review (56 references) of microscopy techniques by p dimension, D is related to particle displacement by Habdas and Weeks11 emphasizes video microscopy, imag- p 2 ing crystals, glasses, and phase transitions, and measure- h(∆x(τ)) i =2Dpτ. (4) 3
The Stokes-Einstein equation relates Dp for spheres of of Brownian particles that follow the Langevin equation. radius R to other parameters, namely Berne and Pecora’s treatment is without error, but the restrictions on the range of validity of their results are kB T not uniformly recognized. Dp = . (5) 6πηR The Langevin equation is a model for an isolated par- ticle floating in a simple solvent. The particle is as- Here kB is Boltzmann’s constant, T is the absolute tem- sumed to be subject to two forces, namely a drag force perature, and η is the experimentally-measured macro- FD(t) = −fv(t) and a ’random’ force F(t). Here f is scopic solution viscosity. When D and R or η are known, the drag coefficient and v(t) is the time-dependent par- this equation may be inverted to yield a microviscosity ticle velocity. The random force is uncorrelated with the ηµ or an apparent hydrodynamic radius rH , respectively, drag force. Also, the random forces at any distinct pair viz. of times are uncorrelated, i.e., hF(t) ·F(t + τ)i = 0 for k T τ 6= 0. The particle’s motion follows from Newton’s Sec- r = B (6) H 6πηD ond Law. The model only specifies statistical properties p of F(t), so only statistical properties of particle displace- and ments can be obtained from the Langevin model. Under these conditions, Doob14 showed in 1942 that: kBT (i) P (∆r, τ) is a Gaussian in ∆r, with mean-square dis- ηµ = . (7) 2 6πRDp placement in one dimension satisfying h∆x (t)i = 2Dt; (ii) the spectrum reduces to For probes in polymer solutions, ηµ may differ markedly from the macroscopically measured viscosity η, typically S(q, τ) = exp(−q2h∆x2(t)i) (9) with ηµ/η ≪ 1. The rH may be much less than the experimentally-determined physical radius R. We denote and (iii) as an absolutely rigorous mathematical conse- ηµ ≈ η and rH ≈ R as Stokes-Einsteinian behavior. The quence, S(q, τ) is a single exponential, namely S(q, τ)= 2 contrary case (ηµ 6= η and rH 6= R) we describe as non- exp(−2Dpq t). These results describe, e.g., the light Stokes-Einsteinian behavior. scattering spectrum of dilute polystyrene spheres in pure In more complex systems, the dynamic structure fac- water, which was an early target for experimental stud- tor is non-exponential. S(q, τ) remains monotonic, with ies. Laplace transform Spectra of probe particles in viscoelastic fluids com- monly are not simple exponentials. In the litera- ∞ ture misinterpretation, eq. 9 is incorrectly claimed– S(q, τ)= dΓA(Γ) exp(−Γt). (8) under the cognomen ’gaussian assumption’ or ’gaussian Z0 approximation’–to be uniformly applicable to light scat- Here A(Γ) ≥ 0 is the relaxation distribution function. In tering spectra, in particular to spectra that do not show a some cases, A(Γ) is qualitatively well-described by a sum single-exponential decay. Is this realistic? In a viscoelas- of relatively separated peaks, commonly termed ’modes’. tic liquid, the elastic moduli are frequency-dependent. This terminology does not imply that there is necessarily Correspondingly, the stress fluctuations in the liquid cre- a 1-to-1 correspondence between individual modes and ate random forces with non-zero correlation times, i.e., individual physical relaxation processes, though such a hF(t)·F(t+τ)i 6= 0 forarangeof τ 6= 0. Correspondingly, correspondence often exists. the displacement distribution function is not a Gaussian, An arbitrary physical A(Γ) can be generated by scat- and h(∆x(τ))2i shows non-diffusive behavior (≁ t1) that tering light from an appropriately-chosen mixture of di- is not characterized by a diffusion coefficient. A contra- lute Brownian probes in a simple fluid. For each Brown- positive statement of Doob’s theorem shows that non- ian species, the mean-square displacement increases lin- exponential spectra correspond necessarily to particles early in time. Correspondingly, every physical S(q, τ) whose motion is not described by eq. 9. corresponds to a system in which the mean-square par- An explicit calculation that correctly expresses S(q, τ) ticle displacement increases linearly in time. However, in terms of all moments h∆x2n(t)i, n ≥ 1, has been a non-exponential S(q, τ) can also arise from systems in made15. S(q, τ) reflects all moments h∆x2ni of P (∆r, τ). which h∆x(τ))2i has more complex behavior. It is im- Except under special conditions not satisfied by probe possible to distinguish these possibilities by examining particles in complex fluids, the higher moments n > 1 spectra taken at a single scattering angle or from spectra make non-trivial contributions to S(q, τ). S(q, τ) and uniformly averaged over a range of q. Thus, it is impos- QELSS single- and multiple-scattering spectra are in gen- sible to determine h(∆x(τ))2i from one light scattering eral not determined by h∆x2i. Equivalently, there are spectrum of an unknown system. pairs of systems having the same S(q, τ), but very differ- Recently, a misinterpretation of the historical litera- ent time dependences for h(∆x)2i. Articles based on the ture on light scattering spectra has emerged. The start- gaussian approximation are considered in Section III.F. ing point is the excellent book of Berne and Pecora13, S(q, τ) is the spatial Fourier transform of the probabil- in particular their treatment of light scattering spectra ity distribution P (∆r, τ) for finding a probe displacement 4
∆r during τ. While a single measurement of S(q, τ) can- 10 1 not be used to compute P (∆r, τ), determination of the complete functional dependence of S(q, τ) on q could in principle be used to determine P (∆r, τ). To the authors knowledge, this has only been done in the trivial case of simple Brownian motion. Particle tracking methods 0
/s) 10 yield directly P (∆r, τ), and can be used to compute com- 2 plex many-time displacement cross-correlations, such as
P (∆r(t,t + τ), ∆r(t + τ,t + τ + θ)), where ∆r(t,t + τ) cm -8 is the probe displacement between t and t + τ. These
cross-correlations, which have not yet been intensively (10 p 10 -1 exploited, are substantially inaccessible to conventional D light scattering spectroscopy. a
III. PHENOMENOLOGY 10 -2 This section describes experiments on the diffusion of 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 rigid probes through polymer solutions and related sys- c (wt %) tems, and related measurements using other techniques, including particle tracking measurements and true mi- 10 2 croscopic rheological measurements. The major subsec- tions of Section III are (A) probe diffusion in polymer solutions, (A.1) probe diffusion in hydroxypropylcellu- lose (HPC) solutions, (B) rotational diffusion of probes, (C) particle tracking methods, (D) true microrheological b 10 1 measurements, (E) probes in polyacrylamide gels, protein / s) gels, and in vivo and in other structured systems, and (F) 2
probe spectra interpreted with the gaussian displacement cm assumption. For simplicity, within each section descrip- -8 tions of the primary literature are sorted alphabetically,
(10 0 except that studies on HPC:water are ordered chronolog- p 10 ically. Measurements on probes in solutions of rigid-rod D polymers and in colloidal systems will be discussed else- where. In generating the results of this Section, experimental findings were taken from tabulated numbers in the origi- 10 -1 nal papers, or were recovered from the original papers by 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 scanning original figures and digitizing the images. Fit- ting curves were generated by non-linear least-squares c (wt %) analysis based on the simplex method, leading to param- eters in Tables I-V and the smooth curves in the Figures. FIG. 1: Fast and slow modes in spectra of (a) 100 nm sul- Figures were generated de novo from digitized numbers fate latex spheres in polyacrylamide solution, and (b) 33 nm and our fitting curves. Most smooth curves represent fits hematite particles in sodium polyacrylate solutions, pH 10, to stretched exponentials in concentration, namely 0.1M NaNO3, both as functions of polymer concentration, after Bremmell, et al.16 ν Dp = Dp0 exp(−αc ). (10) Here ν is a scaling exponent, α is a scaling prefactor, and Dp0 reflect probe behavior at very low polymer concen- 100 nm diameter polystyrene latex (PSL) spheres and tration. 33 nm diameter hematite particles. Matrix solutions in- cluded water: glycerol, aqueous 3 Mda polyacrylamide (PAAM), and aqueous high-molecular weight sodium A. Probe Diffusion in Polymer Solutions polyacrylate (NaPAA). Spectra were fit to a sum of ex- ponentials. PSL spheres in water:glycerol showed single- This section treats experiments that measured probe exponential relaxations whose D scaled linearly with T/η translational diffusion. as the temperature and glycerol concentration were var- Bremmell, et al.16 used QELSS to examine the diffu- ied. Polyacrylamide solutions show marked shear thin- sion of positively-charged 34 nm and negatively-charged ning. 5
10 2
0.8 10 1 p0 p0 /D /D p p D D 0 0.4 10
10 -1 0 10 -3 10 -2 10 -1 10 0 10 1 0 10 20 30 c (g/L) c (g/L)
FIG. 2: Fast (open points) and slow (filled points) diffusion FIG. 3: Dp of 160 nm silica spheres through solutions (top to coefficients of 100 nm polystyrene sulfate spheres in solutions bottom) of 101, 163, 268, and 445 kDa PMMA, as obtained of polyacrylic acid-co-acrylamide in the presence of 0.001 ( ) with QELSS by Brown and Rymden19, and fits to stretched 17 or 1.0 (♦) M NaNO3, from data of Bremmell and Dunstan . exponentials in c.
As seen in Figure 1, in polyacrylamide solutions: Ex- q2. cept at very low polymer concentrations, PSL sphere Brown and Rymden18 used QELSS to examine the spectra are bimodal, usefully characterized by fast and diffusion of PSL spheres through carboxymethylcellulose slow diffusion coefficients Df and Ds. With increasing (CMC) solutions. The primary interest was to treat ad- polyacrylamide concentration, Ds of the 200 nm spheres sorption by this polymer to the latex spheres. Brown and falls 30-fold while η increases by only 20-fold. Ds does Rymden conclude that CMC goes down on the surface in not depend strongly on q. Df shows re-entrance: it first a relatively flat conformation. CMC is a weak polyelec- increases with increasing polymer c and then decreases trolyte. The extend of its binding to polystyrene latex is to below its zero-c value. At elevated c, Df increases pro- complexly influenced by factors including salt concentra- foundly, and more rapidly than linearly, with increasing tion, pH, and probe size and surface chemistry. 2 q . Brown and Rymden19 used QELSS to determine With 68 nm spheres, Df at smaller c increases with the diffusion of 160 nm radius silica spheres and increasing polymer c, though less than the increase with polystyrene random-coil-polymer fractions through poly- 200 nm spheres, while Ds falls with increasing c. With methylmethacrylate (PMMA) in CHCl3, as seen in Fig. hematite particles in sodium polyacrylate solutions, with 3. They also measured the solution viscosity. PMMA concentrations increasing up to 1 wt%, Df and Ds both molecular weights for the sphere diffusion study were 101, increase, Df by ten-fold and Ds by at least three-fold. 163, 268, and 445 kDa. QELSS spectra were uniformly Bremmell, et al.16 refer to speculation that elasticity (so- quite close to single exponentials. For silica spheres, lution viscoelasticity) is related to the observed hyperdif- Dp/Dpo was to good approximation a universal function fusivity. of c[η]. The product Dpη for spheres was independent Bremmel and Dunstan17 examined the diffusion of from c, even though Dp changes by three orders of mag- 100 nm radius PSL spheres in 3 MDa poly(acrylic acid- nitude over this concentration interval. For chain probes, co-acrylamide) at a range of ionic strengths and poly- Dsη increases by a few to 70% with increasing c, the in- mer concentrations. Inverse Laplace transformation of crease in Dsη increasing with increasing matrix molecu- QELSS spectra found a bimodal distribution of relax- lar weight. Brown and Rymden found that concentration ation rates. Figure 2 shows representative results from dependence of the single particle diffusion coefficient, for the smallest and largest ionic strengths examined. In 1 either chain or sphere probes, is to good accuracy given M NaNO3, Df and Ds both increase with increasing c. by a stretched exponential in polymer concentration. 20 In 0.001M NaNO3, Df shows re-entrant behavior, while Bu and Russo used FRAP to measure the diffusion Ds simply decreases with increasing c. Df and Ds both of 55 nm PSL spheres, fluorescein, and eight fluores- show a complex ionic-strength-dependent dependence on cently labeled dextrans (molecular weights 3.9 kDa to 6
2 MDa) through HPC solutions. Nominal matrix molec- 60 18 ular weights were 60, 300, and 1000 kDa. Detailed results were presented for the 300 kDa matrix as seen in Figure 4. Probe particles diffused more rapidly than expected from the macroscopic solution viscosity, with ηµ/η increasing toward 1 as the probe radius is increased. Comparison
40 12 /s) was made by Bu and Russo with the Langevin-Rondelez 2 21 equation , which describes this data well. Dp(c) for each probe fits well to a simple exponential, except for the cm -7 (cP)
largest probe for which more data appears needed. h (10 20 6 p 10 2 D
10 1
) 0 0 -1
s 0 20 40 60 80 100
2 0 10 c (g/L) cm -7 -1 10 FIG. 5: Viscosity of DNA solutions and Dp of green fluores- (10
p cent protein in those solutions, based on data of Busch, et 22
D al. , and fits to stretched exponentials in c. 10 -2
-3 10 60 0 10 20 30 40 c (g/L)
FIG. 4: Dp of fluorescein (0.5 nm), 1.3, 1.7, 2.8, 4.5, 5.8, 8.8, 40 13.3, and 17.9 nm hydrodynamic radius dextrans, and 55 nm polystyrene spheres in aqueous 300 kDa HPC, as obtained 20 (Fick-cP) h
with FRAP by Bu and Russo , and fits to exponentials in c. p D 20 Busch, et al.22 used FRAP to study diffusion of green fluorescent protein (GFP) through glycerol, ficoll, and 160 bp calf thymus DNA solutions. QELSS was used to measure the scattering spectrum of the DNA itself; macroscopic viscosities were obtained using capillary vis- 0 cometers. Representative measurements on DNA solu- 0 50 100 150 200 250 tions appear as Figure 5. In water:glycerol, the probes c (g/L) followed the Stokes-Einstein equation, as seen in Figure
6. In ficoll solutions the probes diffused moderately more 22 rapidly than expected from the solution viscosity. In FIG. 6: Dpη of green fluorescent protein in solutions of glycerol( ), ficoll 70 (△), and DNA (♦), showing near-Stokes- DNA solutions, at elevated c, Dp of the GFP was much Einstein behavior in glycerol and ficoll 70 solutions, and large larger than expected from the solution viscosity. deviations from Stokes-Einstein behavior in DNA solutions. 23 Cao, et al. used QELSS to measure Dp of 32 and 54 nm radius polystyrene spheres, phospholipid/cholesterol vesicles, and multilamellar vesicles diffusing in aqueous than at retarding probe diffusion of polystyrene latex non-crosslinked 65 and 1000 kDa polyacrylamides. Dp spheres, and were more effective at retarding the mo- arose from a relaxation rate that was accurately linear in tion of spheres than at retarding unilamellar vesicles of q2; its concentration dependence was a stretched expo- the same radius. The difference between the vesicle and nential in c. As seen in Figure 7, comparison of Dp with sphere diffusive mobilities was interpreted as arising from the solution fluidity η−1 finds that both polyacrylamides the flexibility of the vesicles. were more effective at increasing the solution viscosity Cheng, et al.24 used FRAP to monitor the diffusion of 7
10
1 /s) 2
/h 0.9 0 , h ,
0.8 cm p0 -7 1 /D p (10
p D 0.7 D
0.6 a
0.5 0.1 0 2 4 6 8 10 0.1 1 10 c (g/L) c (wt %)
1 FIG. 8: Dp of 20 and 70 kDa dextrans in 2MDa aqueous guar solutions as functions of guar concentration, and simple- exponential fits. The larger probe is more effectively retarded 24 0.5 by the solution, after Cheng, et al. . /h 0 0.3 ovalbumin in 40 kDa PEO, and (b) 70 kDa dextran and , h , dendrimer G8 in 200 kDa PEO. For various probes, Dp p0 0.2 falls by factors of 5 to 30 over the observed concentration /D p range. PEO is roughly equally effective at retarding the D diffusion of the 20 kDa dextran and ovalbumin, but is 0.1 more effective at retarding the diffusion of G8 dendrimer b than at retarding the motion of 70 kDa dextran, even though G8 dendrimer and 70 kDa dextran have the same Dp0. 0.05 De Smedt, et al.25 used FRAP to examine diffusion 0 2 4 6 8 10 of fluorescein-labelled dextrans and polystyrene latex c (g/L) spheres through hyaluronic acid solutions. Dextrans had molecular weights 71, 148, and 487 kDa. The hyaluronic acid had Mn and Mw of 390 and 680 kDa. The dex- FIG. 7: Dp/Dp0 of 64 nm polystyrene latex spheres ( ) and tran diffusion coefficients depend on matrix polymer c 65 nm radius unilamellar lipid vesicles (), and normalized as stretched exponentials in c, as seen in Figure 10. fluidity η0/η (•) in (a) 65 kDa and (b) 1 MDa polyacrylamide 23 Hyaluronic acid solutions are somewhat more effective solutions, after Cao, et al. . at retarding the larger dextran probes. Viscosities for these solutions were reported by De Smedt, et al.26. The concentration dependence of η is stronger than the con- extremely small probes (1 ≤ R ≤ 20 nm) in solutions of centration dependence of Dp of the polystyrene spheres, guar galactomannan and polyethylene oxide. The probes which is in turn stronger than the concentration depen- were labelled dextrans (20, 40, 70, and 500 kDa), ovalbu- dence of Dp of the dextrans. Spheres and dextrans both min, and 6th and 8th generation starburst dendrimers. diffuse more rapidly expected from the solution viscosity PEO fractions had molecular weights 40, 200, and 600 and the Stokes-Einstein equation. 27 kDa; the guar had molecular weight 2MDa. Dp of various De Smedt, et al. used confocal laser scanning mi- probe:matrix combinations and η of one polymer solution croscopy (CSLM) and FRAP to study the diffusion of were obtained. Figure 8 shows Dp of 20 and 70 kDa dex- fluorescein-labelled dextran molecules (19 ≤ Mw ≤ 487 trans through 2 MDa guar solutions, the larger dextran kDa) through dextran-methacrylate, comparing Dp of being more effectively retarded by the guar. Figures 9a probes in solutions with probes in swollen and relaxed and 9b compare the diffusion of (a) 20 kDa dextran and cross-linked gels. The ability of the crosslinked gels to 8
10 10
a /s) /s) 2 2 cm cm
-7 1 -7 1 (10 (10
p p D D
0.1 0.1 0 5 10 15 20 25 0 4 8 12 16 20 c (wt %) c (g/L)
10 FIG. 10: Dp of 71.2 ( ), 147.8 (), and 487 (♦) kDa dextrans in Mw = 680 kDa hyaluronic acid solutions as functions of hyaluronic acid concentration, after De Smedt, et al.25.
b
/s) 10 2 cm
-7 1 (10
p /s) D 2 cm
-7 1 (10
p
0.1 D 0 2 4 6 c (wt %)
FIG. 9: Dp of (a) 20 kDa dextran ( ) and ovalbumin () 0.1 and (b) 70 kDa dextran ( ) and dendrimer G8 () in (a) 40 and (b) 200 kDa PEO solutions, and stretched-exponential 0 40 80 120 160 fits, after Cheng, et al.24. c (g/L)
FIG. 11: Dp of 19 ( ), 51 (), 148(♦), and 487 (•) kDa dextrans in dextran methacrylate solutions as functions of retard probe motion increased with gel matrix concen- 27 tration and increasing probe molecular weight, with the dextran methacrylate concentration, after De Smedt, et al. . smallest probe diffusing at a rate independent of the matrix concentration. η, and Dp of each probe, all have stretched-exponential concentration dependences, was always less than or equal to Dp of the same probe as seen in Figure 11. All probes diffuse more rapidly than in a noncrosslinked solution at the same concentration. expected from the solution viscosity. The larger probes Swelling a gel increased Dp, but only at lower levels of are slightly more effectively retarded by the solution. The gel crosslinking. 28 concentration dependence of Dp approaches the concen- Delfino, et al. used QELSS to study probes in aque- tration dependence of η more closely as the molecular ous solutions of carboxymethylcellulose, nominal molec- weight of the probes is increased. Dp of probes in gels ular weight 700 kDa, at concentrations 0.2-11.7 g/L. 9
10 1 10 1
a 10 0 c 10 0 p0 p0 -1 /D /D
p p 10 D D 10 -1 10 -2
10 -2 10 -3 0 4 8 12 0 2 4 6 8 10 c (g/L) c (g/L)
10 0 the spectrum at small c and the slow mode dominated at large c, while for the 102 nm probes, the slow mode 10 -1 always dominated. As seen in Figure 12, except for the slow mode of the largest spheres, which show a weakly stretched-exponential concentration dependence, Dp of
p0 either mode falls approximately exponentially with in- -2
/D creasing polymer c. Increasing c is more effective at re-
p 10 tarding the relaxation of the slow mode than at retarding D the relaxation of the fast mode. Increasing c is more ef- fective at retarding the motion of the larger 47 and 102 10 -3 b nm spheres than at retarding the motion of the small 14nm spheres. Dunstan and Stokes29 employ QELSS to measure diffu- sion of PSL spheres through glycerol and polyacrylamide -4 10 solutions. In glycerol-water, Dp tracked the solution vis- 0 4 8 12 cosity. In polyacrylamide solutions, spectra were fit to c (g/L) a sum of two exponentials. For 100 nm radius sulfate- modified spheres: Ds of the slow exponential fell with increasing polymer c. Df of the fast exponential showed FIG. 12: Fast ( ) and slow (♦) mode D /D 0 for (a) 14, (b) p p re-entrant behavior, with a maximum in Df at an inter- 47 and (c) 102 nm polystyrene sphere probes in aqueous 700 28 mediate polymer concentration considerably larger than kDa carboxymethylcellulose, after Delfino, et al. the overlap concentration, followed at still larger concen- trations by a decline until Df was less than a third of its zero-concentration value. ηµ from the slow exponential The probe particles were 14, 47, and 102 nm radius is greater than η from macroscopic measurements, while PSL spheres. Ubbelohde viscometers and a concentric- ηµ from the fast relaxation is less than η. Ds is substan- cylinder Couette viscometer were used to determine rhe- tially independent from q, while Df of probes in more ological properties of the solutions. From the intrinsic concentrated polyacrylamide solutions increases strongly viscosity [η], the overlap concentration was c∗ =1/[η]= with increasing q. More limited measurements on 34 nm 0.73 g/L, while from the linear slope of ηSP, Rg ≈ 50 radius amidine-modified spheres appear to yield the same nm was found. Probe spectra were fit to an exponential pattern of concentration dependences for Ds and Df . or a sum of two pure exponentials, a fast mode and a Furukawa, et al.30 used FRAP to measure the diffu- slow mode. For the 14 and 47 nm probes, spectra were sion of 19 nm radius polystyrene spheres, 40 and 150 kDa unimodal at smaller c and bimodal at larger c. With 14 dextran, and fluorescein dye through dextran matrix so- nm probes, the fast mode dominated at all concentra- lutions having 40 ≤ Mw ≤ 2000 kDa and concentrations tions. For the 47 nm probes, the fast mode dominated up to 300 or 600 g/L. Dp of dextran chains diffusing 10
1.2
1 a 100 a
0.8 0 0 /D /h s m h D 0.6 10
0.4
0.2 1 0 20 40 60 80 100 0 200 400 600 c (g/L) c (g/L)
10 4 100
3 b 10 2 0
M 2 /h p 10 m h
D 10
10 1 b
10 0 1 1 10 100 1000 0 100 200 300 c (g/L) c (g/L)
FIG. 13: (a) Ds of (•) 40 and (◦) 150 kDa dextrans and FIG. 14: η/η0 for viscosity from macroscopic measurements (b) Dp of 40 (open points) and 150 (filled points) kDa dex- (◦) and ηµ/η0 from probe diffusion using 20 nm radius trans through solutions of 2000 ( ), 500(♦), 110(), 83(⋆), polystyrene latex sphere (N), 150 kDa dextran (), 40 kDa and 40(△) kDa dextrans, after Furukawa, et al.30, and fits to dextran (), and fluorescein (•) probes in (a) 40 and (b) 150 stretched exponentials in c. kDa dextrans, after Furukawa, et al.30, and fits to stretched exponentials in c. through dextran solutions consistently has a stretched exponential concentration dependence; cf. Figure 13. In 350 kDa polymethylmethacrylate in the good solvents 40 kDa dextran solutions, probes other than fluorescein tetrahydrofuran (THF) and N,N-dimethylformamide diffuse at very nearly the rate expected from the solution (DMF), and the theta solvent (at 25o C) dioxane:water. viscosity. In solutions of 150 kDa dextran, all probes Spectra were analyzed with Laplace inversion. For probes diffuse more rapidly than expected from the solution vis- in good solvents, cumulant analysis was also applied. In cosity, but (cf. Figure 14a) ,with increasing probe size ηµ good solvents, a unimodal decay distribution was ob- tends toward the macroscopic η. served. Under Laplace inversion, spectra of probes in Gold, et al.31 studied the diffusion of highly cross- dioxane-water often showed a bimodal distribution of de- linked polystyrene latex spheres through solutions of cay rates. With dioxane:water as the solvent: The slow 11 mode increased as the sample aged. The amplitude of the 100 slow mode varied from small up to equal to the amplitude of the fast mode. The slow mode was more prominent at low angles. Further analysis was based primarily on the relaxation rate of the fast mode. Diffusion coefficients were inferred from slopes of Γ against q2, with typical standard errors in the slope of 1−10% leading to results of Figure 15. For probes in THF and DMF, diffusion is more rapid than expected from the solution viscosities and 0
/h 10 the Stokes-Einstein equation, the Stokes-Einstein equa- h tion failing by 50% or a factor of 2 at elevated polymer concentration. For probes in dioxane:water, probe mo- tion was slower than anticipated from the Stokes-Einstein equation, an effect attributed by Gold, et al. to polymer adsorption by the probe spheres. Gorti and Ware32 made an extensive study of probe diffusion in polyelectrolyte solutions, using FRAP as 1 the primary method, as seen in figure 16. Probe par- 0 20 40 60 80 100 ticles included fluorescein, 19 nm radius polystyrene c (g/L) spheres, and bovine serum albumin. The matrix poly- mers were 70 and 500 kDa polystyrene sulphonates. So- 1 lution viscosities were obtained using Ubbelohde vis- cometers. The dependence of Dp on polymer concen- tration, molecular weight, degree of neutralization, solu- tion ionic strength, and probe size was observed. Poly- mer solutions were more effective at retarding the diffu- sion of larger probes. At fixed polymer concentration,
probe diffusion was slowed by increasing the polymer p0 molecular weight. Measurements covered both the nom- /D
0.1 inal dilute and the nominal semidilute regimes. At low p ionic strength, polystyrene sulphonate solutions experi- D ence an ordinary-extraordinary transition in their QELSS spectra33. For the 1 and 5 mM salt solutions, the tran- sitions occur at 2 and 11 g/L PSS. Probe diffusion mea- surements taken above and below the transition appear to lie on the same line; the ordinary-extraordinary transi- tion does not affect the probe diffusion coefficient. Also, 0.01 −1 Dp does not track the solution fluidity η . Solution 0 20 40 60 80 100 ionic strength has a modest effect on Dp, an increase in c (g/L) I tending to reduce Dp, as seen in Figure 17. An early application of probe diffusion was presented by Jamieson, et al.34, who in the course of reporting on FIG. 15: Reduced viscosities and probe diffusion coeffi- 2.2 MDa xanthan in water, giving measurements of flow cients for solutions of 350 kDa polymethylmethacrylate. Sol- ♦ birefringence, light scattering spectra, and shear viscos- vents are tetrahydrofuran( ), N,N-dimethylformamide( ), and dioxane-water(), after Gold, et al.31. ity, also report limited measurements of Dp of 100 nm polystyrene spheres as polymer c is increased. Dp fell more than thirty-fold over the allowed range of concen- trations, though only for a few data points was Dp sig- small probes, carrageenan solutions and gels differed sub- nificantly less than Do. At the highest concentrations stantially in their effect on probe diffusion, gels being 4 studied, η/η0 attained values as large as 10 . The micro- substantially more effective at retarding probe motion viscosity ηµ inferred from Dp was thus as much as two than were solutions having the same polymer concentra- orders of magnitude less than η. tion. Johansson, et al. also found that the effect of a fixed Johanssson, et al.35 examined the diffusion of low- concentration of polystyrene sulphonate on probe diffu- molecular-weight (0.3-4 kDa) poly-ethylene glycol frac- sion decreases with increasing ionic strength. They pro- tions in κ-carrageenan crosslinked gels and in solutions posed that this difference arises because relatively rigid of 500 kDa sodium polystyrene sulphonate. The probes polymers are more able to affect probe motion than are were radioactively labelled; boundary spreading was used flexible polymers. 36 to determine Dp of dilute probes. Even for these very Konak, et al. report on the diffusion of block copoly- 12
1 1 p0 p0 /D /D p p
D D 0.1
0.1
0.01 0 5 10 15 20 0 5 10 15 20 c (g/L) c (g/L)
FIG. 16: Dp of fluorescein ( ), bovine serum albumin (), FIG. 17: Dp of 19 nm radius PSL in 500 kDa aqueous and 19 nm radius PSL (♦) in 500 kDa aqueous NaPSS: 10 polystyrene sulfonate solutions at various ionic strengths: mM phosphate buffer, and fits to stretched exponentials, after 0.001 ( ), 0.005 (), 0.01 (△), 0.02 (+), and 0.05 (♦) M 32 Gorti and Ware . phosphate ion, and fits of Dp to stretched exponentials in c, after Gorti and Ware32. mer micelles through solutions of linear polystyrene, as shown in Figure 18. The block copolymer was 12 a 74 kDa Kraton G-1650 polystyrene-hydrogenated polybutadiene-polystyrene that forms 4.8 MDa 23 nm 10 radius micelles. The matrix polymers were 110 and 200 kDa polystyrenes having Mw/Mn = 1.4. The mi-
/s) 8 celle diffusion coefficient Dp and the viscosity η were 2 reported. At intermediate polystyrene concentrations,
Dp was larger than expected from η and Do, but at the cm -8 6 largest c examined Dpη returned to its low-concentration value. The largest-c re-entrant behavior was only ob- (10 served at a single point, so it is impossible to say whether p 4 D Dpη/Dp0η0 ≈ 1 represents the general largest-matrix- concentration behavior, or whether at concentrations 2 larger than those studied Dp is smaller than expected from η. In a series of three papers, Lin and Phillies37,38,39 de- 0 termine diffusion of polystyrene spheres through aque- 0 20 40 60 80 ous solutions of non-neutralized poly-acrylic acid (PAA). Lin and Phillies37 report on the diffusion of nominal 38 c (g/L) nm carboxylate-modified spheres in solutions of 300 kDa
PAA at polymer concentrations 0.37 ≤ c ≤ 171 g/L. FIG. 18: Dp of 23 nm radius Kraton G-1650 micelles in so- They measured Dp as a function of temperature at 11 lutions of 110 ( ) and 200 () kDa polystyrenes in dioxane, polymer concentrations and η of PAA solutions at various after data of Konak, et al.36. polymer concentrations. At each polymer concentration Dp tracked T/η with reasonable accuracy. These prelimi- nary measurements were extended in a further study. Lin mer concentration. For each sphere size, Dp(c) has a and Phillies38 compared the diffusion of probe spheres stretched-exponential concentration dependence. (radii 20.4-1500 nm) in 300 kDa PAA, for solution con- The apparent hydrodynamic radius rH (Figure 19) of centrations 0.0113-145 g/L. As seen in Figure 19, in each the smaller spheres grew modestly with increasing poly- solution Dp/Dpo fell markedly, for large spheres by as mer concentration, rH of the largest spheres fell, and much as three orders of magnitude, with increasing poly- rH of intermediate spheres is nearly independent from c. 13
10 1 10 0
-1 10 10 0 p0 p0
/D -2 /D p 10 p D D 10 -1 a 10 -3
10 -4 10 -2 0.01 0.1 1 10 100 0.0010.01 0.1 1 10 100 1000 c (g/L) c (g/L)
1000 10 0 p0 /D p (nm) -1 h D r 100 10 b
10 10 -2 0 50 100 150 0.001 0.01 0.1 1 10 100 c (g/L) c (g/L)
FIG. 19: Dp/Dp0 and rh of 20 ( ), 80 (), 620 (♦) and 1500 FIG. 20: Dp/Dp0 of 20 ( ), 80(), 620 (△), and 1500 (♦) nm (△) nm radius spheres in aqueous 300 kDa non-neutralized radius carboxylate-modified polystyrene spheres in solutions 38 polyacrylic acid, after Lin and Phillies . Dp/Dp0 → 1 at of (a) 50 kDa, and (b) 1 MDa non-neutralized polyacrylic small c; data and fits were shifted vertically for clarity. acid, and fits of Dp to stretched exponentials in c, after Lin and Phillies39 . Lin and Phillies interpreted their results as arising from polymer adsorption (to form a layer 10-20 nm thick) on sition, η changing from a stretched-exponential concen- the spheres, coupled with a failure of the Stokes-Einstein tration dependence to a power-law concentration depen- equation (spheres diffuse faster than expected from the dence at c ≈ 1.4 g/L PAA. In 50 kDa (Figure 20a) and solution viscosity), especially for the larger spheres. 1 MDa PAA (Figure 20b), Dp(c) for each probe size fol- Finally, Lin and Phillies39 examined the diffusion of lows a stretched exponential in c. In 50 kDa PAA solu- 20.4, 80, 620, and 1500 nm radius carboxylate-modified tions, Dp and η are related by the Stokes-Einstein equa- polystyrene spheres in solutions of 50 kDa and 1 MDa tion. Dp in 1 MDa HPC solutions does not show Stokes- polyacrylic acid. They also measured the macroscopic so- Einsteinian behavior. Dpη/Dpoηo increases by four or- lution viscosity η. In 50 kDa PAA, η follows a stretched ders of magnitude with increasing c, as seen in Figure exponential in c. 1 MDa PAA shows a viscometric tran- 21. 14
Extended searches of the spectrum on the time scale Lin40 used QELSS to study diffusion of 155 and 170 on which the spectrum would have decayed, if the Stokes- nm nominal radius titania spheres as probe particles in a Einstein equation were correct, found no sign of a spec- melt of 7500 Da polyethylene oxide, at temperatures from tral relaxation; the spectrum had already decayed to the 85 to 160 C. Over this temperature range, Dp changes by baseline. Lin and Phillies proposed that that the discrep- nearly two orders of magnitude. Comparison was made ancy between Dp and the macroscopic η arises from shear with the shear-dependent viscosity obtained using a cone- thinning in the probe diffusion process, namely the micro- and-plate viscometer. The observed microviscosities were viscosity found on the time and distance scales probed by substantially less than the measured viscosity. An ex- the polystyrene spheres is far less than the macroscopic trapolation procedure based on the apparent activation viscosity η. energy inferred from the temperature dependence of η In 50 kDa HPC, the second cumulant of S(q,t) was was used to estimate an effective shear rate for probe dif- − substantially independent of c. In 1 MDa PAA, the spec- fusion ≥ 104 s 1, corresponding viaγ ˙ ∼ D/L2 to probe tral second cumulant was nearly constant for c
h mers, at small c the probe R was comparable to ξ. From r the ultracentrifuge experiments, the polymer solutions are approximately equally effective at retarding the sed- 0.1 imentation of smaller and larger probes. For the larger probes, Dp/Do and so/s have nearly the same concen- tration dependences; the polymers retard sedimentation and probe diffusion to approximately the same extent. For the smaller probes, the polymers may be more effec- 0.01 tive at retarding probe diffusion than at retarding probe sedimentation, though only for a few concentrations is the effect large. Nehme, et al., interpreted this behavior as a time scale 0.001 effect: When probe motion is adequately rapid, the poly- 0 4 8 12 16 20 mer is less able to move out of the way of the probes. The c (g/L) sedimentation coefficients came close to being a univer- sal function of the overlap concentration c∗. Polymer FIG. 21: Hydrodynamic radius of 20 ( ), 80 (), and 620 adsorption in this system is substantial; the thickness of (♦) nm radius spheres in aqueous 1 MDa polyacrylic acid, the adsorbed polymer layer on each probe was estimated 39 after Lin and Phillies as 20 nm. Dp from QELSS consistently tracked the so- . lution η, but was about 15% slower than expected from 15
10 -7 10 2
a a /s) 2 0 10 -8 10 1 h/h (cm p D
10 -9 10 0 0 4 8 12 16 20 0 20 40 60 80 100 c (g/L) c (g/L)
10 -7 10 -7
b b 10 -8 /s) /s) 2 2 10 -8 (cm (cm p p D D 10 -9
10 -9 10 -10 0 4 8 12 16 20 0 20 40 60 80 100 c (g/L) c (g/L)
FIG. 22: Dp from QELSS ( ) and inferred (♦) from the FIG. 23: (a) η/η0 and (b) Dp of 200 nm spheres in solu- friction factor from the sedimentation coefficient of (a) 80 tions of (♦) 215 kDa and ( ) 1100 kDa polystyrene in N,N- and (b) 40 nm PSL in solutions of 90 kDa poly-L-lysine, after dimethylformamide, after Onyenemezu, et al.43. Nehme, et al.42 . methods, all giving good agreement. Polymer molecu- lar weights were 215 and 1100 kDa, with polydispersities η in a concentration-independent way. 1.05 and 1.07, respectively. Polymer concentrations ap- Onyenemezu, et al.43 studied viscosity and probe dif- proaching 100 and 60 g/L, respectively, were attained, fusion (209 nm radius divinylbenzene-styrene spheres) in corresponding (cf. Figure 23a) to η/ηo as large as 20 or polystyrene: N,N-dimethylformamide solutions. Solu- 100 and 0.1 ≤ c[η] ≤ 10. Figure 23b shows Dp and fits to tion viscosities were measured with Cannon-Ubbelohde stretched exponentials in c. At larger scattering angles, viscometers. Shear rate effects were studied rheometri- a second, much faster relaxation interpreted as polymer cally; solutions showed Newtonian behavior at the shear scattering was found. Onyenemezu, et al., assert that rates found in the viscometers. QELSS spectra were ob- within experimental error the Stokes-Einstein equation is tained at three scattering angles, and fit using cumulant always closely obeyed in their systems; cf. Figure 24 for expansions and inverse Laplace transform (CONTIN) the modest non-Stokes-Einsteinian behaviors that their 16
4 1
3 0 h/h
0 0.1
h/h 2 p p0 D /D p D
0.01
1 0 20 40 60 80 100 0 100 200 300 c (g/L) c (g/L)
FIG. 24: Dpη/η0 of 200 nm spheres in solutions of (♦) 215 kDa FIG. 25: Dp/Dp0 of 20 nm (open points) and 230 nm (filled and ( ) 1100 kDa polystyrene in N,N-dimethylformamide, points) radius probes, and η/η0 (+, ×), with solutions of 70 after Onyenemezu, et al.43. (squares) and 500 (circles) kDa aqueous dextran, and fits to stretched exponentials (lines), after Phillies, et al.45. data reveal. Phillies44 reports on the diffusion of bovine serum albu- examines probe diffusion in solutions of not-quite-dilute min through solutions of 100 kDa and 300 kDa polyethy- polystyrene sulfonate. lene oxides. Dp depended measurably on the probe con- In Ref. 46, Phillies, et al. examined probes (19 ≤ R ≤ centration. At elevated polymer c and low protein con- 380 nm) diffusing through aqueous non-neutralized and centration, Dp was as much as a third faster than ex- neutralized low-molecular-weight (5 ≤ M ≤ 470 kDa) − pected from the c-dependent solution fluidity η 1. With poly-acrylic acid. The parameter space is quite large, increasing protein concentration, Dp fell toward values with possible effects of concentration, molecular weight, − expected from the macroscopic η 1. This study pushed and degree of neutralization, to to mention solution ionic the technical limits of then-current light-scattering in- strength. Phillies, et al.46 were only able to sample the strumentation. behaviors encountered in these systems. 45 Phillies, et al. observed probe diffusion in aqueous In 2/3 neutralized 5 kDa PAA, D/D0 fell exponentially dextran. Dextran concentrations covered 0 ≤ c ≤ 250 with increasing polymer concentration, but was indepen- g/L, using 9 different dextran samples. Probes were dent of salt concentration (0 ≤ I ≤ 0.1 M NaCl) and polystyrene spheres with radii 21 and 230 nm; solution very nearly independent of probe radius. Probes in 150 viscosities were obtained using thermostatted capillary kDa PAA showed a more complex dependence on these viscometers. Multiple tests confirmed that probe scat- parameters: For polymer concentrations 0.1-20 g/L, Dp tering completely dominated matrix polymer scattering. was relatively independent from I for at higher salt con- Representative measurements of Dp and η for two poly- centrations (I ≥ 0.01 M), but fell by a third to two- mer molecular weights appear in Fig. 25, together with thirds as I was reduced from 0.01 M toward zero added fits to stretched exponentials in c. Except for the smaller salt, the decline being larger at elevated polymer con- spheres in solutions of the very largest dextran, Stokes- centration. Probes in largely-neutralized 150 kDa have Einsteinian behavior was uniformly observed, Dp and stretched-exponential dependences on polymer c to good −1 η having very nearly the same concentration depen- accuracy, for concentrations out to 10 g/L and D/Do dences exp(−αcν ). This paper represented the first ex- as small as 0.25, except that at very low (c ≤ 0.2g/L) amination of the dependence of α and ν on Mw for probe polymer concentration Dp is larger than expected from diffusion through an extensive series of homologous poly- a fit of a stretched exponential exp(−αcν ) to the higher- mers. concentration measurements. Careful analysis revealed 46,47,48 1/2 Phillies, et al. report three studies of probe that at very low concentrations Dp = Dp0(1 − ac ) and diffusion in polyelectrolyte solutions. Two examine similarly for η. Also, the non-Stokes-Einsteinian behav- probes in dilute and concentrated solutions of high47 iors found by Lin and Phillies37,38,39 in non-neutralized and low46 partially-neutralized poly-acrylic acid. One48 PAA are also seen in partially-neutralized PAAs. 17
1.2 1
0.8
/s) 0.8 /s) 2 2 cm cm -7 -7 0.6 (10 (10 p 0.4 p D D 0.4
0 0.2 0 1 2 3 4 5 0 0.04 0.08 0.12 c (g/L) I (M)
FIG. 26: Dp of 20 nm radius probes in 596 kDa PAA, 60 FIG. 27: Dp of 20 nm probes in 1 g/L 596 kDa PAA, at % neutralized, at ionic strengths 0 (•), 0.01 ( ), 0.02 (), neutralizations 60% (◦), 85% (×), and 100% (•), as functions and 0.1 (+) M, after Phillies, et al.47, and fits to stretched of ionic strength I, and fits to stretched exponentials in I, exponentials (lines). after Phillies, et al.47.
In a separate study, Phillies, et al.47 examined the dif- Carboxylate-modified polystyrene spheres are charge- fusion of 21, 52, 322, and 655 nm radius carboxylate- stabilized. To prevent aggregation on the time scale of modified polystyrene spheres through 596 kDa Mw the experiments, the solvent included 1 mM NaOh and partially-neutralized poly-acrylic acid, using light scat- 0.1 or 0.32 g/L sodium dodecyl sulfate. Figure 28 shows tering to determine Dp. The dependences of Dp and representative data, namely Dp for each sphere size in the solution viscosity on c, solution ionic strength I 178 and 1188 kDa polystyrene sulfonate, and fits to sim- (0 ≤ I ≤ 0.1 M), and fractional neutralization A of ple exponentials in c. These results are interpreted below. 50,51,52 the polymer (0.6 ≤ A ≤ 1) were determined. Dp has Phillies, et al. report studies of the temperature a stretched-exponential dependence on c (Figure 26) and dependence of probe diffusion through various polymer a stretched-exponential dependence exp(−aIβ) on I for solutions. These studies constitute a response to a cri- β < 0. Probes generally diffuse faster than predicted tique, made to the author at several conferences, of us- by the Stokes-Einstein equation, the prediction of the ing the stretched-exponential concentration dependence. Stokes-Einstein equation being approached more nearly Critics complained that data had not been ”reduced rel- with lower polymer neutralization, larger solvent ionic ative to the glass temperature”. To examine this issue, 50,51,52 strength, and larger probes. The dependence of the ap- Phillies, et al. measured η and Dp for probes in parent hydrodynamic radius of the probes on solution pH various solutions at a series of temperatures. The sys- (and, implicitly, polymer neutralization) is much more tems studied included (i) 20.4 nm radius carboxylate- pronounced for the smaller 21 and 52 nm spheres than modified polystyrene sphere probes in 2/3 neutralized for the larger 322 and 655 nm spheres. If one believed low-molecular-weight 5 and 150 kDa poly-acrylic acid that the diffusion of probe particles, whose sizes exceed with and without added 0.1 M NaCl50, (ii) the same all of the length scales of the polymer solution, were gov- probes in 2/3 neutralized intermediate-molecular-weight erned by the macroscopic viscosity, then for this large-M 596 kDa poly-acrylic acid51 with or without added 0.1 polymer the longest length scale is at least 52 nm and M NaCL or added surfactant, and (iii) 34 nm nominal perhaps larger than 322 nm. radius polystyrene spheres in solutions of dextrans hav- 52 Phillies, et al.48 examined polystyrene spheres, radii ing 11.8 ≤ Mw ≤ 542 kDa . Polymer concentrations 7, 34, and 95 nm, diffusing through aqueous polystyrene reached 100, 45, or 20 g/L, respectively, for the three sulfonate. The purpose of the experiments was to de- poly-acrylic acids and 300 g/L for the dextrans. 50 termine the initial slope α of Dp against polymer c for For probes in solutions of low-molecular weight poly- various polymer M. Polymers had 7 molecular weights acrylic acid, the temperature dependence of of Dp seen with 1.5 ≤ Mw ≤ 1188 kDa. To minimize low-salt poly- in Figure 29 is entirely explained by the temperature de- electrolyte anomalies, the solvent included 0.2 M NaCl. pendence of the solvent viscosity ηs and Walden’s rule 18
3
1 /s) /s) 2 2 2 cm cm -7 -7 (10 (10 p p 1 D D
0.1 0 0 4 8 12 16 20 0 200 400 600 800 c (g/L) T/h (K/cP)
FIG. 28: Dp of 7 (△), 34 (♦), and 95 ( ) nm radius FIG. 29: Dp of 20 nm radius probes as a function of T/η in 2/3 polystyrene spheres in solutions of 178 (open points) and 1188 neutralized 5kDa polyacrylic acid at polymer concentrations (filled points) kDa polystyrene sulfonate, and fits to simple 0(◦),5(), 25 (•), 50 (△), and 100 (N) g/L, and linear fits at exponentials in c, after Phillies, et al.48. each c showing simple Walden’s rule behavior, after Phillies, et al.50.
Dp ∼ T/ηs. Furthermore, at each polymer concentra- These data thus reject the possibility of adjusting Dp at tion η/ηs is very nearly independent from T . This result o different c for a hypothesized variation in T − Tg with c. includes data for 2-65 C and fourteen polymer concen- 51 trations at two polymer molecular weights. There was no Separately, Phillies, et al. also fit their data at each polymer concentration to residual temperature dependence of Dp to be explained by glass temperature issues. The notion that Dp is sen- T T 2 sitive to T − T other than through η is rejected by this Dp = Dp0(1 + A1 + A2( ) + ...), (12) g s η η data. On the other hand, if hydrodynamic interactions between the polymer chains and the probes were primar- both with A2 = 0 forced and with A2 as an additional free ily responsible for controlling Dp, it would be reasonable parameter. Adding A2 as a further free parameter had no for Dp(T ) to depend linearly on the solvent viscosity, as effect on the RMS error in the fits, indicating that there is observed. Note that the chain monomer mobility is was no non-linear dependence of Dp on T/η. Phillies, also reasonable expected to scale linearly with ηs, so the et al. conclude that their data were consistent with hy- observations of Reference 50 are equally consistent with drodynamic models in which solvent-mediated forces are reptation-type dynamic models. the dominant forces between polymer chains, and were In ref. 51, Phillies, et al. examined probe diffusion equally consistent with reptation-type models in which through solutions of intermediate-molecular-weight poly- the solvent viscosity determines the monomer friction acrylic acid, comparing the temperature dependence of η constant. 52 and Dp with Walden’s rule and with the Vogel-Fulcher- Phillies and Quinlan used light scattering spec- Tamman equation. For the solution viscosity, this equa- troscopy to obtain the T -dependence of Dp of 34 nm tion is radius carboxylate-modified polystyrene spheres in solu- tions of various dextran fractions, 11.8 ≤ M ≤ 542 kDa B w η = η0 exp(− ). (11) at concentrations up to 300 g/L. Solution viscosities were T − Tg also measured. Representative measurements appear in Figure 31. Measurements of Dp were fit to a modified Here η0, B, and Tg are material-dependent parameters; Vogel-Fulcher-Tamman equation B is sometimes replaced with BT¯ g. Walden’s rule was fol- lowed accurately by all of these data. The VFT equation B Dp = Dp0T exp(− ). (13) uniformly describes the temperature dependence of the T − Tg measurements accurately. Fits of measurements covering all c studied to eq. 11, using the same Tg but different B and to eq 12. A very weak deviation from Walden’s rule at different c, gave excellent results as seen in Figure 30. was observed, Dp increasing with increasing T faster than 19 0.16
2 0.12 /s) 2 /s) 2 1.5 cm
-7 0.08 cm -7 (10
1 p (10 D p
D 0.04 0.5
0 0 200 400 600 800 0 20 40 60 T/h (K/cP) T (C) s
FIG. 31: Dp of 34 nm radius probes against T/ηs in aqueous FIG. 30: Dp of 20 nm radius probes as a function of T in 2/3 200 g/L 83.5 kDa dextran ( ) and in 100 g/L 542 kDa dex- neutralized 596 kDa polyacrylic acid:0.1 M NaCl: 0.1 wt% tran (), showing the linear dependence of Dp on T/ηs, after Triton X-100 at concentrations 0 (◦),3(),6(△), 8(♦), and Phillies, et al.52. 15 (▽) g/L and fits at each c to the Vogel-Fulcher-Tamman 51 equation using the same Tg at every c, after Phillies, et al. .
and larger spheres, Ds of the polymer rises and η of the mixture decreases with increasing sphere concentration. expected from the viscosity. The deviations, which are Shibayama, et al.54 examine the diffu- at most twice the random scatter in the measurements, sion of polystyrene probe particles in poly(N- increase with increasing c but are independent of Mw. isopropylacrylamide) gels and solutions using QELSS. Phillies and Quinlan note that the deviations are consis- The solution spectra are bimodal, with both relax- tent with a slight change in solvent quality with increas- ation rates scaling linearly in q2. A faster, relatively ing dextran monomer concentration. Equation 13, with concentration-independent mode corresponds to motion Tg the same for all solutions, fits almost all measurements of the polymer chains, while a slower mode corresponds to within 2% rms error. The apparent glass temperature to probe diffusion. As a result of the preparation of water:dextran solutions, as inferred from the modi- method, solutions at each concentration involved fied VFT equation, is independent of dextran molecular polymers having a different average molecular weight. weight and dextran concentration. The measurements Over the observed range of polymer concentrations, a 0.6 here serve to exclude any hypothesis that the strong con- reduced probe diffusion coefficient Dp/M changed centration dependence of Dp arises from a strong concen- by several orders of magnitude. Shibayama, et al. tration dependence of Tg, namely (i) Tg is independent propose that the probe diffusion coefficient should scale of c, and, alternatively, (ii) after removing the temper- with concentration and polymer molecular weight as −1.75 ature dependence of ηs, there is next-to-no remnant T - Dp/Dpo ∼ (c/c∗) , based on the corresponding 55 dependence of Dp available to be interpreted as a varia- prediction in Shiwa for polymer self-diffusion. Spectra tion in T − Tg. of probes in true crosslinked gels were also bimodal, but Roberts, et al.53 used PFGNMR to examined a model with increasing crosslinker concentration the amplitude liquid filled-polymer system, formed from silica nanopar- of the probe mode is suppressed to zero. ticles suspended in monodisperse poly(dimethylsiloxane). Turner and Hallett1 used QELSS to measure the diffu- Silica particles had diameters 0.35 and 2.2nm; polymers sion of carboxylated styrene butadiene spheres, diameters had molecular weights 5.2 and 12.2 kDa, with Mw/Mn of 93, 183, 246 nm, in solutions of dextrans having nomi- 1.07 and 1.03, respectively. These are actually not probe nal molecular weights 20, 70, 150, 500, and 2000 kDa measurements; the volume fractions of probes and ma- at polymer concentrations up to 20 g/l. As seen in Fig- trix polymers were both always substantial. Ds of the ure 32, the normalized diffusion coefficient Dp/Dpo was small silica particles and the 5.2 kDa polymer in a mix- very nearly insensitive to probe diameter, but changed ture both fall linearly with increasing polymer volume more than three-fold as polymer concentration c was in- fraction for polymer volume fraction φ in the range 0.2 creased. To good approximation, Dp/Dpo = exp(−ac). to 0.95. In contrast, in a mixture of the larger polymer Dp at fixed c depends appreciably on dextran molecular 20
2 ionic surfactant Triton X-100, which suppresses polymer binding, eliminated the complexity. Probe spheres in surfactant-containing mixtures showed a much simpler behavior, namely the apparent hydrodynamic radii of the 1 a probes fell smoothly with increasing polymer c. The de- gree of failure of the Stokes-Einstein equation increased markedly with increasing probe radius. By using mea-
p0 surements on probes in PEO:Triton X-100 to supply cal- 0.5
/D ibrating factors, Ullmann and Phillies quantitated the p substantial degree of polymer adsorption by probes in D surfactant-free solutions.
0.2 10 2
0.1 0 5 10 15 20 1 10 c (g/L)
2 (cP) h 10 0 1 p0 0.5 -1
/D 10 p
D 0.1 1 10 100 1000 0.3 c (g/L)
0.2 b FIG. 33: η of aqueous 7.5 (△), 18.5 (•), 100 (), and 300 ( ) kDa PEO and stretched-exponential fits, after Ullmann, et al.57. 0.1 0 5 10 15 20 Ullmann, et al.57 extended Ullmann and Phillies56 to c (g/L) study viscosity (Figure 33) and diffusion (Figure 34) of 20.8, 51.7, 322, and 655 nm diameter carboxylate- modified PSL spheres in solutions of 7.5, 18.5, 100, and FIG. 32: Dp/Dp0 of (a) 93 ( ), 183(), and 246 (△) nm bu- 300 kDa polyethylene oxides. Dp is substantially modi- tadiene spheres in aqueous 2 MDa dextran, and (b) 246 nm fied by the addition of Triton X-100, which is believed spheres in solutions of 20 ( ), 70 (), 150 (△), 500 (•), and to suppress polymer binding by the spheres. D fol- 2000 (♦) kDa dextran, all as functions of dextran concentra- p tion, following Turner and Hallett1. lows a stretched exponential in polymer concentration, except that 655 nm diameter spheres in the 18.5 kDa polymer show reentrant behavior, with Dp first climbing 50% above Do and then declining markedly. The rela- weight, a increasing with increasing M. The microvis- tionship between Dp and η is complex in the lower-M cosity inferred from Dp agreed well with the viscosity polymers; at large M, Dp follows a stretched exponen- measured with a rotating drum viscometer. tial in polymer concentration. Dp/Do depends relatively Ullmann and Phillies56 made a study of the diffusion weakly on probe radius. of polystyrene latex spheres through polyethylene oxide: Ullmann, et al.58 studied with QELSS the diffusion of water. These data are discussed with the following pa- 52, 322, and 655 nm radius polystyrene spheres in so- per. They used the macroscopic η to compute effective lutions of bovine serum albumin (BSA) in 0.15M NaCl, hydrodynamic radii for their PSL sphere probes (radii pH 7.0. They compared Dp with η as determined us- 21-655 nm), finding that rh had a complex dependence ing capillary viscometers, as seen in Figure 35. Dp of on polymer concentration. Addition of 0.01% of the non- the two larger spheres had a stretched-exponential de- 21
10 1 10 -1
a 0
/s) 10 /s) 2 2
cm cm -2
-7 -7 10
(10 -1 (10 p 10 p D D c
10 -2 10 -3 0.1 1 10 100 1000 0.1 1 10 100 1000 c (g/L) c (g/L)
10 0 10 -1
-1
/s) 10 /s) 2 2
cm cm -2
-7 -7 10
(10 -2 (10 p 10 p D b D d
10 -3 10 -3 0.1 1 10 100 1000 0.1 1 10 100 c (g/L) c (g/L)
FIG. 34: Dp of (a) 20.8, (b) 51.7, (c) 322, and (d) 655 nm PSL in aqueous 7.5 (•), 8 (△), 100 (), and 300 ( ) kDa 57 effects. PEO and stretched-exponential fits, after Ullmann, et al. . Wattenbarger, et al.61 used FRAP to examine the dif- fusion of bovine serum albumin (BSA) through solu- tions of a 160 base-pair DNA at DNA concentrations pendence on c. The Stokes-Einstein relation failed, Dpη 2-63 g/L and NaCl concentrations 0.01 and 0.1M. DNA of the 322 and 655 nm spheres increasing with increasing molecules had lengths ca. 56 nm, identified as being ap- c. Dp of the 52 nm spheres showed re-entrant behav- proximately 1 persistence length. As seen in Figure 36, ior, Dp at first increasing above its value in pure solvent Dp was approximately exponential in c. Increasing the and then returning to the values expected from η. At solution ionic strength increases Dp/Dpo, especially at very small c, Dp of the 52 nm spheres showed a local larger DNA concentrations. minimum. Dilution experiments showed that the mini- Won, et al.62 report on the diffusion of 200 mum arose from aggregation of partially-protein-coated nm radius PSL spheres in solutions of 1.3 MDa spheres due to the BSA in solution. When spheres were poly(vinylmethylether): toluene using QELSS, as shown diluted from concentrated protein solution to more di- in Figure 37. Additional measurements of Dp made with lute protein solution, Dp of the 52 nm spheres returned forced Rayleigh scattering were in very good agreement linearly to its zero-c value, with no sign of aggregation with the QELSS data. PVME concentrations reached 22
1.2 1 1 0.9 0 h/h 0.8 0.8 p0 /D
p
D p0 0.6 0.7
/D d p D a 0.4 0.6
0.2 0.5 0 40 80 120 160 200 0 10 20 30 40 c (g/L) c (g/L)
FIG. 35: Solutions of bovine serum albumin, showing their FIG. 36: Probe diffusion coefficient61 of bovine serum al- viscosity (•) and self-diffusion coefficient (♦) (from refs. 59, bumin in DNA solutions containing 0.01 (◦) or 0.1 (♦) M 60), and Dp of 322 () and 655 (◦) nm polystyrene spheres, NaCl, and fits to stretched exponentials, after Wattenbarger, and fits (lines a-d, respectively) to stretched exponentials in et al.61. protein c, from data of Ullmann, et al.57. 1x10 -7 up to 100 g/l, i.e., c[η] up to 36. QELSS spectra were -8 generally unimodal; a small amplitude slow mode was 1x10 sometimes seen. The decline in Dp with increasing c was followed over more than four orders of magnitude. 1x10 -9 The product Dpη/Dpoηo shows re-entrant behavior, first climbing to nearly 3.5, and then falling back to 1.0 by p0 -10 10 g/L PVME. A comparison of PSL spheres in this /D
p 1x10
system with spheres in polystyrene:dimethylformamide D found that D/Do is to first approximation a universal -11 function of c[η], except for the concentrations at which 1x10 re-entrant departure from Stokes-Einstein behavior is found. Tracer diffusion of linear and star polystyrene 1x10 -12 molecules through PVME solutions has also been stud- ied extensively63,64,65,66,67,68. -13 69 1x10 Zhou and Brown measured Dp of stearic-acid-coated silica spheres in polyisobutylene (PIB): chloroform using 0.1 1 10 100 QELSS, as seen in Figure 38. Comparison was made c (g/L) with PFGNMR determinations of polymer self-diffusion and QELSS measurements of polymer mutual diffusion. FIG. 37: Diffusion coefficient62 of 200 nm polystyrene spheres Polymer molecular weights ranged from 57 kDa to 4.9 2 in 1.3 MDa PVME: toluene and fit of filled points to a MDa. Sphere motion was diffusive, with a q -dependent stretched exponential, after Won, et al.62. linewidth, leading to diffusion coefficients seen in Figure 38. Inverse Laplace transform of the spectra clearly re- solved a weak peak due to polymer mutual diffusion from firm that in viscous simple liquids the Dp measured an intense peak taken to reflect probe motion. Probe with light scattering tracks η, no matter whether η is diffusion coefficients very nearly tracked the solution vis- changed by varying the temperature or the composition cosity, D η increasing very slightly over the range of con- 70 p of the liquid. Phillies and Fernandez confirmed Dp of centrations studied. polystyrene spheres in water depends linearly on T/η, Many of the above studies compare Dp with the macro- showing that PSL spheres do not swell or contract with 71 scopically measured viscosity η. A series of studies con- changing T . Phillies measured Dp of bovine serum al- 23
10 0 work was assisted by the practical matters that HPC samples are available with a wide range of molecular weights, dissolve well in water, and have interesting ther- modynamic properties, including a transition from good to theta solvent behavior with increasing temperature as -1 well as a liquid-to-liquid-crystal transition at extremely 10 high polymer concentration. Much of the later work
p0 was motivated by the viscosity measurements of Phillies 75
/D and Quinlan , who observed η(c) for solutions of 300 p kDa, 1 MDa, and 1.3 MDa HPC over wide concentration D ranges. Phillies and Quinlan75 found an unusual visco- -2 10 metric transition not observed in most polymers studied with optical probe diffusion. At first, work focused on finding evidence corroborating the reality of the transi- tion. Later work focused on the search for a physical interpretation of this transition. 10 -3 Phillies and Quinlan75 established that up to a tran- 0 20 40 60 80 100 sition concentration c+ the concentration dependence of c (g/L) η of HPC solutions follows accurately a stretched expo- nential
ν FIG. 38: Diffusion coefficient of 160 nm silica spheres in so- η = ηo exp(−αc ). (14) lutions of 57 ( ), 95 (), 182 (△), 610 (•), and 1900 () kDa polyisobutylene, and fits to stretched exponentials, after At concentrations c>c+, the concentration dependence Zhou and Brown69. of η follows equally accurately a power law
η =ηc ¯ x. (15) bumin and 45.5 nm radius PSL in water:glycerol and the same PSL spheres in water:sorbitol for temperatures Here α is a scaling prefactor, ν and x are scaling ex- 5-50 C and concentrations as high as 86 wt% glycerol ponents, and ηo andη ¯ are dimensional prefactors. To and 65 wt% sorbitol. Phillies found probe diffusion in avoid model-dependent phrasings, Phillies and Quinlan these mixtures followed accurately the Stokes-Einstein termed the c
lutions. Relatively complex dependences of D on con- p0 p 0.5
centration were suppressed by the addition of TX-100. /D In the presence of TX-100, simple stretched-exponential p D concentration dependences were observed, but the second spectral cumulant still increased with increasing polymer concentration. 0.2
10 2 10 -1 0 10 20 30 c (g/L)
2 0 10 1 h/h 10 0 b
p0 0.5 /D 0 p 10 D 0 10 20 30 40 c (g/L) 0.2
79 FIG. 39: Viscosity of Mw 110 ( ), 140 (), 450 (△), and 850 (•) kDa HPC, and fits of η(c) to exponential concentration 10 -1 dependences. 0 4 8 12 16 20 c (g/L) Yang and Jamieson79 examined the diffusion of polystyrene spheres in HPC, measuring the shear vis- cosity η (Figure 39), and using QELSS to obtain Dp (Figure 40). Polymer molecular weights based on static tions were ≤ 4 wt%. Light scattering spectra and zero- light scattering measurements were 110, 140, 450, and shear viscosities were determined at a series of polymer 850 kDa; sphere diameters were 120, 210, and 350 nm. concentrations. Addition of Triton X-100 dramatically Polymer concentrations extended up to 40 g/l. Yang and markedly reduced the hydrodynamic radii determined Jamieson found Dp to be independent of polymer con- using QELSS. Dp of nonaggregated spheres very nearly − centration over a range of not-quite-zero concentrations. tracks η 1, with a modest 12% deviation for the larger At larger concentrations, Dp fell as a stretched exponen- spheres; however, only two measurements with η/η0 > 10 tial in c. With Triton X-100 added to suppress polymer were reported. 81 binding, Dp(c) was larger than expected from η. The Mustafa and Russo applied QELSS to measure the ratio of the macroscopic and microscopic viscosities η/ηµ diffusion of 90.7 nm radius polystyrene latex through so- increases with increasing c and M. A modest depen- lutions of high-molecular-weight (1 MDa) hydroxypropy- dence of η/ηµ on probe R is seen, small probes having lcellulose at concentrations up to 1.5 g/L. Comparison the smaller ηµ. was made between solutions to which Triton X-100 had Russo, et al.80 report on HPC solutions with PSL been added in order to eliminate polymer binding to the spheres as optical probes. Polymer M was 292 kDa; probes, and solutions to which Triton X-100 had not been sphere radii were 40 and 90 nm. Polymer concentra- added. QELSS spectra were profoundly bimodal, the fast 25
82 2 Phillies and Clomenil made a detailed study of the diffusion of 67 nm PSL spheres in 139 kDa HPC at 10oC (good-solvent conditions) and 41oC (near-theta-solvent conditions), as seen in Figure 41. At each temperature, 0 10 c Dp depended on polymer concentration via a stretched exponential exp(−αcν ). The exponent ν was 1 under theta conditions but 0.74 under good-solvent conditions,
p0 showing probe diffusion can be sensitive to solvent qual- 0.5
/D ity. The dependence of ν on solvent quality is not en- p tirely useful as a test of models for polymer dynamics: D The observed dependence is equally predicted by the Langevin-Rondelez21, Altenberger83, and hydrodynamic 84,85 0.2 scaling treatments.
10 6 10 -1 0 4 8 12 16 20 10 5 c (g/L) 10 4 FIG. 40: Diffusion coefficient of (a) 120, (b) 210, and (c) 350 nm diameter PSL in solutions of Mw 110 ( ), 140 10 3 (), 450 (△), and 850 (•) kDa HPC, and fits of Dp(c) to (cP)
stretched-exponential concentration dependences, after Yang h and Jamieson79 . 10 2
10 1 mode being half or a quarter as intense as the slow mode. q2 measurements confirmed that the slow mode is diffu- 0 sive, and indicated that the fast mode is probably diffu- 10 sive. For both modes, the Stokes-Einstein equation fails, probe motion being up to two-fold faster than expected 0.01 0.1 1 10 100 from the macroscopic solution viscosity. c (g/L)
1 FIG. 42: η of 139 (◦) and 1280 kDa (, displaced two-fold ver- tically for clarity) hydroxypropylcellulose at temperatures 10 C (filled points) and 39 C (open points) and fits to stretched exponentials (smooth curves) and power laws (straight lines), after Phillies, et al.86.
/s) 0.1 2 Phillies, et al.86 measure diffusion of 67 nm PSL
cm spheres via QELSS in 139, 146, 415, and 1280 kDa hy-
-7 droxypropylcellulose (HPC) at temperatures 10, 25, and 39o C, for polymer concentrations up to 100 g/l. Shear (10
p viscosities were measured for the same solutions. Aque- 0.01 D ous HPC is a good-solvent system at 10C and approaches a pseudotheta transition at 41 C, so varying the temper- ature allows a variation in solvent quality. η (Figure 42) was found to have a stretched-exponential concentration dependence up to an apparent transition concentration 0.001 c+, and an apparent power-law concentration dependence 0 20 40 60 80 100 at higher concentrations. These observations led to the c (g/L) much more systematic study of Phillies and Quinlan75. Spectra were analysed with Koppel’s method of cumu- lants. Individual spectra also fit well to stretched expo- FIG. 41: Dp of 67 nm polystyrene spheres in aqueous 139 kDa nentials in time. Over the observed concentration regime, HPC at 10 (◦) and 39 (•) C, after Phillies and Clomenil82 . Dp fell by 1.5 to 2.5 orders of magnitude, as seen in Fig- 26
1 1
0.1
/s) 0.1 2 p0 /D cm p -7 0.01 D 0.01 (10 p D 0.001 0.001 0.0001 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 weight fraction HPC c (g/L)
FIG. 45: Dp of fluorescein dye in hydroxypropylcellulose so- FIG. 43: Dp of 67 nm polystyrene spheres in aqueous 139 (◦) lutions, showing a simple exponential dependence of Dp on and 415 kDa () hydroxypropylcellulose at 10 C (filled points) c, with no discontinuity at the liquid crystal phase transition and 39 C (open points) and fits to stretched exponentials near weight fraction 0.4, after Mustafa, et al.87 86 (smooth curves), after Phillies, et al. . .
100 polymer M. S(q,t) was obtained with an experimental S/N of 300-1000. A careful search was made for a very fast spectral mode that might have perturbed cumulant fits; none was found. Later work from the same group found that S(q,t) for probes in HPC:water is generally bimodal. Ref. 86 was found to have examined the more
0 prominent shorter-lived relaxational mode. h 87
p0 10 Mustafa, et al. used fluorescence recovery after
D photobleaching to examine diffusion of fluorescein dye h/
p through 130, 292, and 855 kDa HPC solutions. Dp was
D measured at various grating spacings and temperatures at HPC concentrations from dilute solution up through the HPC liquid:liquid crystal phase transition. As seen in Figure 45, Dp decreases approximately exponentially 1 in polymer concentration but is independent of polymer molecular weight. There is no discontinuity in Dp(c) on crossing the HPC liquid:liquid crystal phase transition. 1 10 100 Ngai and Phillies88 extend the Ngai-Rendell coupling c (g/L) model89 to treat probe diffusion and polymer dynamics in polymer solutions. This is not an experimental paper; it forces extant experimental data to confront a particu- FIG. 44: Dpη/Dp0η0 of 67 nm polystyrene spheres in aqueous lar theoretical model, which in the paper was extensively 139 (◦), 146 (), 415 (♦), and 1280 () kDa HPC at 10 C, 86 reconstructed to treat the particular experimental meth- after Phillies, et al. . ods under consideration. Ngai and Phillies consider zero- shear viscosities and optical probe diffusion spectra for HPC solutions, extracting from them the Ngai-Rendell ure 43. Over the same range of c, Dpη/Dp0η0 increased model relaxation time τo and coupling exponent n. Opti- by factors of several to several hundred, the increase be- cal probe spectra can be used to obtain n in four indepen- ing largest for the largest-M polymer (as seen in Figure dent ways, namely from η(c) and from the concentration, 44), and larger at 10oC than at 39oC. In fits to Williams- time, and wave-vector dependences of g(1)(q,t). The four Watts spectral forms, θ and β fall as stretched expo- paths from η and g(1)(q,t) lead to consistent values for n, nentials in c, the decline being more dramatic at larger especially for larger probes and higher polymer concen- 27 trations at which the scaling-coupling arguments invoked haviors fell into two classes, distinguished by probe size. in the paper are most likely to be valid. Figure 46 shows sample results, namely θs against probe 90 Phillies and LaCroix used QELSS to study diffusion radius R at different concentrations: for small probes, θs of four PSL sphere species in aqueous 300 kDa HPC at is very nearly independent of c, while for large probes concentrations up to 15 g/L. Spheres had diameters 21, θs decreases sharply with increasing c. The characteris- 102, 189, and 760 nm. Field correlation functions were tic length separating large and small probes, which ap- fit to the sum of a weak, fast initial exponential and a pears to be the same at all polymer concentrations, is ap- stretched exponential exp(−θtβ). The fast initial expo- proximately the size of a complete polymer chain, probes nential was never more than 2.5% of g(1)(q,t). β and the having a diameter less than the polymer hydrodynamic relaxation pseudorate θ both had stretched-exponential radius being ’small’ and probes having a diameter larger concentration dependences, the dependences being the than the polymer radius of gyration being ’large’. A simi- stronger for the larger probes. For the 21 nm spheres, lar distinction between small and large probes is seen for θη increased with increasing c; for the large spheres, θη β of the slow mode, with β ≈ 1 for large probes and was very nearly independent from c. The relaxation time β decreasing with increasing c for small (R ≤ 35 nm) τ = θ−1/β depends on c less strongly than η does, so that probes. On the other hand, θ of the fast mode is nearly for smaller spheres τ/η falls with increasing c. For larger independent of concentration regardless of probe size. spheres, τ/η is approximately independent of c. However, Streletzky and Phillies92 present an approach to in- because these spectra follow exp(−(t/τ)β) with β 6= 1, terpreting the QELSS spectral mode structure for opti- it would be erroneous to write τ = Dq2 for some D. cal probes diffusing in HPC:water, based on their prior In consequence, the constancy of τ/η represents pseudo- work. They consider only probes in the HPC solutionlike Stokes-Einsteinian behavior, not Stokes-Einsteinian be- regime, whose phenomenology is sharply divided between havior, for τ. These results are subsumed in the later ’small’ and ’large’ probes, small and large being defined studies of Streletzky and collaborators, discussed below. relative to the size of the polymer chains. The crossover from small- to large- probe behavior is found at the same probe radius at all polymer concentrations. They inter- pret three time-scale-separated modes to be: (i) a large- -2 10 probe ’slow’ mode that appears to reveal particle motion in a viscous medium, (ii) large-probe ’fast’ and small- probe ’slow’ modes that occur on the same time scale, 10 -3 and appear to be related to chain internal dynamics, and (iii) a small-probe ’fast’ mode. The large-probe ’slow’ mode is approximately governed by the solution viscos-
s ity, and goes linearly to zero as q2 → 0. Later work by Q -4 10 these authors established that the ’fast’ and ’slow’ modes were better described as ’sharp’ and ’broad’, depending on their time stretching exponent β, because ’fast’ and 10 -5 ’slow’ do not consistently describe the mode lifetimes. The Ngai scaling-coupling model only works for modes in category (ii), which for both small and large probes is the shorter-lived of the two modes. -6 10 Streletzky and Phillies93 compare optical probe diffu- 20 50100 200 500 sion and viscosity measurements on HPC:water.The im- 2R (nm) mediate question treated by Streletzky and Phillies93 was whether the transition is real, whether the apparent tran- sition at c+ is simply an artifact of ref. 75’s numerical FIG. 46: Sharp mode pseudorate constant versus probe size fitting procedures, or as a third alternative whether the at HPC concentrations 0 ( ), 0.5(),1(♦),2(△),3(▽), 3.75 (⋆), 4(•),5(),6(), and 7 (N) g/L, showing a demarkation stretched exponential behavior is merely a numerical ap- between small and large probe behavior for R between 20 and proximant lacking deep physical meaning. Streletzky and 93 30 nm, after Streletzky and Phillies91. Phillies proceeded by using QELSS to measure the dif- fusion of large (50, 87, and 189 nm diameter) PSL probes 91 in aqueous 1MDa HPC, including concentrations above Streletzky and Phillies use QELSS to study the dif- and below c+. After testing alternatives, spectra were fusion of 14-455 nm diameter PSL spheres in 1MDa fit to the sum of a fast stretched exponential and a slow hydroxypropylcellulose-water for concentrations 1 ≤ c ≤ pure exponential 7 g/L. Probe spectra fit well to a sum of two stretched ex- β (1) βf ponential modes exp(−θt ), characterized in this paper g (q, τ)= Ao(Af exp(−θf c )+(1 − Af ) exp(−θc). as ’slow’ and ’fast’. The dependences of spectral fitting (16) parameters on probe radius, polymer concentration, wave At c+, every single spectral parameter changes its con- vector, and solution viscosity were examined. Probe be- centration dependence. As seen in Figure 47, below c+, 28
10 -1 1.0
0.8 10 -2
0.6
b -3 b b
Q 10 0.4
10 -4 0.2
10 -5 0.0 0 5 10 15 0 4 8 12 16 c (g/L) c (g/L)
FIG. 47: Broad mode pseudorate constant θf against polymer FIG. 48: Broad mode stretching exponent βf against polymer concentration for 50 (•), 87 (♦), and 189 (+) nm probes, concentration for 50 (•), 87 (♦), and 189 (+) nm probes, showing the change in concentration dependence near c+ ≈ 6 showing the change in concentration dependence near c+ ≈ 6 g/L, after Streletzky and Phillies93 . g/L, after Streletzky and Phillies93 .
(1) θf is independent of c; at larger c, θf decreases as a early parts of the slope of g (q,t) rather than the inte- stretched exponential in c. As seen in Figure 48, βf de- grated lifetime, and ’fast’ and ’slow’ do not map one-to- clines with increasing c until c+ is reached; at higher con- one to short- and long-lived, the original paper identified centrations βf is nearly independent of c. The fast-mode the modes for which coupling-scaling works as the ’fast’ fractional amplitude Af is nearly zero at c = 0; it climbs mode of small probes and the ’slow’ mode of large par- with increasing c until it reaches Af ≈ 0.4 − 0.6, which ticles, leaving it less clear which physics was applicable it does at a concentration near c+. θ follows stretched when. exponentials in c; near c+, for the 87 and 189 nm probes Streletzky and Phillies eventually demonstrated that there is a sudden change in the slopes of the θ(c) curves. these results are consistent with the earlier papers of + 90 88 At c , the correlations of θ and θf with η both change. Phillies and Lacroix and Ngai and Phillies . The + θf η climbs near-exponentially with increasing c until c demonstration consists of showing that Phillies and is reached; at larger c, θf η increases weakly with increas- Lacroix had only been able to study the shorter-lived ing c. The slope dθη/dc also changes near c+, though mode of probes in 300 kDa HPC, for which the Ngai- 88 less markedly than the slope of θη changes. θ and θf are Phillies model is correct; in both systems, it is the both linear in q2. Streletzky and Phillies conclude that shorter-lived mode that follows coupling-scaling. the marked changes in the concentration dependences of Streletzky and Phillies95 make a systematic presenta- + the optical probe diffusion parameters near c prove the tion of their findings on probe diffusion in aqueous HPC. physical reality of the solutionlike-meltlike transition. Spectra were found to be sums of two stretched exponen- Streletzky and Phillies94 attempt to apply the Ngai- tials. Their principle results are: (i) probes in the solu- Phillies88 coupling/scaling model for probe diffusion to tionlike c
40 concentration of these structures increases at low tem- perature, leading to glass formation. In polymer solu- tions studied by Phillies, et al., solution properties are very nearly athermal at accessible temperatures. Increas- 30 ing concentration replaces decreasing temperature as the factor driving structure formation.
20 B. Probe Rotational Diffusion I/c
By examining the polarized and depolarized light scat- tering spectra at a series of scattering angles, or by mea- 10 suring the depolarized spectrum at zero scattering angle, it is possible to infer the rotational diffusion coefficient Dr of probe particles that depolarize scattered light. Camins and Russo99 used zero-angle depolarized 0 light scattering to study the rotational diffusion of 0 4 8 12 16 poly(tetrafluoroethylene) colloidal probes in gelling poly- c (g/L) acrylamide. The probes were optically-anisotropic pro- late ellipsoids of modest (3:1) axial ratio, the effective hydrodynamic radii being 140 ± 40 nm, dispersed in FIG. 50: Normalized intensity I/c of aqueous HPC solutions, 97 polyacrylamide systems undergoing gelation. During after O’Connell, et al. . gelation, the depolarized relaxation slows and broadens markedly, changing from a near-exponential relaxation 10 5 to a stretched exponential with exponent β ≈ 0.5. In- terpretation of spectra in terms of mobile and immobile 4 probe fractions suggests that most probes become immo- 10 bile during the gelation process, but at least some are at least somewhat free to move. 3 Cheng and Mason100 discuss the extension of optical ) 10 -1 probe diffusion methods to study rotational diffusion in (s 900 kDa poly(ethylene oxide). The optical probe was 2 10 a micron-size α-eicosane microdisk whose angular ori-
10 ics in which ’hydrodynamic effects surpass any due to topological constraints’, even in solutions having nearly a hundred times the viscosity of water. Hill and Soane103 examine the rotational diffusion of collagen molecules in aqueous 4 Mda polyethyleneoxide 1 using the electrooptic Kerr effect. Collagen is a rigid rod,
-1 with diameter and length of ca. 1.5 and 170 nm, respec-
h tively. In a Kerr effect experiment, a linearly polarized , , r laser beam is passed through a sample on which an elec- , D ,
p tric field has briefly been imposed. Hill and Soane cite 24
D kV/cm as their largest attainable electric field. Partial 0.1 induced alignment of the molecules in the cell leads to an induced birefringence and thence to a change in the transmission of the polarized light, through a quarter- wave plate and second optical polarizer, into the detector. The relaxation time for the Kerr effect is determined by 0.01 rotational diffusion of the weakly aligned objects in the 0 4 8 12 16 20 Kerr cell. The collagen preparations had a bimodal mass c (wt%) distribution, so two relaxation times were observed, both strongly concentration dependent. Jamil and Russo104 studied translational and rota- FIG. 52: Dp ( ) and Dr () of tobacco mosaic virus in 647 tional diffusion of poly(tetrafluoroethylene) latex in aque- −1 kDa dextran solutions, and η ( ) of those solutions, after ous sodium polystyrenesulfonate (NaPSS). Spectra were Cush, et al.101 , and (lines) fits to stretched exponentials in c. single exponentials. Dp and Dr were extracted from the q2 dependence of the spectral linewidths. Addition of 10 NaCl causes the probes to aggregate; further addition of NaPSS reverses the salt-induced probe aggregation, even though it appears that NaPSS does not bind to the latex particles. Dr tracks the changes in viscosity of the solu- tion attendant to addition of polymer, in the sense that ηDr is independent of polymer concentration. However, 1 at fixed c, ηDr does decrease as the salt concentration is -1
h increased. , , r Koenderink, et al.105 examine the motion of perfluo-
, D , rinated hydrocarbon spheres through xanthan solutions. p
D Depolarized QELSS spectra were measured at a series 0.1 of angles and fit to second-order cumulant expansions. The spheres had radii 93 nm; xanthan molecular weight was 4 MDa. Koenderink, et al., measured solution vis- cosity, shear thinning, storage and loss moduli, transla- tional and rotational diffusion coefficients Dp and Dr of 0.01 the probes, and probe sedimentation coefficient s, and 0 10 20 30 40 made an extensive and systematic comparison with the c (wt%) available theoretical background. As seen in Figure 54, xanthan is far more effective at increasing the solution viscosity than at hindering translational or rotational dif- FIG. 53: Dp ( ) and Dr () of tobacco mosaic virus in fusion of the probes, and is more effective at hindering − 428 kDa ficoll solutions, and η 1 () of those solutions, after sedimentation than at hindering diffusion. The observed 101 Cush, et al. , and (lines) fits to stretched exponentials in c. variation of Dpη and Drη with c in xanthan solutions is much larger than the variation observed for probes in many other systems. dextran solutions. The authors also studied the molecu- Le Goff, et al.106 polymerized actin fragments with var- lar weight dependence of Dr and Dp at fixed 14.6 wt% ious degrees of labelling, using a protocol which led to dextran concentration, finding Dr and Dp have power- ends made of heavily-labelled actin and an extremely long law dependences on polymer Mw for 40 ≤ Mw ≤ 2750 center segment that was sparsely labelled or unlabelled. kDa. Cush, et al. reason from the minimal issues with Fluorescence microscopy was used to track the motion Stokes-Einsteinian behavior for Dr and Dp that their of the labelled ends, determining thereby the distribu- data support a continuum picture of polymer dynam- tion and time dependence of the fluctuating end-to-end 32
10 3 1 1
0.1 10 0 S)
/f 2 /h 0
f 10 h t t (m 0.01 100
0.001 1000 10 1 0.01 0.1 0 0.02 0.04 0.06 0.08 0.1 c (wt %) f
FIG. 54: Dp(), Dr( ), and s(♦) of 92.5 nm radius spheres FIG. 55: Rotational relaxation times of 68 ( ), 128 (♦), and in 4 MDa xanthan, all reported [left axis] as nominal normal- 170 () nm PBLG rods in solutions of 62 nm silica spheres as 107 ized drag coefficients f0/f, and shear viscosity η/η0(•) [right a function of sphere concentration , and simple-exponential 105 axis], after data of Koenderink, et al. , and fits to stretched fits to τr(c). exponentials in c.
various latex particles containing magnetite inclusions distance and the angular reorientation of the end-to-end through solutions of 70 kDa polystyrene sulfonate. Most vector. Proof-of-principle data was obtained. Repeat work was done on 50 nm radius particles. Extensive pre- of these experiments in solutions at a series of concen- liminary work was needed to characterize the available trations should provide extremely valuable information latex-magnetite probes, which tend to be rather polydis- about translation, rotation, and internal motion of some- perse unless they are fractionated. Addition of the poly- what rigid polymer chains. mer reverses salt-induced aggregation of the probes. On Phalakornkul, et al.107 used transient electric birefrin- increasing the polymer concentration the translational and rotational diffusion coefficients D and D both fell. gence to examine rotational relaxation of poly-(γ-benzyl- t r α-L-glutamate) (PBLG) in solutions of silica spheres. At fixed polymer concentration, addition of NaCl did not have an obvious effect on D but tended to reduce D . PBLG preparations had lengths 68, 128, or 170 nm. t r Spheres had radius 31 nm. Birefringence spectra were At fixed salt or polymer concentration, the number of polymer or salt concentrations (respectively) that were analyzed with the Laplace inversion program CONTIN, which found consistent fast and slow modes and an oc- examined was not large, so it is difficult to make detailed inferences about the effect of the solutes on probe diffu- casional very weak, very fast relaxation. At very low sphere concentration, the slow mode reduces to rod ro- sion. tation. The fast mode appears to correspond to the first bending mode of the PBLG. The slow mode of the short rods had a concentration-independent relaxation time τs. C. Particle Tracking Methods Increasing the rod length greatly increases the concentra- tion dependence of τs. For the longest rods, τs increases In an extremely important paper, Apgar and Tseng, et three-fold as sphere volume fraction is increased from 0.0 al.109 measured the distribution of mean-square displace- to 0.08, while the relaxation time of the first internal ments for probes in a variety of solutions. They used mode increases less strongly with increasing c. η and all fluorescence microscopy to track fluorescently labelled rod relaxation times had simple exponential concentra- microspheres diffusing through actin solutions. Spheres tion dependences (Figure 55). The concentration depen- with radii in the range 0.43-0.6 µm were used. Tracking dence of η is stronger than the concentration dependence was used to determine the mean-square displacements as of τs of the short rods, but weaker than the concentration functions of time for a large number of probe particles, dependence of τs of the long rods. and to obtain statistics for mean-square deviations over Sohn, et al.108 used polarized and depolarized light each of several time intervals. In homogeneous water- scattering spectroscopy to examine the diffusion of glycerol solutions, the distribution of mean-square dis- 33 placements is Gaussian. In solutions of dilute actin, con- displacements perpendicular to the local chain tend to centrated F-actin, and F-actin/fascin, the distributions cease to increase. However, the authors note that D are very definitely not Gaussian. Correspondingly, anal- parallel to the chains, as measured by them, implies a yses of measurements on particle motion in F-actin so- solvent viscosity an order of magnitude larger that the lutions, if based on an assumption of Gaussian particle actual viscosity of water. That is, the apparent bead displacements, have a doubtful starting point. mobility along the chain axis is very different from the Chen, et al.110 used video microscopy to measure the expected bead mobility in pure solvent. motion of single particles and pairs of particles through Gardel, et al.114 used video microscopy to track 230, a viscoelastic solvent, namely λ-DNA in water. Mea- 320, and 420 nm radius carboxylate-modified spheres dif- surements were made of one- and two-particle diffusion fusing in solutions of F-actin. Single-particle-motions for spheres with diameters 0.46, 0.97, and 2.0 µm and in- and correlated-motion of pairs of particles were com- terpreted using ’two-point microrheology’111, which com- puted. F-actin solutions are considerably more effec- pares the diffusion coefficient of a single particle with the tive at retarding the motion of large particles than at relative diffusion of pairs of particles. Interpretation was retarding the motions of small particles, especially at made in terms of a model involving polymer depletion long times. Relative motions of particles were obtained zones around each particle. for particles separated by as much as 100µm. Single- Crocker, et al.112 describe a novel physical technique particle motions sometimes show longer-time plateaus. in which video microscopy is used to capture the posi- Correlated-pair displacements increase linearly in time tions of pairs of diffusing particles at an extended se- with no plateau, but at the longest times studied the mean-square correlated pair motions are not yet larger ries of times. Cross correlations in the particle displace- ′′ ments yield a range-dependent cross-diffusion coefficient than the observed plateaus. G (ω) from one-particle and Dij (r, t), where r the distance between particles i and two-particle measurements differ by factors of three or more, the two-particle values being in relatively close j, and t is the elapsed time during which the displace- ′′ ment occurs. The Laplace transform of Dij is taken to agreement with macroscopic measurements of G (ω) be inversely proportional to a frequency-dependent vis- made with a parallel-plate stress-controlled rheometer. cosity G˜(s)/s. The method was applied to study 100 Gittes, et al.115 observed the overdamped thermal dis- and 235 nm radius spheres in glycerol, guar gum, and F- placements of single 0.9µm silica spheres in F-actin and actin solutions. In glycerol solutions, the single-particle polyacrylamide gels. Microscopic observation was ap- and cross-diffusion coefficients agree with each other. In plied to isolate single particles, whose motions were de- viscoelastic guar solutions, there are factor-of-two dis- tected interferometrically. The power spectrum of the 2 agreements in the storage and loss moduli inferred from displacements h(∆x)ω i was used to compute the shear the single-particle and cross-diffusion coefficients and a modulus, which has both real and imaginary parts. Over generalized Stokes-Einstein equation; G′ and G′′ from a frequency range 0.1 ≤ ν ≤ 3 · 103 Hz, G′(ω) and ′′ Dij (r, t) are much closer to macroscopic measurements. G (ω) both showed power-law frequency dependences Crocker, et al.112 conclude that their two-point pair diffu- ∼ ωz with z ≈ 0.76. sion method has ”underlying validity”, while ”single par- Goodman, et al.116 studied the motion of 1 µm ticle microrheology provides qualitatively different mod- ′ fluorescent-labelled carboxylate-modified polystyrene la- uli and completely fails to detect the crossover [of G tex spheres through solutions of circular-supercoiled and ′′ 112 and G ]”. Crocker, et al.’s implicit conclusion is that linearized (single cut at controlled location) DNAs, size single-particle microrheology does not provide valid rhe- 2.96, 5.4, and 10.3 kilobasepairs (1.8-6 MDa; contour ological measurements. lengths 1, 1.84, 3.57 µm), using particle tracking meth- Dichtl and Sackmann113 studied the diffusion of indi- ods. The linear and matching circular DNAs are unam- vidual actin chains to which substantial numbers of 35 biguously identical in molecular weight. While issues of nm fluorescent latex or 17 nm gold beads had been at- chain configuration can arise in studies of synthetic cir- tached, employing a confocal laser scanning microscope cular polymers, the circular DNAs here are unambigu- to track bead motions with three-dimensional resolution. ously not knotted or topologically crosslinked. The spa- Bead positions fluctuated; bead motions could be decom- tial resolution on particle resolution was 5 nm. Video posed into components parallel and perpendicular to the monitoring was limited to a 30 Hz time resolution. From visualizable polymer chain. Measurements were confined particle-tracking, time-dependent mean-square displace- to time intervals much shorter than the terminal relax- ments were determined as functions of DNA molecular ation time of the solutions. Fluctuations perpendicular weight, concentration and circularization. Solution mi- to the chain axis had a Gaussian distribution whose width croviscosities η were inferred from the particle motions varied from bead to bead by nearly an order of magni- via a generalized Stokes-Einstein equation. At these rela- tude. Fluctuations parallel to the tube axis had a non- tively long times and small viscosities (η< 20 cP), mean- Gaussian bimodal distribution. At short times, mean square displacements after convective corrections were in square displacements parallel and perpendicular to the most solutions nearly linear in time. In some linear DNA local chain are nearly equal. At longer times, displace- solutions, the mean-square displacement was not simply ments parallel to the chain continue to increase, while diffusive. In these cases, h∆x2i was replaced with a nom- 34
24 100
16 0 /h
m 10 (cP) m h h 8
0 1 0 0.4 0.8 1.2 1.6 2 10 -4 10 -3 10 -2 10 -1 10 0 c (g/L) c (g/L)
FIG. 56: ηµ from mean-square displacement of 500 nm radius FIG. 57: Microviscosity of polystyrene sulfonate combs via PSL through circular DNAs ( , , ♦) (masses 2.96, 5.40, and particle tracking of 274 nm radius polystyrene latex probes, 10.31 kilobasepairs, respectively) and their single-cut linear and fits to stretched exponentials. Polystyrene sulfonates dif- derivatives (•, +, ), after Goodman, et al.116. fer in structure, namely (a) ( ) 2.1 Mda chain with 24 side branches, (b) (♦) same backbone as (a) but 3/8 as many side chains, (c) (•) double the backbone length of (a) but 5/8 as many side chains, and (d) () double the backbone length of inal creep microcompliance that is linearly proportional (a) and ≫6 times as many side chains, together with forced 2 (Γ(t) ∼ h∆x (t)i) to the time-dependent mean-square fits to stretched exponentials, after Papagiannapolis, et al.119 . displacement. Goodman, et al.116 report evidence that particle mi- crotracking can also provide evidence about solution mi- particle on which a weak laser beam has been focused. croviscoelasticity that is more detailed than that pro- The technique is directly responsive to motion in a plane, vided by macroscopic measurements, for example, details variations in the intensity of the light scattered in various of spatial variations in microviscoelastic parameters that directions acting as an optical lever to amplify the time- could be ascribed to solution microheterogeneity. The dependent particle motions. They compare mechanical microviscosities for the linear DNAs were as much as and DWS measurements of G′ and G′′ in the same so- several times larger than the microviscosity of the cor- lutions. Direct measurement from their graph indicates responding circularized DNA. c[η] was never larger than the two methods gave values for viscoelastic parameters 6, but one certainly does not see the reptation prediction that agree to within perhaps 30%, and sometimes better. ηring ≫ ηlinear at concentrations and molecular weights Papagiannapolis, et al.119 used diffusing wave spec- reached here. troscopy and video particle tracking to observe the dif- Lau, et al.117 use video tracking to measure the two- fusion of polystyrene latex spheres through solutions of particle cross-diffusion-coefficients of naturally-occurring fully-neutralized polystyrene sulfonate comb polymers. particles in the intracellular medium. Comparison is Probes had 548 nm diameter for particle tracking mea- made between one-particle and two-particle measure- surements and 112nm diameter for the DWS measure- ments, each giving a time course of a mean-square dis- ments. The combs differed roughly two-fold in their placement, in the same cell at the same time. Cell in- main-chain length, in the number of branches, or in the teriors are characterized by extensive chemical reactions, length of the branches. Video tracking observed motion and are not equilibrium systems. The observed behav- during times ≥ 3 · 10−2s, during which the mean square iors are more complex than would be obtained in a simple displacement increased linearly in time, permitting de- viscoelastic medium. For example, in these cells the rel- termination of Dp and thencefrom the solution viscosity ative displacement ∆r of two particles depends on time ηµ/ηo at different polymer concentrations. From Figure as h∆r2(t)i∼ t1.5. 57, changing the number of sidechains at fixed concentra- 118 Mason, et al. examine a single 520 nm diameter tion has a very limited effect on ηµ, while doubling the sphere diffusing through 3 wt% 5 MDa polyethylene ox- length of the backbone at fixed monomer concentration ide:water and DNA solutions, using laser deflection par- very markedly increases ηµ. Fits of ηµ to stretched ex- ticle tracking to determine the motion of the single probe ponentials in c are quite unsatisfactory at low concentra- 35 tion, the measured ηµ(c) being indistinguishable from the centrations, measurements of the distribution P (∆R) of solvent viscosity up to polymer concentrations at which mean-square displacements against time found that dif- the best-fit stretched exponential predicts ηµ(c) should ferent particles all had the same P (∆R). In contrast, in be readily distinguishable from unity. concentrated gliadin solutions P (∆R) for different parti- Schnurr, et al.120 demonstrate a novel interference mi- cles showed a wide range of different time dependences. croscopy method for studying probe motion. In their Correspondingly, at low (250 g/L) gliadin concentrations technique, mesoscopic beads are suspended in a solution P (∆R) at fixed time is a gaussian, but at large (400 g/L) or gel, their images are identified through a microscopic, gliadin concentrations P (∆R) is extremely non-gaussian. and optical interferometry through the microscope stage taking each bead to be one arm of the interferometer was used to track bead motion. The power spectral density of D. True Microrheological Measurements the displacements and the Kramers-Kronig relation are applied to infer the imaginary and real part of the com- This section considers true microrheological measure- plex shear modulus of the gels. Here silica beads with ments. In a true rheological measurement, external forces radii 0.25-2.5 µm were studied in F-actin and polyacry- or displacements are imposed, and consequent displace- lamide gels. ments and/or forces are measured. True microrheological 121 Tseng and Wirtz used particle tracking based on measurements, not to be confused with ’microrheology’ video light microscopy to measure the motion of 485 nm studies of Brownian motion, differ from classical macro- radius polystyrene spheres in solutions of F-actin and scopic rheological studies in that the probes or appara- α-actinin. Mean-square displacements were determined tus parts function on a mesoscopic length scale. If one as a function of diffusion time and characterized statis- believes that probe diffusion measurements could be in- tically. Storage and loss moduli and phase angle were verted to obtain η, G′, or G′′, it becomes interesting to computed from the displacements. The distribution of compare those viscosities and moduli both with the cor- displacements generally did not have a Gaussian form; responding quantities measured in classical instruments its mean square displacement at long times increased less and also with the same quantities measured with true mi- rapidly than linearly in time. The observed displacement crorheological instruments build on the same size scale as distributions depend in a complex way on the F-actin and the diffusing probes. α-actinin concentrations. Amblard, et al126 used video microscopy and magnetic Valentine, et al.122 used single particle and two-particle tweezers to study probe motion in viscoelastic F-actin tracking methods to examine the diffusion of carboxylate- systems. The probes were polystyrene spheres having modified polystyrene spheres in fibrin-F-actin-scruin net- radii of 75, 760, and 750 nm. Video microscopy deter- works. The objective was to study the effect of mod- mines particle positions; the tweezers can apply a con- ifying the probe surface chemistry. Different spheres stant force to a particle. The F-actin filaments at con- were uncoated, heavily coated with bovine serum albu- centration 0.1 g/l had an estimated length of 20 µm, min, or surface modified by systematic coupling of large persistence length ca 14 µm, and a mesh size ξ ∼ 1µm. numbers of short methoxy-poly(ethylene glycol) groups. Video microscopy determined the mean-square displace- Particle motion was examined using video-tracking mi- ment as a function of time. Magnetic tweezers allowed croscopy. Valentine, et al., found that the dynamic be- application of a fixed force, permitting determination of havior of the particles changed very substantially as sur- displacement under the influence of an applied force, also face treatments were changed. However, particles whose as a function of time. With small beads (d/ξ < 1), the single-particle displacement distributions differ consider- mean-square distance travelled during diffusion followed ably due to different surface coatings can have very sim- hx2(t)i ∼ t1. For large beads (d/ξ > 3), over a range of ilar two-particle displacement correlations. times 0.03-2.0 s, Amblard, et al., found hx2(t)i∼ tq, with Xu, et al.123,124 used mechanical rheology, particle q =0.76±0.03 for motion with an external driving force, tracking microrheology, and DWS to characterize the and with q =0.73±0.01 for free diffusion. For driven mo- rheological properties of actin filaments. Experimental tion, the apparent drag on the probe increases after the interests included the concentration dependence of the probe has travelled 10-20 µm. Use of probe surface coat- elastic modulus and the frequency dependence at high ings including streptavidin, surfactant, or bovine serum frequency of the magnitude of the complex viscoelastic albumin had no effect on other results. Specific surface modulus. DWS spectra were interpreted by invoking the interactions were concluded not to affect these findings Gaussian assumption. greatly. Here we see a direct demonstration that the Xu, et al.125 used video tracking microscopy to ob- drag processes for driven motion and for thermal mo- serve the diffusion of 970 nm diameter fluorescent tion can be the same: The mean-square displacements polystyrene spheres through water:glycerol and aqueous for both processes scale approximately as t3/4.The time wheat gliadin solutions. Increasing the gliadin concentra- dependence of the thermal motion is not ∼ t1, so the un- tion slows probe diffusion. In glycerol, the distribution derlying thermal particle motion is not simple Brownian of particle displacements was a well-behaved gaussian. diffusion. For probes in gliadin suspensions: At low gliadin con- Bishop, et al.127 studied the driven rotational mo- 36 tion of small spheres inside a model for the intracellular 10 medium. The spheres were 1-10µm birefringent vaterite (CaCO3) crystals. The driving force is provided by illu- minating the probes with circularly polarized light, and measuring the degree of polarization of the light after it has passed through the sample. The rotation rate is obtained by examining the transmission of one linear po- larization of the incident light. Bishop, et al., examined micro rotational motion of their probes within a drop of hex- /G" 1 ane and in bulk water, finding that their measured mi- croviscosities were in good agreement with the viscosities macro measured macroscopically. G" Hough and Ou-Yang128 report using optical tweezers to drive the motion of a single 1.58 µm silica micro- sphere through solutions of 85kDa end-capped (C12H25-) polyethylene oxide in water. The tweezer position was driven with piezoelectrically controlled mirrors at fre- 0.1 quencies as large as 40,000 rad/s. Measurements of the 0.01 0.1 1 10 magnitude A and phase δ of the sphere oscillations as n (Hz) functions of frequency were inverted, using the model that the sphere is a forced damped harmonic oscillator, ′ ′′ ′′ ′′ to obtained G (ω) and G (ω). The dynamic moduli were FIG. 58: Gmacro/Gmicro of F-actin solutions, including pure ”...quite different from those obtained by a macroscopic water ( ), 2 g/L F-actin solutions with random lengths (), ♦ rheometer, and are sensitive to surface treatment of the and 2 g/L F-actin solutions with ℓ = 10µm ( ), using data from Schmidt, et al.130 , showing that true moduli measured bead.” macroscopically and microscopically are unequal. Keller, et al.129 introduce an oscillatory magnetic bead rheometer, in which 2.25µm latex beads contain- ing imbedded Fe2O3 are placed in an oscillating mag- netic field, and the bead positions are tracked with video Figure 58 shows representative parts of Schmidt, et ′′ ′′ microscopy. The amplitude and phase shift of the bead al.’s measurements of Gmacro/Gmicro. Over a wide range ′ ′′ ′′ motion are determined, allowing calculation of G (ω) and of frequencies, Gmacro/Gmicro for F-actin solutions is in G′′(ω). The paper was a proof of principle demonstrating the range 2-8. Schmidt, et al.130 cite Maggs131 as predict- a method for making measurements at higher frequencies ing similar results. Maggs examined spherical nanoparti- (up to 40 Hz), so only a single solution of nondilute F- cles diffusing through intertwined actin filaments in solu- actin was examined. tion. Maggs’ model treats probes bending filaments and Schmidt, et al.130 compare microscopic and macro- distorting the local mesh, using scaling arguments and scopic measurements of the storage and loss moduli of the presence of two independent length scales, namely F-actin and gelsolin solutions. Schmidt, et al. exam- the persistence length and the mesh size. The model does ined F-actin solutions with 0.5 ≤ c ≤ 2 g/L, and con- not include hydrodynamic interactions between probes trolled average lengths 1.5 ≤ ℓ ≤ 10µm, all for frequen- and the mesh. The apparent storage and shear moduli cies 0.004-4 Hz. The microscopic particles were 4.5µm are seen to be sensitive to the apparatus length scale. paramagnetic beads. These results are from true mi- Schmidt, et al.132 studied the rheological properties crorheological measurements: the microspheres were sub- of solutions of fd-virus, using classical mechanical and ject to a known external force and the amplitude and magnetic tweezers rheometry to determine G′ and G′′. phase shift of their motions relative to the driving force Macroscopic and microscopic measurements are reported were obtained. The macroscopic probe was a rotating to be in reasonable agreement. Comparison was made disk rheometer. With a viscous small-molecule liquid, with some modern theoretical calculations133; the ob- microrheology and macrorheology agree as to the mea- served frequency dependence of G′ at low frequency was sured viscosity. Moduli of F-actin solutions measured much weaker than predicted. Comparison was also made microscopically were substantially smaller than moduli with studies of actin solutions, permitting separation of measured with a rotating disk rheometer. The network fundamental physical properties from peculiarities of par- relaxation time estimated from microscopic data is the ticular chemical systems. Schmidt, et al., observe that same as or substantially larger than the time measured actin aggregation is extremely sensitive to a wide variety with the macroscopic instrument. The frequency depen- of proteins and other environmental factors. They pro- dences from microrheological and from macrorheological pose that a practical way to avoid these challenges to the measurements are somewhat similar, but are clearly not use of F-actin solutions as a model for testing theories of the same: The microrheological measurements typically polymer solution dynamics is to use alternative physical show stronger dependences of G′ and G′′ on ω. systems, such as the fd-virus that they studied, as sound 37 models for testing theories of polymer solution dynamics. polymerization initiator. The fraction concentration of From the published literature, in small-molecule liq- bisacrylamide crosslinker was varied from zero (leading uids true microrheology and classical measurements give to linear polyacrylamide solutions) to 5% (leading to a the same results. In many though not all polymer solu- strong gel). Under fixed optical conditions, the extent tions, true microrheology and classical rheometry do not to which the correlation function decayed at long times agree, the viscosity measured with true microrheology decreased with increasing crosslinker concentration and generally being smaller than the viscosity determined by with decreasing scattering angle, indicating that above a classical, macroscopic rheometer. 1.6% bisacrylamide more and more particles are confined, Particle tracking gives direct access to correlation func- especially over longer distances. Nishio, et al. make an tions h∆R~(t1,t2)∆R~(t3,t4)i. These functions in turn inversion of their data to determine the distribution of give access to the memory functions for the Langevin- effective pore lengths for different probe radii (and, im- equation ’random’ force in ways that light scattering plicitly, pore diameters) showing that the distribution is spectroscopy does not. Bandwidth for video tracking quite wide. presently limits that technique, but bandwidth improve- Park, et al.137 used holographic relaxation spec- ments are a matter of time, and alternative paths to de- troscopy to measure Dp of a dye and a labelled protein 120 termining displacements ∆R~(t1,t2) exist . through polyacrylamide gels as a function of polyacry- lamide concentration c. The holographic method mea- sures diffusion over distances orders of magnitude larger E. Probes in Gels and Biological Systems than any structure in the gel. QELSS was used to in- fer a nominal correlation length ξ for the gels from their True gels are not solutions. However, some mod- apparent diffusion coefficient D via els of polymer solution dynamics invoke analogies with kBT polymer motion through gels, at least on favorable time D = , (21) 6πηξ scales, so it is clearly worthwhile to compare probe mo- tion through polymer solutions with polymer motion where kB is Boltzmann’s constant, T is the absolute tem- through true gels. Much work has focused on (1) cross- perature, and η is the solvent viscosity. On comparison linked polyacrylamide gels and (2) cross-linked actin and with literature data on diffusion by D2O, sucrose, and other protein gels, but there are also results on (3) probes urea through the same medium, Park, et al find from in the complex interior of living cells and other gelling their measurements that the probe diffusion coefficient systems. Dp depends on c and probe radius R via
ν δ Dp = Dpo exp(−ac R ) (22) 1. Probes in Polyacrylamide Gels for ν ≈ 0.94 and δ ≈ 0.59. That is, Park, et al. found Allain, et al.134 studied the diffusion of 0.176 that real polyacrylamide gels are size filters that selec- µm diameter polystyrene spheres through solutions of tively retard the diffusion of larger probe particles. Park, acrylamide/N,N’-methylene bisacrylamide during irre- et al. also find from QELSS measurements on gels that −0.64 versible gel formation. QELSS was used to measure the ξ ∼ c , leading them to note that Dp/Dpo is to good relaxation spectra of the probes, which dominated the approximation a function of the single variable RC/ξ. scattering by their host solutions. The probe diffusion Reina, et al.138 used QELSS to measure the diffusion −1 time τ ∼ Dp was obtained from a second order cumu- of 25 and 50 nm radius PSL probe particles in polyacry- lant fit to the spectra. During the gelation process, τ of lamide solutions and cross-linked polyacrylamide gels the probes increased 120-fold with increasing duration of over a range of polymer concentrations and scattering an- the gelation process. Over the same time interval, the gles. In simple solutions, the probe spectrum is close to a macroscopic solution viscosity increased only 12-fold, so single exponential, whose decay rate falls with increasing the probe diffusion time was not simply proportional to polymer c until the gel threshold is reached. Above the the macroscopic viscosity. threshold, a second slow mode appears; the relaxation Matsukawa and Ando135 used PFG NMR to study rates of both modes then increase with increasing poly- polyethylene glycol and water diffusing through fully mer concentration. At the threshold, particle trapping swollen cross-linked polyacrylamide gels. PEG molec- becomes evident from the lack of complete relaxation of ular weights were 4.25, 10.89, and 20 kDa. Dp was taken the scattering spectrum. Above the threshold, a complex to be proportional to Dpo exp(−κR), where κ is a gel q-dependence of the spectrum determined by the probe length scale that could be varied by changing the degree diameter and the gel mesh spacing is observed. of swelling. Suzuki and Nishio139 examined 60 nm radius PSL Nishio, et al.136 used QELSS to monitor the diffusion spheres in polyacrylamide gels, determining the extent of polystyrene latex spheres in polyacrylamide: water. to which the spectrum decays toward zero as t → ∞, Probe radii were 25 and 50 nm. Probes were added to the and the dependence of the spectrum on monomer concen- solutions prior to adding the ammonium persulphonate tration and scattering angle. Spectra were polymodal. 38
1.2 heavily overlapped F-actin filaments. The experimental focus was studying nine probes in each of a few solu- tions, so concentration dependences for Dp were exam- ined imprecisely. Probes diffusing through solutions of 1 globular particles showed a D/Do that was only weakly dependent on probe radius Rh. For probes in these so- lutions, D/Do was always larger than ηo/η, Dη/Doηo
p0 increasing with increasing solution viscosity. In con-
/D 0.8 trast, D/Do of probes diffusing through F-actin solutions p falls markedly with increasing probe radius, but extrap- D olates to D/Do = 1 as Rh → 0. Figure 59 shows the nine probes diffusing through heavily overlapped actin 0.6 matrix solutions. A joint fit of all measurements gave 0.55 0.38 D/D0 =1.21 exp(−0.067R c ), as shown. Hou, et al.141 compared their results with findings of Luby-Phelps, et al.142,143 on probes in the cytoplasm of 0.4 living cells. In living cells, D/Do depends strongly on 0 10 20 30 probe radius Rh but extrapolates to D/Do ≈ 0.3 as R (nm) Rh → 0. In contrast to results in in vivo probes, probes in simple ficoll and bovine serum albumin solutions have a D/Do that is substantially independent of Rh. Probes 141 FIG. 59: Diffusion of ficoll probes through solutions of in F-actin solutions have D/Do ≈ 1 in the limit of small overlapped F-actin chains at actin concentrations 1 ( ), Rh. However, probes in a mixture of F-actin and con- 3(), 5 (♦), 8 (△), and 12 () g/L, and a fit to a single centrated (7-10%) ficoll or concentrated (7%) BSA show joint stretched exponential in c and R. the same properties as probes in cytoplasm, showing that probe diffusion in cytoplasm is plausibly governed by the simultaneous presence of a network phase and concen- Qualitative properties of spectra were interpreted from trated globular macromolecules. physical models. Madonia, et al.144 prepared nongelling and gelling hemoglobin solutions and used QELSS to watch the dif- fusion through them of 43 and 250 nm radius probe par- 2. Probes in Protein Gels ticles. At the onset of gelation, Dp of the 250 nm probes falls rapidly. Under identical experimental conditions, Fadda, et al.140 used QELSS and static light scatter- as gelation proceeds Dp of the 43 nm spheres increases ing to monitor the diffusion of 225 nm radius polystyrene slightly. The gel structure functions as a size filter, trap- latex probes through gelatin solutions during the gela- ping the larger particles but not obstructing significantly tion process. Static light scattering determined the par- the motions of the smaller particles. ticle radius. The presence of a deep first minimum in Newman, et al.145 used QELSS to monitor the dif- I(q) showed that the particles were highly monodisperse. fusion of polystyrene latex spheres through F-actin so- The spectrum hI(0)I(τ)i of the probes was monitored as lutions. Four probe species with radii of 55, 110, 270, a function of time t after quenching from high to low and 470 nm were employed. Actins were polymerized temperature. The spectrum was approximated as a pure with MgCl2, KCl, or CaCl2; however, all samples stud- exponential at small τ and a stretched exponential in τ ied were liquids, not gels. Prior to polymerization, Dp at large τ. When gelation set in, the short-time decay be- in the actin solutions was not significantly different from came faster, the long-time decay became very long, and Dp of the same probe in pure water. On addition of a normalized measure of spectral intensity fell. Compar- salt to start the polymerization process, Dp of the latex ison was made with particular models of gel formation. spheres began to fall, and the second cumulant of the Hou, et al.141 used probe diffusion measured with probe spectra began to increase, showing that the distri- FRAP to examine several physical models of the intra- bution of probe displacements is no longer a Gaussian. cellular medium. While simpler model systems contain- Changes in spectral parameters stop after several hours. ing either globular particles or long chains did not re- Dp was determined for all probes, actin concentrations produce the behavior of probes moving in vitro, probes 1 ≤ c ≤ 20µM, and several scattering angles. Dp in so- diffusing through a mixture of globular and long-chain lutions polymerized with Mg or Ca followed a stretched ν δ proteins did show most physical properties seen with in exponential Dp/Dp0 = exp(−αc R ) for ν =1.08 ± 0.09 vivo probe diffusion studies. The probe particles were and δ =0.73 ± 0.05. In contrast, for probes in solutions a series of fluorescently-tagged size fractionated ficolls. polymerized with K, one finds δ ≈ 0. The background matrix included concentrated globular Newman, et al.146 used QELSS to examine the diffu- particles (ficolls or bovine serum albumin(BSA) ) and/or sion of PSL spheres (radii 55, 105, and 265 nm) through 39 solutions of polymerized actin. The actin concentration erogenous. While the motion of some particles showed was fixed at 0.65 g/L throughout the experiments. Ad- complete trapping, the particle never moving far from dition of gelsolin, a protein that interacts with actin its starting point, the motions of other particles showed to shorten the actin filaments, was used to determine only partial trapping: particles alternately were trapped the effect of actin filament length on probe diffusion at within restricted regions and made rare saltatory jumps constant total actin concentration. Analysis of QELSS to other trapping regions. Diffusion by 250, 320, and 500 spectra indicated that if little gelsolin was present the nm radius polystyrene spheres was compared with the actin network trapped many probe particles, at least over diffusive exponent γ of h(∆x)2i ∼ tγ for networks at a long distance scales. Trapping was more effective for the series of F-actin concentrations, showing that γ is a uni- larger probe particles. On shortening the actin filament versal function of R/ξ, with probe radius R and network length by adding gelsolin, the probe diffusion coefficient mesh size ξ. increased and the trapping vanished. The inferred micro- viscosity fell from 5-20 times that of water when actin fil- aments were very long (no added gelsolin) down to twice 3. Probes in Living Cells and Other Systems that of water at the largest gelsolin concentration ex- amined. After addition of gelsolin, changing the probe Arrio-Dupont150 used a fluorescence recovery tech- radius five-fold had little effect on the inferred microvis- nique to measure the diffusion of a series of fluorescently- cosity. Addition of gelsolin changed modestly the probe labeled proteins through the interior of cultured muscle spectrum line shape: the second spectral cumulant was cells. Probe radii extended from 1.3 to 7.2 nm. Cellular large at very low gelsolin concentration, when some par- interiors are size filters, selective obstructing the diffusion ticles were trapped, but dropped a great deal when all of the larger probes. The retardation of the diffusion of probes were free to diffuse. larger globular proteins by the cell interiors is substan- 147 Schmidt, et al. , as part of an extremely system- tially more extensive than the retardation of similarly atic study of actin dynamics, examined the diffusion of large dextrans by the same interiors151, perhaps because 35, 50, and 125 nm radius latex spheres through fully dextrans are flexible and globular proteins are inflexible. polymerized actin networks. The actin networks were Kao, et al.152 used fluorescence recovery after photo- also characterized using QELSS, FRAP, and electron mi- bleaching and picosecond resolution fluorescence polar- croscopy. Sphere diffusion through the networks followed ization measurements to determine translational and ro- ν δ Dp = Do exp(−αc R ) with prefactor α ≈ 1, and ex- tational diffusion coefficients and fractional fluorescence ponents ν ≈ 1 and δ ≈ 2. For actin concentrations recovery levels of fluorescein derivatives in Swiss 3T3 fi- 0 < c ≤ 2 mg/ml, 125 nm radius spheres were retarded broblasts and viscous solutions. Careful analysis permit- by up to sixfold, while the concentration dependence of ted separate determination of the fluid-phase viscosity, Dp of the smaller spheres was weak. Over the range of the level of probe binding, and the inhibition of probe radii studied, namely R comparable with the mesh spac- diffusion due to the cell volume occupied by the relatively ing seen in an electron micrograph, heavily interlaced immobile cytomatrix. The combination of these effects networks of actin molecules act as size filters, selectively accounts for the retardation of small-molecule solutes by retarding the motion of the larger probe particles. cellular interiors. Stewart, et al.148 used holographic relaxation spec- Luby-Phelps143 and collaborators used FRAP to study troscopy (HRS) to monitor the diffusion of dye and the diffusion of a series of fractionated ficolls, both in tagged proteins through fibrin gels. In an HRS exper- the cytoplasm of 3T3 cells and in concentrated (10-26%) iment, a pair of crossed laser beams are used to bleach protein solutions. Ficoll fractions had hydrodynamic di- a holographic pattern in solution, creating an index of ameters 60-250 A˚. Dp/Do of ficolls in protein solutions refraction grating formed from unbleached molecules, was independent of the size of the ficoll molecules. That and perhaps separate gratings formed from bleached, is, protein solutions are not size filters; they retard to photochromically modified, or photochemically bound approximately the same fractional extent the diffusion molecules. Stewart, et al. found that small dye molecules of small and large probe particles. In contrast, cell cy- diffused through fibrin gels as if they were in pure water. toplasm is a size filter. Dp/Do of ficolls in cytoplasm The diffusion of labelled BSA molecules was retarded by falls sixfold as the ficoll diameter is increased from 60 to the gel, though less so if the gel was formed in the pres- 500 A˚. For the largest ficoll, FRAP finds that a third of ence of Ca+2, while in contrast the photobleaching reac- the ficoll particles are trapped by cytoplasm; only 2/3 of tion caused some labelled immunoglobulin G molecules the particles are able to diffuse through substantial dis- to bind to the fibrin gel and become immobile. tances. Luby-Phelps, et al. propose that objects larger Wong, et al.149 study diffusion of probe particles in than 500 A˚ or so are mechanically prevented from diffus- F-actin networks, using video tracking microscopy to ob- ing through cell cytoplasms. serve particle motion. Mean-square displacements were Luby-Phelps, et al.153 describe a fluorometric method measured directly. They were not diffusive: h(∆x)2i using two homologous dyes for determination of the local was not ∼ t1. Examination of the motions of individ- solvent viscosity, and apply the technique to determine ual particles showed that particle dynamics were het- the viscosity of water in the cytoplasm of a cell. The 40
fluorometric behavior of one dye (but not the other) is Dasgupta, et al.157 used DWS and QELSS to study sensitive to the local solvent mobility, while both dyes are polystyrene spheres diffusing through 200 and 900 kDa approximately equally responsive to other environmental polyethylene oxide solutions. A wide range of polymer influences. The dyes were bound to ficoll for microinjec- concentrations, 15c∗ − 45c∗ was examined, taking c∗ of tion purposes. It was found that the solvent viscosity the two polymers to be 0.48 and 0.16 wt%, respectively. is not greatly different from the viscosity of bulk water, Polystyrene probes had radii 230, 320, 325, 485, and 1000 resolving a long and well-known literature dispute. nm, the PSL being carboxylate-modified except for the Seksek, et al.154 studied the diffusion of labelled dex- 325 nm probes, which were sulphate modified. QELSS trans and ficolls within fibroblasts and epithelial cells us- spectra were interpreted by invoking the ’Gaussian’ as- ing FRAP. Dextrans had molecular weights 4, 10, 20, 40, sumption, even though the nominal (∆x(t))2 was clearly 70, 150, 580, and 200 kDa. Ficolls were size-fractionated, not linear in t. QELSS spectra were truncated and only four fractions being used for further studies. Probes reported for times longer than 10 mS. were microinjected into cells. With large probes, the cy- Gisler and Weitz158 used diffusing wave spectroscopy toplasm and nucleus can be studied separately. Small to examine the motion of polystyrene spheres through F- probes pass through the nuclear membrane, so nucleus actin solutions. The Gaussian assumption was invoked and cytoplasm had to be studied simultaneously. In to convert DWS spectra to nominal mean-square parti- MDCK cells, Dp/Dpo did not depend on the size of the cle displacements. The inferred displacements were then probes. Seksek found that under some conditions pro- used to infer viscoelastic properties of F-actin gels. cesses other than translational diffusion could lead to Heinemann, et al.159 used diffusing wave spectroscopy photobleaching recovery. to examine 720 nm polystyrene latex spheres mov- Probe diffusion measurements are often said to be re- ing through aqueous solutions of potato starch, using lated to the phenomena that control the release of medi- γ-dodecalactone to induced aggregation of the starch cal drugs from semirigid gels. Shenoy and Rosenblatt155 molecules. DWS spectra were inverted by invoking the provide an example in which release times are mea- gaussian approximation. Inverted spectra were used to sured directly. They examined succinylated collagen and generate nominal values for the storage and loss moduli hyaluronic acid matrices, separately and in mixtures, us- as functions of frequency. ing bovine serum albumin and dextran as the probes Kao, et al.160 used diffusing wave spectroscopy, in a whose diffusion was to be measured. specialized instrument in which the digital correlator is replaced by a Michelson interferometer, to examine the diffusion of 103-230 nm radius colloidal spheres over the F. Probe Spectra Interpreted with the Gaussian first 20 nS of their displacements. The Gaussian approx- Assumption imation for particle displacements in quasielastic scatter- ing was invoked. This section notes a series of papers, sometimes linked Kaplan, et al.161 use DWS to study structure forma- under the cognomen microrheology, that rely on the as- tion in alkyl ketene dimer emulsions. Spectra were in- sertion that the incoherent structure factor for the diffu- terpreted by invoking the Gaussian assumption. These sion of dilute probes through a viscoelastic matrix may in systems are used in paper manufacture as sizing agents, 2 2 general be rewritten via hexp(iq · rii = exp(−q hr i/2), but lead to technical difficulties if they gel. The scat- the so-called ’gaussian assumption’. As noted above, tering particles were intrinsically present in the system. this assumption is incorrect except in a certain very The time evolution of spectra over a period of weeks was special case. The special case in which this assertion observed. Within a few days, systems that were going to is correct is the case in which the incoherent structure gel could readily be distinguished from those that would factor is characterized by a single pure exponential, so remain stable. It is important to emphasize that even if that − log(hexp(iq · rii) is linear in t. Interpretations a measurable is not simply related to underlying physical based on the Gaussian assumption, when applied to non- properties, the measurable may still represent a valuable exponential spectra, are therefore highly suspect. practical analytical tool for industrial purposes. Bellour, et al.156 used DWS to measure probe diffusion Knaebel, et al.162 used DWS, static light scattering, of 0.27, 0.5, 1, and 1.5 µm radius polystyrene spheres small-angle neutron scattering, and mechanical rheom- in solutions of cetyltrimethylammonium (CTA) bromide etry to characterize an alkali-soluble emulsion system. and sodium hexane sulphonate, in some cases after ion These solutions exhibit shear thinning and thixotropic exchange removal of the Br−. CTA salts in aqueous so- behavior. The low-shear viscosity increases dramatically lutions form giant extended flexible micelles. At elevated with increasing concentration, up to a limiting concen- surfactant concentrations, the micelles are nondilute and tration beyond which η increases more slowly with in- potentially entwine. However, unlike regular polymers, creasing c. The DWS measurements were analyzed by the micelles have a lifetime for a mid-length scission pro- invoking the Gaussian assumption. cess that effectively permits micelles to pass through each Lu and Solomon163 use DWS to measure the diffu- other. At elevated surfactant concentrations, the DWS sion of 0.2 to 2.2 µm polystyrene spheres and 0.25 µm spectrum had a bimodal relaxation form. colloidal silica beads in solutions of hydrophobically- 41 modified ethoxylated urethane, which is an associating a 0.01 volume fraction solution of 497 nm diameter PSL polymer, at polymer concentrations 0-4 wt%. Mechanical spheres diffusing freely in water, showing excellent agree- rheology measurements were made on the same systems. ment between the measured spectrum and the predicted DWS measurements were analyzed using the Gaussian theoretical form for the DWS spectrum, if the photon displacement assumption. From the reported nominal transport mean free path ℓ∗ was used as a fitting param- h∆r2(t)i, the underlying single-scattering spectrum was eter. Pine, et al., report that ℓ∗ from their dynamic mea- not a pure exponential. surements was consistently smaller than inferred from Mason and Weitz164 applied DWS to study the diffu- static backscattering measurements. Comparison was sion of 420 nm polystyrene spheres, at large concentra- made with mixtures of 312 and 497 nm diameter spheres tion in ethylene glycol (φ =0.56), at lower concentration (volume fractions 0.01, 0.04, respectively) that had been (φ =0.02), and in a 15 wt % solution of 4 MDa polyethy- deionized to produce a colloidal glass. The DWS spec- lene oxide. The Gaussian assumption was used to con- trum changed markedly on formation of the glass. vert g(1)(q,t) to a mean square particle displacement and Popescu, et al.171 present a novel QELSS apparatus, thence to a ’time-dependent diffusion coefficient’. A gen- based on their earlier analysis172, that uses extremely- eralized Stokes-Einstein equation translated the diffusion short-coherence-length visible light in its measurements. coefficient into microscopic storage and loss moduli. In Popescu, et al.’s method, light from a superlumines- Narita, et al.165 used DWS and QELSS to study probe cent diode with a coherence length of 30µm was sent per- diffusion in solutions of 95 kDa polyvinylalcohol (PVA) pendicular to the window surface into a scattering vol- and crosslinked PVA gels. The probes were 107 and 535 ume. Light, backscattered from particles that are located nm PSL spheres. QELSS spectra were visibly bimodal at within a few coherence lengths of the surface, combines all polymer concentrations studied. The relaxation time coherently with light back-reflected by the window, al- of the fast QELSS mode was diffusive. The slow mode lowing measurement in heterodyne mode of the QELSS showed a stronger-than-q2 dependence of its linewidth. spectrum, even of highly scattering samples. Popescu’s DWS spectra lacked visible long-time shoulders, and were analysis invokes the Gaussian assumption ’for times much interpreted using the Gaussian assumption. shorter than the decaying time of the autocorrelation Nisato, et al.166 use QELSS and DWS to examine 85 function’. and 107 nm polystyrene latex probe spheres in poly- Sohn, et al.173 present a theoretical analysis of acrylic acid gels that had been chemically-crosslinked by Popescu, et al.171,172’s microvolume QELSS apparatus, addition of methylene bis-acrylamide. DWS spectra were based on invoking the Gaussian assumption. The possi- reported for forward and backward scattering geometries, bility of working directly in frequency domain, and de- nominally giving determinations of motion on two differ- scribing the spectrum as a sum of Lorentzians rather than ent length scales. QELSS spectra had a long-time limit a sum of exponentials, is examined. that was far above the baseline calculated from the mean Rojas-Ochoa, et al.174 make a systematic study of scattering intensity. The Gaussian assumption was used DWS spectra of monodisperse interacting hard sphere to infer from the long-time limit of the dynamic struc- systems. Small-angle neutron scattering was used to de- ture factor a long-time limit of the mean displacement termine the sphere size and the static structure factor, h∆r2(∞)i; the assumption was also applied to interpret which was significantly perturbed by interparticle inter- DWS spectra. actions. The photon transport mean free path was deter- Palmer, et al.167 used DWS to study the motions of mined directly by measuring the optical transmittance of 480 nm radius latex spheres in actin and actin:α-actinin samples of various thicknesses. The hydrodynamic radius solutions. They reported spectra, nominal mean-square of the spheres was obtained via QELSS applied to dilute displacements, time-dependent diffusion coefficients, and samples. DWS was used to measure particle motions magnitude of the complex modulus as inferred from a over a few to a few hundred microseconds. The solvent generalized Stokes-Einstein equation. A Gaussian as- was Newtonian. Under these conditions, which would sumption was used to interpret spectra. not arise in a viscoelastic matrix solution, the Gaussian Palmer, et al.168 used DWS to examine the diffusion of assumption is applicable. Comparison of this no-free- 480 nm radius polystyrene beads through F-actin, con- parameter determination of Dp with predictions of or- centrations 0.42-6.89 g/l. The Gaussian assumption was thodox theory for the concentration dependence of Dp invoked to analyze spectra in terms of mean-square dis- found excellent agreement. Thus, under the very restric- placements and inferred frequency dependent viscoelas- tive conditions under which the theoretical models for ′ ′′ tic moduli. The authors also measured G and G with a Dp and diffusing wave spectroscopy are applicable, the mechanical rheometer, and compare with the DWS data. model gives good results. Pine, et al.169 presented the first experimental demon- Romer, et al.175 used DWS to study gel formation in stration of DWS as applied to complex fluids, based on colloid preparations. Gelation was induced by varying the earlier study of Maret and Wolf170 on dynamic light the solution ionic strength, using enzymatic degradation scattering in the multiple-scattering limit. The Gaus- of a neutral organic compound to create a slow change sian assumption was implicit in their use of the earlier uniform across the sample in the ionic strength of the work of Maret and Wolf170. DWS was first applied to solution. The Gaussian assumption was invoked to con- 42 vert DWS spectra to determinations of the mean-square using the ”gaussian approximation” are identified. displacement of individual particles, even though parti- As itemized in Tables I-V, probes have included cles in gelling systems have long-time correlations in the polystyrene spheres, divinylbenzene-styrene spheres, sil- forces on them, so that the nominal h(∆r)2i from DWS ica spheres, tobacco mosaic virus, bovine serum albumin, for identical particles in a gelling system is not linear in ovalbumin, starburst dendrimers, fluorescein, hematite time. Romer, et al.176 present additional DWS and clas- particles, unilamellar vesicles, micelles, ficols, PBLG sical mechanical rheology measurements on the same sys- rods, and low-molecular weight dextrans. Polystyrene tem, showing that DWS and rheometric properties show spheres, ficols, and dextrans were used in the bulk of the dramatic changes after similar elapsed times, the elapsed probe measurements. Matrix polymer chains include a time being characteristic of the system’s sol-gel transi- wide range of water-soluble polymers. Only limited data tion. exist on probes in solutions of the organophilic polymers Rufener, et al.177 used DWS to study concentrated G- that form the staple for the remainder of the polymer actin that had been systematically polymerized into F- literature. actin. The Gaussian assumption was invoked to interpret Rotational motion measurements require the use of ori- spectra of polystyrene sphere probes. Spectra of small entationally anisotropic objects. Rodlike particles in- probes decayed to the expected baseline. Spectra of large clude tobacco mosaic virus, collagen, actin fragments, probes decayed only part way to the expected baseline, and PBLG. Optically anisotropic spheres, the anisotropy leading to the inference that large but not small spheres arising from oriented internal domains or magnetite in- were trapped by the gel network. clusions, were examined by Cush, et al.101,102 and Sohn, van der Gucht, et al.178 used QELSS to study the et al.173. diffusion of 125 and 250 nm radius modified silica Results on probe diffusion fall into three phenomeno- particles in solutions of the self-assembling monomer logical classes. In the first two classes, diffusion is use- bis(ethylhexylureido)toluene. They also measured the fully characterized by a single relaxation time and hence low-concentration viscosity, dynamic shear moduli, and a well-defined probe diffusion coefficient. In the third static light scattering intensity. Concentration-driven class, light scattering spectra are more complex. The self-association gives η(c) a peculiar form: η(c) is nearly classes are: ∗ constant out to a crossover concentration c = 0.11g/L (i) Systems in which Dp decreases as c is increased, monomer, and then increases as a power law in c. QELSS with dDp/dc monotonically growing more negative with spectra were relatively unimodal at low polymer concen- increasing c. tration, but gain a slow mode at larger c. The concentra- (ii) Systems showing re-entrant concentration behav- tion at which the slow mode appears is nearly an order of ior, in which over some concentration range Dp first ∗ magnitude larger than c . Light scattering spectra were increases with increasing c and perhaps then decreases interpreted using the Gaussian assumption even when again. spectra were clearly not single exponentials. (iii) Systems whose spectra have bimodal or trimodal van Zanten, et al.179 used DWS to examine 195, 511, relaxations, corresponding to the relaxation of probe con- 739, 966, and 1550 nm diameter PSL spheres diffusing in centration fluctuations via several competing modes. 330 kDa polyethylene oxide:water solutions for PEO con- We begin with systems having a well-defined Dp. centrations 0.2 ≤ c ≤ 15 wt%, the Gaussian assumption being invoked. A cone and plate viscometer was used to study transient creep and dynamic oscillatory responses. B. Temperature Dependence
In systems in which the solvent quality does not change IV. SYSTEMATICS strongly with temperature, Dp scales with temperature as T/ηs, ηs being the (temperature-dependent) solvent A. Systems viscosity. This result applies equally to probes in solu- tions containing no polymer, in which probe-solvent in- Here we have examined the literature on the diffusion teractions govern diffusion, and to probes in concentrated of probes through polymer solutions. Nearly 200 probe polymer solutions, in which probe-polymer interactions size: polymer molecular weight combinations were ex- however mediated must dominate the diffusion process. amined at a range of polymer concentrations. There is Studies confirming this result include Bremmell, et al.16 a solid but not extremely extensive body of work on the on probes in water:glycerol and Phillies, et al.50,51,52 on temperature dependence of probe diffusion in polymer probes in aqueous polyacrylic acids and aqueous dex- solutions. A half-dozen studies of probe rotational mo- trans. Phillies and Quinlan52 observed slight deviations tion and more than a dozen reports based on particle from simple T/ηs behavior for probes in water:dextran, tracking are noted, along with a half-dozen sets of true which they interpret in terms of a temperature depen- microrheological measurements, in which mesoscopic ob- dence of the solvent quality. jects perform driven motion in polymer solutions. Two Because Dp in general simply tracks the temperature dozen studies of diffusion in which spectra are interpreted dependence of T/ηs, one infers that solvent-based hy- 43 drodynamic forces play a dominant role in probe diffu- scopically, so that Dpη depends strongly on polymer con- sion. Unfortunately, this inference says very little about centration and polymer molecular weight. In almost all the nature of polymer solution dynamics, because mod- systems showing deviations from Stokes-Einsteinian be- els for polymer solution dynamics largely agree on pre- havior, Dpη increases with increasing c, sometimes by dicting an inverse dependence of relaxation rates on the large factors. Probe particles generally diffuse faster than solvent viscosity: Changing the solvent viscosity changes expected from the solution viscosity. From the experi- the mobility of individual polymer subunits, the so-called mental standpoint, this sign for the discrepancy between monomer mobility, thus changing the rate at which poly- Dp and η leads to a marked simplification, because the mer chains can form or release hypothesized chain entan- obvious experimental artifacts such as probe aggregation glements. Changing the solvent viscosity equally changes and polymer adsorption all lead to probes that diffuse the strength of probe-chain hydrodynamic interactions, too slowly, not too swiftly as actually observed. thus changing the forces coupling the motion of a probe In simple fluids, the probe diffusion coefficient follows and of nearby chains. Temperature dependence studies the Stokes-Einstein equation with Dp ∼ 1/η. If the do rule out one entire class of approaches, namely ap- Stokes-Einstein equation remained valid in polymer so- proaches referring to reduction relative to a glass tem- lutions, Dp would accurately track the solution viscosity, perature Tg. Analyses of Dp uniformly compare mea- so that measuring Dp would simply be a replacement surements made at fixed T , and not at fixed T − Tg. Tg for classical rheological measurements. Indeed, the early 1 is claimed to depend very strongly on c, so measurements work of Turner and Hallet found that ηµ inferred from at fixed T and different c are claimed to have very dif- Dp is very nearly equal to η measured classically with a ferent T − Tg. It was proposed that comparisons should rotating drum viscometer. In contrast, a substantial mo- be made at fixed T − Tg, not fixed T , and comparisons tivation driving Lin and Phillies37,38,39 to extend their made at fixed T are not valid. To determine Tg, thereby probe diffusion work was that in their systems ηµ was allowing comparisons at fixed T − Tg albeit different T , very definitely not equal to the macroscopic η, even for 50,51,52 Phillies, et al. measured Dp at fixed c over a range very large probes. The differences between the works of of T . After removing from Dp the dependence of ηs on Turner and Hallet, and of Lin and Phillies, apparently T , the remnant T -dependence of Dp could have revealed reflect differences between chemical systems. how Tg depends on c. Comparing results at fixed T − Tg Probes in small-M polymer solutions generally show would eliminate any c-dependence of Dp arising from the 50,51,52 Stokes-Einsteinian behavior with ηµ ≈ η, including c-dependence of Tg. Phillies, et al. actually found 19 160 nm spheres in 101-445 kDa PMMA:CHCl3 for that after removing from Dp the dependence of ηs on η/ηs up to 10, 20 and 230 nm probes in aque- T , there is no remnant T -dependence of Dp to inter- ous 20 kDa dextran45, and 20-1500 nm spheres in pret. Correspondingly, either Tg is independent of c, or aqueous 50 kDa polyacrylic acid39. On the other the notion ’reduction relative to the glass temperature’ is hand, 49 and 80 nm probes in aqueous 90 kDa poly- fundamentally inapplicable to probe diffusion in polymer L-lysine42 show small c-independent deviations from solutions. In either case, the data serve to reject the use Stokes-Einsteinian behavior. Stokes-Einsteinian behav- of glass temperature corrections as a part of interpreting ior is also found in some large-M systems. Onyene- Dp(c). mezu, et al.43 find Stokes-Einsteinian behavior within Changes in temperature may also change the solvent experimental accuracy for 1100 kDa polystyrene solu- quality and hence the radius of each polymer coil. Clome- 1 82 tions having η/ηs as large as 100. Turner and Hallett nil and Phillies examined probes in HPC:water at dif- 45 and Phillies, et al. reveal ηµ/η ≈ 1 in dextran solu- ferent temperatures, including temperatures at which tions even with M as large as 2 MDa. In most non- water is a good solvent for HPC and temperatures ap- dilute polymer solutions, non-Stokes-Einsteinian behav- proaching the pseudotheta transition. Corresponding to ior is found20,22,23,24,25,26,30,31,32,34,38,39,42,47,56,57,58,79, the change in solvent quality with temperature, the de- often leading at large c to ηµ ≪ η and rH ≪ R. A par- pendence of Dp on polymer concentration is markedly ticularly conspicuous example of non-Stokes-Einsteinian temperature dependent. Unfortunately, as detailed in behavior appears in Fig. 21, based on Lin and Phillies39, Ref. 82, the observed dependence is equally consistent involving 20-620 nm probes in aqueous 1MDa polyacrylic with most models of polymer solutions, so these results acid. do not sort between different models of polymer dynam- Much early work on the relationship between D ics. p and η referenced the pioneering study of Langevin and Rondelez21 on sedimentation in polymer solutions. Langevin and Rondelez propose that small sedimenting C. Probe Diffusion and the Solution Viscosity particles experience the solvent viscosity, but adequately large sedimenting particles experience a much larger drag The relationship between Dp and the solution viscosity proportional to the macroscopic solution viscosity. The was a primary motivation for early studies of probe dif- distinction between small and large particles is deter- fusion. In some systems though not others, Dp does not mined by a ratio R/ξ, R bring the probe radius and scale inversely as the solution viscosity measured macro- ξ being a hypothesized characteristic correlation length 44
”...dependent on concentration...but independent of the a considerably narrower range. Whenever Dp and c were molecular weight M.” Langevin and Rondelez cite un- varied over a sufficient range to support a credible log-log 180 published results of deGennes as predicting for the plot of Dp against c (e.g., Figures 13b, 19, 20, and 37), sedimentation coefficient s the apparent slope d log(Dp)/d log(c) decreases monoton- ically with increasing c, precisely as expected for data s δ ν = exp(−AR c )+ ηs/η. (23) that follow a stretched exponential in c. s0 If Dp followed a power law in c, a plot of log(Dp) Here A is a molecular-weight-independent constant and against log(c) would reveal a straight line. It is possi- δ is a scaling exponent. ξ ∼ 1/(Acν ) is the distance be- ble on the aforementioned figures to draw straight lines tween the hypothesized entanglement points of the tran- tangent to the observed Dp(c) curves, but such straight sient statistical lattice seen in the deGennes model181 for lines manifestly are simply local tangents. Furthermore, polymer solutions. For small particles, this form predicts a log-log plot of a stretched exponential in c, no matter its s ≈ s0, while for large particles the exponential is domi- parameters, always shows a smooth curve, while on the nated by ηs/η, even for ηs/η ≪ 1, so that s is determined same plot a power-law is a straight line. Measurements by the solution’s macroscopic viscosity. The Langevin- that actually follow a power law in c therefore cannot be Rondelez equation21 is a special case of general assertions described accurately with a force-fitted stretched expo- that: polymer solutions have a longest distance scale ξ, nential, and vice versa. properties measured over distances s ≫ ξ necessarily re- Relationships between α, ν, and M for probes in an flect macroscopic system properties, and therefore Dp of extensive series of homologous polymers were studied by large (R ≫ ξ) probes must follow the Stokes-Einstein Phillies, et al.45 (PSL in dextran: water) and by Phillies, equation. If this proposal were correct, the polyacrylic et al.48 (PSL in polystyrenesulfonate:water.) Represen- acid:water systems in which the Stokes-Einstein equation tative results on these systems appear as Figures 25 and fails badly would be in this regime R ξ, implying a sur- 28, respectively. Figure 60 shows α and ν for probes prising ξ of hundreds of nanometers. in dextran:water45. Over the observed molecular weight 0.84 Langevin and Rondelez report sedimentation experi- range, α ∼ Mw . With increasing M, ν decreases ments leading to ν ≈ 0.62, δ ≈ 1 being consistent with from 1.0 to slightly more than 0.5. The authors deter- limited measurements, and A ∼ M 0 in the limit of low mined the polymer molecular weight distributions using probe concentration. Langevin and Rondelez do not treat aqueous size-exclusion chromatography; measured poly- diffusion explicitly. The same drag coefficient determines dispersities Mw/Mn ranged from 1.16 to 2.17. No effect s and Dp, leading to Dp/Dp0 ∼ s/s0, implying that eq. of polydispersity on probe diffusion was identified. These 23 should also govern probe diffusion. Bu and Russo20 results show that α depends strongly on M, and serve to examine the diffusion of extremely small probes (0.5-55 reject any suggestion that polymer dynamics are strongly nm) in HPC:water. In one of the few published explicit sensitive to the detailed shape of the polymer molecular tests of eq. 23 as applied to diffusion, Bu and Russo find weight distribution, as opposed to being determined by the dependence of Dp/Dp0 on R to match Langevin and averaged molecular weights. Rondelez’s prediction. Small probes diffuse much more Phillies, et al.48 sought to determine the initial slope rapidly that expected from η; for larger probes ηµ ap- of Dp(c) for probes in a series of of homologous monodis- proaches η. perse polymers. The intent was to test the Phillies- A result A ∼ M γ for γ 6= 0 is entirely inconsistent with Kirkitelos49 hydrodynamic calculation of α. Measure- the physical model invoked by Langevin and Rondelez in ments were fit to straight lines, simple exponentials, and their ansatz for their equation. However, in large num- stretched exponentials in c, which describe these data bers of probe:polymer systems Dp/Dp0 depends strongly out to progressively larger c. The Phillies-Kirkitelos49 on M. It is at present unclear whether the strong M- calculation, which parallels hydrodynamic calculations dependence of Dp/Dp0 would disappear with sufficiently of the concentration dependence of Ds of hard spheres, small probes, in which case eq. 23 could be correct, or has no free parameters. The predicted α depends on whether Dp/Dp0 is determined by an equation similar to probe radius ap, chain thickness ac, and M. So long 23 but in which A depends on M. as ap ≫ ac, the calculated α is very nearly insensitive to ap and ac. As seen in Figure 61, for M ≥ 38 kDa good agreement is found between the measurements and D. Concentration Dependence of Dp the no-free-parameter calculation. For M ≤ 20 kDa, the experimental α is too large, a deviation interpreted by As seen above, in almost but not quite every system Phillies, et al.48 as appearing because short polystyrene whose relaxations can be characterized by a single diffu- sulfonate chains are not well-approximated as random sion coefficient Dp, the concentration dependence of Dp is coils. described by a stretched exponential in polymer concen- Accurate determination of α and ν from non-linear tration. The fitting parameters α and ν of the stretched least squares fits is challenging, because errors in these exponentials are found in Tables I-V. It is apparent that parameters are strongly anticorrelated. To determine α α varies over a substantial range, and that ν varies over and ν accurately, one needs accurate measurements of 45
1 0.1
0.1 a a 0.01
0.01 0.001 10000 100000 1000000 1 10 100 1000
M (Da) Mw (kDa)
FIG. 61: α, the initial slope of Dp(c) against M for 7 ( ), 34 (•), and 95() nm radius polystyrene spheres in aqueous 1 polystyrene sulfonate: 0.2 M NaCl, and (line) the no-free- parameter prediction for α(M) in this system, after Phillies, et al.48.
With these exclusions, Figures 62 exhibit α and ν as
n 0.8 functions of M for probes in a range of polymer solu- tions. The solid line α ∼ M 0.84 is taken from Figure 65, Ref. 60; it represents a best fit to the measurements on dextran solutions. With few exceptions, α is seen to be closely correlated with the molecular weight of the 0.6 matrix polymer over two orders of magnitude in its de- pendent and independent variables. From Figure 62, one sees ν ≈ 1 at lower polymer molecular weight, and more scattered values at larger M. Probes in polyisobutylene 10000 100000 1000000 are noteworthy for retaining ν ≈ 1 up to the largest M M (Da) examined. The solid line in Figure 62 corresponds to the dependence of ν found for the not entirely physically dissimilar polymer self-diffusion coefficient. As seen in FIG. 60: Parameters α and ν from fits of Dp of 20.4 (◦) Figure 65 of Ref. 12, ν from D is ≈ 1.0 at lower M, and 230 () nm spheres in dextran solutions, and η of those s ν declining to 0.5 by M ≈ 500 kDa. α and ν are thus seen solutions, to exp(−αc ), against dextran Mw, showing α ∼ M 0.84 with ν decreasing from 1.0 toward 0.5 over the observed to be correlated with matrix molecular weight M, a cor- molecular weight range. relation that will be predicted by a valid physical model for these parameters. A few systems show re-entrant behavior in which a relaxation rate increases rather than decreasing with in- Dp(c) at quite small, not just large, concentrations, as creasing c. Most observations of reentrant behavior in- well as over a wide range of Dp(c)/Dp(0). Only a lim- volve systems showing bimodal spectra. For example, ited number of studies seem to attain these conditions. Bremell16,17, and Dunstan and Stokes29 report for low- Furthermore, the systematic review above reveals sys- ionic-strength solutions a slow probe mode that slows tems with multimodal spectra, systems in which Dp(c) with increasing c and a fast probe mode whose relaxation exhibits re-entrant behavior, and some measurements on rate first increases and then decreases with increasing c. polyelectrolyte and rodlike polymer solutions, none of With the same polymer and elevated (0.1, 1.0 M) ionic which appear exactly comparable to mesoscopic probes strengths, the relaxation rates of both modes increase diffusing in solutions of near-random-coil neutral poly- and perhaps plateau with increasing c. Ullmann, et al. mers. do report a probe in polyethylene oxide:water whose sin- 46
HPC:water, and by Papagiannapolis, et al.119 (who find the plateau in ηµ rather than Dp). 1 E. Particle Size Effects
The evidence above on the effect of probe size reduces 0.1 to two general statements with a qualification. First, true gels, e.g., polyacrylamide gels, protein gels, or cellular in-
α teriors, are indeed size filters, as witness the results of, e.g., Luby-Phelps, et al.142,143. True gels are substan- 0.01 tially more effective at retarding the diffusion of larger probes than at retarding the diffusion of smaller probes. Second, to first approximation polymer solutions are rel- atively weak size filters. As witness the results of Hou, et al.141, polymer solutions are only slightly more effec- 0.001 tive at retarding the diffusion of smaller probes that at 1 10 100 1000 10000 retarding the diffusion of larger probes. However, the ap- M (kDa) proximation that polymer solutions are weak size filters breaks down for very small probes, as studied by Bu and 2 Russo20. In a polymer solution, a very small probe may be considerably more mobile, relative to its size, than a large probe would be. It is legitimate to ask if rheology, and probe diffusion 1 using sufficiently large albeit mesoscopic probes, should 0.8 measure the same viscosity. It is noteworthy that the few true microrheological measurements on forced motion 0.6 of micron-scale probes indicate that true microrheology
n and classical rheology do not always measure the same 0.4 quantities. In simple liquids130 and in rigid-rod fd-virus solutions132, macroscopic and true microscopic measure- ments of viscosity are consistently in good agreement. On 0.2 the other hand, in low-molecular-weight 85 kDa polyethy- lene oxide solutions, dynamic moduli measured with a 1.6 µm probe are128 ’quite different from those obtained by a macroscopic rheometer’. In F-actin solutions, true mi- 0.1 crorheological moduli are substantially smaller than their 1 10 100 1000 10000 macrorheological counterparts130, typically by factors of M (kDa) 2-8. Only very limited true microrheological measure- ments exist for neutral random-coil polymers in good solvents, so generalization appears hazardous. Because FIG. 62: α and ν of exp(−αcν ) as functions of M, for meso- 19 30 microscopic and classical macroscopic rheological instru- scopic probes in solutions of PMMA (N) , dextran (◦) ments sometimes give different values for η, there is no 45 ♦ 39 43 (•) , polyacrylic acid ( ) , polystyrene(△) , polyisobuty- reason to expect that η from D will agree with η from lene (+), and bovine serum albumin (⋆)58 (plotted based on µ p 0 84 a classical macroscopic instrument, rather than agreeing its hydrodynamic radius). The lines are α ∼ M . , based on probes in dextran:water45, and ν = ν(M) based on Fig. 65 of with the very different η from a microscopic instrument. Ref. 12.
F. Comparison of Probe Diffusion and Polymer gle relaxation rate has a true maximum with increasing Self-Diffusion c. However, Ullmann, et al.’s data were from an old- style linear correlator, so their spectra might have been Phillies, Brown, and Zhou182 re-examine results of bimodal, the second peak passing unseen. Brown and collaborators19,69 on probe diffusion by sil- In a very few systems, an apparent low-c plateau is ica spheres and tracer diffusion of polyisobutylene (PIB) observed, in which Dp is nearly constant over a range of chains through PIB: chloroform solutions. These com- low concentrations and then begins to fall above some parisons are the most precise available in the literature, low-concentration limiting c. Good examples of this be- in the sense that all measurements were made in the havior are supplied by Yang and Jamieson79 for probes in same laboratory using exactly the same matrix polymer 47 samples. Comparisons were made between silica sphere lutions have a dominant characteristic length scale ξ, ξ probes and polymer chains have very nearly equal Dp being approximately the size of a polymer chain, and not and Dt in the absence of PIB. For each probe sphere and some shorter distance. In particular, while small and probe chain, the concentration dependence of the single- large probes both show bimodal spectra, probes having particle diffusion coefficient is accurately described by a R < ξ and R > ξ show very different concentration de- stretched exponential in c. For large probes (160 nm sil- pendences for their spectral parameters. The observed ica spheres, 4.9 MDa PIB) in solutions of a small (610 characteristic length ξ is approximately 2Rg, the diame- kDa) PIB, Dp(c)/Dt(c) remains very nearly independent ter of a random chain coil. ξ depends weakly or not at of c as Dp(c) falls 100-fold from its dilute solution limit. all on polymer concentration. In contrast, the hypothe- In contrast, with small probes (32 nm silica spheres, 1.1 sized characteristic length scale between chain entangle- MDa PIB) in solutions of a large polymer (4.9 MDa PIB), ments is a distance, smaller than a polymer chain, that Dp(c)/Dt(c) depends very strongly on c. Between 0 and depends strongly on polymer concentration. Taken to- 6 g/L matrix PIB, Dt falls nearly 300-fold, while over the gether these results serve to reject interpretations of the same concentration range Dp falls no more than 10-fold. viscoelastic solutionlike:meltlike transition in this system Dp was followed out to the considerably larger concen- as representing a transition to reptation dynamics. How- tration at which it, too, had fallen 300-hundred-fold from ever, it is not certain if chain dynamics in HPC:water are its small-matrix-concentration limit. This result is some- representative of generic polymer dynamics, or if special what surprising relative to some theories of polymer dy- chemical effects unique to HPC are important, so it ap- namics, because in non-dilute polymer solutions polymer pears premature to generalize these interpretations to all chains are predicted181 to have available modes of mo- polymer solutions. tion, such as reptation, that are predicted to be denied to rigid spheres. That is, in solutions of a very large (4.9 MDa) poly- V. SUMMARY mer, at concentrations large enough that Dp was reduced 300-hundred-fold, the matrix polymer was far more effec- The experimental literature on the motion of meso- tive at obstructing the motions of rigid probes than at scopic probe particles through polymer solutions is sys- obstructing the motion of long flexible chains. One could tematically reviewed. The primary focus is the study propose that at still larger c the trend in Dp(c)/Dt(c) of diffusive motion of small probe particles. Compar- reverses with increasing c, but there is absolutely no sign ison is made with literature data on solution viscosi- of such a phenomenon in the published experiments. In ties. A coherent description was obtained, namely that the 4.9 MDa system, c[η] was only taken out to 2 or the probe diffusion coefficient generally depends on poly- ν 6, so it could reasonably be argued that the observed mer concentration as Dp = Dp0 exp(−αc ). One finds phenomenology only refers to non-dilute non-entangled that the scaling prefactor α depends on polymer molec- systems, and that a different phenomenology might be ular weights as α ∼ M 0.8. The scaling exponent ν encountered in entangled polymer solutions. 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, G. Systems Having Multimodal Spectra Dpη typically increases markedly with increasing poly- mer concentration and molecular weight. In some sys- While multimodal probe spectra have been reported tems, e.g., hydroxypropylcellulose:water, the observed in several systems, only in hydroxypropylcellulose: wa- probe spectra are bi- or tri-modal. Extended analysis ter is there a highly systematic examination of the full of the full probe phenomenology implies that hydrox- range of experimental parameters. Because only one sys- ypropylcellulose solutions are characterized by a single, tem showing multimodal spectra has been studied sys- concentration-independent, length scale that is approxi- tematically, one cannot be certain to what extent one is mately the size of a polymer coil. In a very few systems, examining specific chemical effects rather than a general one sees re-entrant or low-concentration-plateau behav- phenomenology. A few specific points seem most worth iors of uncertain interpretation. From their rarity, these repeating: Probe and probe-free HPC solution spectra behaviors are reasonably interpreted as corresponding to are both bimodal or trimodal, with modes on the same specific chemical effects. True microrheological studies time scales. However, the probe and polymer modes are examining the motion of mesoscopic particles under the not the same: The concentration dependences of a probe influence of a known external force are also examined. mode and the corresponding polymer mode can be op- The viscosity determined with a true microrheological posite in sign. measurement is in many cases substantially smaller than A wide range of phenomena indicate that these so- the viscosity measured with a macroscopic instrument.
∗ To whom inquiries should be sent [email protected] 1 D. N. Turner and F. R. Hallett. A study of the diffusion 48
of compact particles in polymer solutions using quasi- (2000), 5932-5937. elastic light scattering. Biochimica et Biophysica Acta, 23 X. Cao, R. Bansil, D. Gantz, E. W. Moore, N. Niu and N. 451 (1976), 305-312. H. Afdhal. Diffusion behavior of lipid vesicles in entangled 2 D. A. Weitz and D. W. Pine. Diffusing-wave spectroscopy. polymer solutions. Biophys. J., 73 (1997), 1932-1939. In Dynamic Light Scattering, ed. W. Brown, (Oxford, UK: 24 Y. Cheng, R. K. Prud’homme and J. L. Thomas. Diffu- Oxford University Press, 1993), pp. 652-720. sion of mesoscopic probes in aqueous polymer solutions 3 M. J. Saxton and K. Jacobson. Single particle tracking: measured by fluorescence recovery after photobleaching. applications to membrane dynamics. Annu. Rev. Biophys. Macromolecules, 35 (2002), 8111-8121. Biomol. Struct., 26 (1997), 373-399. 25 S. C. De Smedt, A. Lauwers, J. Demeester, Y. Engel- 4 G. D. J. Phillies, G. S. Ullmann, K. Ullmann and T.- borghs, G. De Mey and M. Du. Structural information H. Lin. Phenomenological scaling laws for ’semidilute’ on hyaluronic acid solutions as studied by probe diffusion macromolecule solutions light scattering by optical probe experiments. Macromolecules, 27 (1994), 141-146. particles. J. Chem. Phys., 82 (1985), 5242-5246. 26 S. C. De Smedt, P. Dekeyser, V. Ribitsch, A. Lauwers and 5 G. D. J. Phillies and K. A. Streletzky. Microrheology of J. Demeester. Viscoelastic and transient network proper- complex fluids via observation of tracer microparticles. ties of hyaluronic acid as a function of the concentration. Recent Res. Devel. Phys. Chem., 5 (2001), 269-285. Biorheology, 30 (1994), 31-42. 6 J. L. Harden and V. Viasnoff. Recent advances in DWS- 27 S. C. De Smedt, T. K. L. Meyvis, J. Demeester, P. Van based micro-rheology. Curr. Op. Coll. Interf, Sci., 6 Oostveldt, J. C. G. Blonk and W. E. Hennink. Diffu- (2001), 438-445. sion of macromolecules in dextran methacrylate solutions 7 M. J. Solomon and Q. Lu. Rheology and dynamics of par- and gels as studied by confocal scanning laser microscopy. ticles in viscoelastic media. Curr. Op. Coll. Interf. Sci., 6 Macromolecules, 30 (1997), 4863-4870. (2001), 430-437. 28 I. Delfino, C. Piccolo and M. Lepore. Experimental study 8 F. C. MacIntosh and C. F. Schmidt, Microrheology. Curr. of short- and long-time diffusion regimes of spherical par- Op. Coll. Interf. Sci., 4 (1999), 300-307. ticles in carboxymethylcellulose solutions. Eur. Polym. J., 9 A. Mukhopaday and S. Granick. Micro- and nanorheology. 41 (2005), 1772-1780 Curr. Op. Coll. Interf. Sci., 6 (2001), 423-429. 29 D. E. Dunstan and J. Stokes. Diffusing probe measure- 10 Y. Tseng, T. P. Kole, S.-H. J. Lee and D. Wirtz. Local ments in polystyrene latex particles in polyelectrolyte so- dynamics and viscoelastic properties of cell biological sys- lutions: deviations from stokes-einstein behavior. Macro- tems. Curr. Op. Coll. Interf. Sci., 8 (2002), 210-217. molecules, 33 (2000), 193-198. 11 P. Habdas and E. R. Weeks. Video microscopy of colloidal 30 R. Furukawa, J. L. Arauz-Lara and B. R. Ware. Self- suspensions and colloidal crystals. Curr. Op. Coll. Interf. diffusion and probe diffusion in dilute and semidilute so- Sci., 7 (2002), 196-203. lutions of dextran. Macromolecules, 24 (1991), 599-605. 12 George D. J. Phillies. Self and tracer diffusion of polymers 31 D. Gold, C. Onyenemezu and W. G. Miller. Effect of in solution, arXiv:cond-mat/0403109 (3 March 2004). solvent quality on the diffusion of polystyrene latex 13 B. J. Berne and R. Pecora. Dynamic Light Scattering spheres in solutions of poly(methylmethacrylate). Macro- (New York, NY: Wiley, 1976), especially Chapter 5. molecules, 29 (1996), 5700-5709. 14 J. L. Doob. The brownian movement and stochastic equa- 32 S. Gorti and B. R. Ware. Probe diffusion in an aqueous tions. Annals Math., 43 (1942), 351-369. polyelectrolyte solution. J. Chem. Phys., 83 (1985), 6449- 15 G. D. J. Phillies. Interpretation of light scattering spectra 6456 . in terms of particle displacements, J. Chem. Phys., 122 33 S. C. Lin, W. I. Lee and J. M. Schurr. Biopolymers, 17 (2005), 224905 1-8. (1978), 1041. 16 K. E. Bremmell, N. Wissenden and D. E. Dunstan. Dif- 34 A. M. Jamieson, J. G. Southwick and J. Blackwell. Dy- fusing probe measurements in newtonian and elastic so- namical behavior of xanthan polysaccharide in solution. J. lutions. Adv. Coll. Interf. Sci., 89-90 (2001), 141-154. Polymer Sci.: Polymer Phys. Ed., 20 (1982), 1513-1524. 17 K. E. Bremmell and D. E. Dunstan. Probe diffusion mea- 35 L. Johansson, U. Skantze and J.-E. Loefroth. Diffusion surements of polystyrene latex particles in polyelectrolyte and interaction in gels and solutions. 2. Experimental re- solutions of varying ionic strength. Macromolecules, 35 sults on the obstruction effect. Macromolecules, 24 (1991), (2002), 1994-1999. 6019-6023. 18 W. Brown and R. Rymden. Interaction of carboxymethyl- 36 C. Konak, B. Sedlacek and Z. Tuzar. Diffusion of block cellulose with latex spheres studied by dynamic light scat- copolymer micelles in solutions of a linear polymer. tering. Macromolecules, 20 (1987), 2867-2873. Makromol. Chem., Rapid Commun., 3 (1982), 91-94. 19 W. Brown and R. Rymden. Comparison of the transla- 37 T.-H. Lin and G. D. J. Phillies. Translational diffusion of tional diffusion of large spheres and high molecular weight a macroparticulate probe species in salt-free poly(acrylic) coils in polymer solutions. Macromolecules, 21 (1988), acid:water. J. Phys. Chem., 86 (1982), 4073-4077. 840-846. 38 T.-H. Lin and G. D. J. Phillies. Probe diffusion in 20 Z. Bu and P. S. Russo. Diffusion of dextran in aqueous poly(acrylic acid): water. Effect of probe size. Macro- hydroxypropylcellulose. Macromolecules, 27 (1994), 1187- molecules, 17 (1984), 1686-1691. 1194. 39 T.-H. Lin and G. D. J. Phillies. Probe diffusion in poly- 21 D. Langevin and F. Rondelez. Sedimentation of large acrylic acid:water — effect of polymer molecular weight. colloidal particles through semidilute polymer solutions. J. Coll. Interf. Sci., 100 (1984), 82-95. 40 Polymer, 19 (1978), 875-882. T.-H. Lin. Diffusion of TiO2 particles through a 22 N. A. Busch, T. Kim and V. A. Bloomfield. Tracer dif- poly(ethylene oxide) melt. Makromol. Chem., 187 (1986), fusion of proteins in DNA solutions. 2. Green fluorescent 1189-1196. protein in crowded DNA solutions. Macromolecules, 33 41 S. Mangenot, S. Keller and J. Raedler. Transport of nu- 49
cleosome core particles in semidilute DNA solutions. Bio- bovine serum albumin:water. J. Coll. Interf. Sci., 105 phys. J., 85 (2003), 1817-1825. (1985), 315-324. 42 O. A. Nehme, P. Johnson and A. M. Donald. Probe diffu- 59 K. M. Keller, E. R. Canales and S. I. Yum. Tracer and sion in poly-l-lysine solution. Macromolecules, 22 (1989), mutual diffusion coefficients of proteins. J. Phys. Chem., 4326-4333. 75 (1971), 379-387. 43 C. N. Onyenemezu, D. Gold, M. Roman, and W. G. 60 R. G. Kitchen, N. N. Preston and J. D. Wells, J. Polym. Miller. Diffusion of polystyrene latex spheres in linear Sci., 55 (1976), 39. polystyrene nonaqueous solutions. Macromolecules, 26 61 M. R. Wattenbarger, V. A. Bloomfield, B. Zu and P. (1993), 3833-3837. S. Russo. Tracer diffusion of proteins in DNA solutions. 44 G. D. J. Phillies. Diffusion of bovine serum albumin in Macromolecules, 25 (1992), 5263-5265. a neutral polymer solution. Biopolymers, 24 (1985), 379- 62 J. Won, C. Onyenemezu, W. G. Miller and T. P. Lodge. 386. Diffusion of spheres in entangled polymer solutions: a 45 G. D. J. Phillies, J. Gong, L. Li, A. Rau, K. Zhang, L.- return to Stokes-Einstein behavior. Macromolecules, 27 P. Yu and J. Rollings. Macroparticle diffusion in dextran (1994), 7389-7396. solutions. J. Phys. Chem., 93 (1989), 6219-6223. 63 T. P. Lodge and L. M. Wheeler. Translational diffusion 46 G. D. J. Phillies, T. Pirnat, M. Kiss, N. Teasdale, D. of linear and 3-arm-star polystyrenes in semidilute so- Maclung, H. Inglefield, C. Malone, L.-P. Yu and J. lutions of linear poly(vinylmethylether). Macromolecules, Rollings. Probe diffusion in solutions of low-molecular- 19 (1986), 2983-2986. weight polyelectrolytes. Macromolecules, 22 (1989), 4068- 64 T. P. Lodge and P. Markland. Translational diffusion 4075. of 12-arm star polystyrenes in dilute and concentrated 47 G. D. J. Phillies, C. Malone, K. Ullmann, G. S. Ullmann, poly(vinyl methyl ether) solutions. Polymer, 28 (1987), J. Rollings and L.-P. Yu. Probe diffusion in solutions of 1377-1384. long-chain polyelectrolytes. Macromolecules, 20 (1987), 65 T. P. Lodge, P. Markland and L. M. Wheeler. Tracer dif- 2280-2289. fusion of 3-arm and 12-arm star polystyrenes in dilute, 48 G. D. J. Phillies, M. Lacroix and J. Yambert, Probe diffu- semidilute, and concentrated poly(vinylmethylether) so- sion in sodium polystyrene sulfonate - water: experimen- lutions. Macromolecules, 22 (1989), 3409-3418. tal determination of sphere-chain binary hydrodynamic 66 L. M. Wheeler, T. P. Lodge, B. Hanley and M. Tir- interactions. Journal of Physical Chemistry, 101 (1997), rell. Translational diffusion of linear polystyrenes in di- 5124-5130. lute and semidilute solutions of poly(vinylmethylether). 49 G. D. J. Phillies and P. C. Kirkitelos, Higher-order hydro- Macromolecules, 20 (1987), 1120-1129. dynamic interactions in the calculation of polymer trans- 67 L. M. Wheeler and T. P. Lodge. Tracer diffusion of port properties, Journal of Polymer Science B: Polymer linear polystyrenes in dilute, semidilute, and concen- Physics, 31 (1993), 1785-1797. trated poly(vinylmethylether) solutions. Macromolecules, 50 G. D. J. Phillies, A. Saleh, L. Li, Y. Xu, D. Rostcheck, M. 22 (1989), 3399-3408. Cobb and T. Tanaka. Temperature dependence of probe 68 J. Won and T. P. Lodge. Tracer diffusion of star-branched diffusion in solutions of low-molecular-weight polyelec- polystyrenes in poly(vinylmethylether) gels. J. Polym. trolytes. Macromolecules, 24 (1991), 5299-5304. Sci. Polym. Phys. Ed., 31 (1993), 1897-1907. 51 G. D. J. Phillies, D. Rostcheck and S. Ahmed, Probe dif- 69 Zhou Pu and W. Brown. Translational diffusion of large fusion in intermediate-molecular-weight polyelectrolytes: silica spheres in semidilute polymer solutions. Macro- temperature dependence. Macromolecules, 25 (1992), molecules, 22 (1989), 890-896. 3689-3694. 70 A. C. Fernandez and G. D. J. Phillies. Temperature de- 52 G. D. J. Phillies and C. A. Quinlan. Glass temperature ef- pendence of the diffusion coefficient of polystyrene latex fects effects in probe diffusion in dextran solutions. Macro- spheres. Biopolymers, 22 (1983), 593-595. molecules, 25 (1992), 3110-3316. 71 G. D. J. Phillies. Translational diffusion coefficient of 53 C. Roberts, T. Cosgrove, R. G. Schmidt and G. V. Gor- macroparticles in solvents of high viscosity. J. Phys. don. Diffusion of poly(dimethylsiloxane) mixtures with Chem., 85 (1981), 2838-2843. silicate nanoparticles. Macromolecules, 34 (2001), 538- 72 P. Wiltzius and W. van Saarloos. Absence of increase in 543. length scale upon approaching the glass transition in liq- 54 M. Shibayama, Y. Isaka and I. Shiwa. Dynamics of probe uid glycerol. J. Chem. Phys., 94 (1991), 5061-5063. particles in polymer solutions and gels. Macromolecules, 73 Cited by Ref. 72 as Yu. F. Kiyachenko and Yu. I. Litvinov, 32 (1999), 7086-7092. JETP Lett., 42 (1985), 257. 55 Y. Shiwa. Hydrodynamic Screening and Diffusion in En- 74 G. D. J. Phillies and D. Clomenil. Lineshape and tangled Polymer Solutions. Phys. Rev. Lett., 58 (1987), linewidth effects in optical probe studies of glass-forming 2102-2105. liquids. J. Phys. Chem., 96 (1992), 4196-4200. 56 G. S. Ullmann and G. D. J. Phillies. Implications of the 75 G. D. J. Phillies and C. A. Quinlan. Analytic structure failure of the Stokes-Einstein relation for measurements of the solutionlike-meltlike transition in polymer solution with QELSS of polymer adsorption by small particles. dynamics. Macromolecules, 28 (1995), 160-164. Macromolecules, 16 (1983), 1947-1949. 76 G. D. J. Phillies. Range of validity of the hydrodynamic 57 G. S. Ullmann, K. Ullmann, R. M. Lindner and G. D. scaling model. J. Phys. Chem., 96 (1992), 10061-10066. J. Phillies. Probe diffusion of polystyrene latex spheres 77 G. D. J. Phillies. Quantitative prediction of α in the in poly(ethylene oxide):water. J. Phys. Chem. 89 (1985), scaling law for self-diffusion. Macromolecules, 21 (1988), 692-700. 3101-3106. 58 K. Ullmann, G. S. Ullmann and G. D. J. Phillies. Optical 78 W. Brown and R. Rymden. Diffusion of polystyrene la- probe study of a nonentangling macromolecule solution— tex spheres in polymer solutions studied by dynamic light 50
scattering. Macromolecules, 19 (1986), 2942-2952. 9903-9913. 79 T. Yang and A. M. Jamieson. Diffusion of latex spheres 97 R. O’Connell, H. Hanson, and G. D. J. Phillies. Neu- through solutions of hydroxypropylcellose in water. J. tral Polymer Slow Mode and Its Rheological Correlate. Coll. Interf. Sci., 126 (1988), 220-230. J. Polym. Sci. B. Polym. Phys., 43 (2005), 323-333. 80 P. S. Russo, M. Mustafa, T. Cao and L. K. Stephens. In- 98 S. A. Kivelson, X. Zhao, D. Kivelson, T. M. Fischer and teractions between polystyrene latex spheres and a semi- C. M. Knobler. Frustration-limited clusters in liquids. J. flexible polymer, hydroxypropylcellulose. J. Coll. Interf. Chem. Phys., 101 (1994), 2391-2397. Sci., 122 (1988), 120-137. 99 B. Camins and P.S. Russo. Following polymer gelation by 81 M. Mustafa and P. S. Russo. Nature and effects of nonex- depolarized dynamic light scattering from optically and ponential correlation functions in probe diffusion experi- geometrically anisotropic latex particles. Langmuir, 10 ments by quasielastic light scattering. J. Coll. Interf. Sci., (1994), 4053-4059. 129 (1989), 240-253. 100 Z. Cheng and T. G. Mason. Rotational diffusion microrhe- 82 G. D. J. Phillies and D. Clomenil. Probe diffusion in ology. Phys. Rev. Lett., 90 (2003), 018304 1-4. polymer solutions under θ and good conditions. Macro- 101 R. Cush, D. Dorman and P. S. Russo. Rotational and molecules, 26 (1993), 167-170. translational diffusion of tobacco mosaic virus in ex- 83 A. R. Altenberger, M. Tirrell, and J. S. Dahler, J. Chem. tended and globular polymer solutions. Macromolecules, Phys., 84 (1986), 5122. 37 (2004), 9577-9584. 84 G. D. J. Phillies. Dynamics of polymers in concentrated 102 R. Cush, P. S. Russo, Z. Kucukyavuz, Z. Bu, D. Neau, solution: the universal scaling equation derived. Macro- D. Shih, S. Kucukyavuz and H. Ricks. Rotational and molecules, 20 (1987), 558-564. translational diffusion of a rodlike virus in random coil 85 G. D. J. Phillies and P. Peczak. The ubiquity of stretched- polymer solutions. Macromolecules, 30 (1997), 4920-4926. exponential forms in polymer dynamics. Macromolecules, 103 D. A. Hill and D. S. Soane. Measurement of rotational 21 (1988), 214-220. diffusivity of rodlike molecules in amorphous polymer ma- 86 G. D. J. Phillies, C. Richardson, C. A. Quinlan and S. Z. trices by the dynamic kerr effect. J. Polymer Science B, Ren. Transport in intermediate and high molecular weight 27 (1989), 2295-2320. hydroxypropylcellulose/water solutions. Macromolecules, 104 T. Jamil and P. A. Russo. Interactions between 26 (1993), 6849-6858. colloidal poly(tetrafluoroethylene) latex and sodium 87 M. B. Mustafa, D. L. Tipton, M. D. Barkley, P. S. poly(styrenesulfonate). Langmuir, 14 (1998), 264-270. Russo. and F. D. Blum. Dye diffusion in isotropic and liq- 105 G. H Koenderink, S. Sacanna, D. G. A. L. Aarts and A. uid crystalline aqueous (hydroxypropyl)cellulose. Macro- P. Philipse. Rotational and translational diffusion of flu- molecules, 26 (1993), 370-378. orocarbon tracer spheres in semidilute xanthan solutions. 88 K. L. Ngai and G. D. J. Phillies. Coupling model analy- Phys. Rev. E, 69 (2004), 021804 1-12. sis of polymer dynamics in solution: probe diffusion and 106 L. Le Goff, O. Hallatschek, E. Frey and F. Amblard. viscosity. J. Chem. Phys., 105 (1996), 8385-8397. Tracer studies on F-actin fluctuations. Phys. Rev. Lett., 89 K. L. Ngai. In Disorder Effects In Relaxation Processes ed. 89 (2002), 258101 1-4. R. Richert and A. Blumen, (Berlin, Germany: Springer- 107 J. G. Phalakornkul, A. P. Gast and R. Pecora. Rotational Verlag (1994)). dynamics of rodlike polymers in a rod/sphere mixture. J. 90 G. D. J. Phillies and M. Lacroix. Probe diffusion in Chem. Phys., 112 (2000), 6487-6494. hydroxypropylcellulose–water: Radius and line-shape ef- 108 D. Sohn, P. S. Russo, A. Davila, D. S. Poche and M. L. fects in the solutionslike regime. J. Phys. Chem. B, 101 McLaughlin. Light scattering study of magnetic latex par- (1997), 39-47. ticles and their interaction with polyelectrolytes. J. Coll. 91 K. A. Streletzky and G. D. J. Phillies. Translational Interf. Sci., 177 (1996), 31-44. diffusion of small and large mesoscopic probes in 109 J. Apgar, Y. Tseng, E. Federov, M. B. Herwig, S. C. Almo hydroxypropylcellulose-water in the solutionlike regime. and D. Wirtz. Multiple-particle tracking measurements of J. Chem. Phys., 108 (1998), 2975-2988. heterogeneities in solutions of actin filaments and actin 92 K. A. Streletzky and G. D. J. Phillies. Relaxational bundles. Biophys. J., 79 (2000), 1095-1106. mode structure for optical probe diffusion in high molec- 110 D. T. Chen, E. R. Weeks, J. C. Crocker, M. F. Islam, R. ular weight hydroxypropylcellulose. J. Polym. Sci. B, 36 Verna, J. Gruber, A. J. Levine, T. C. Lubensky and A. (1998), 3087-3100. G. Yodh. Rheological microscopy: local mechanical prop- 93 K. A. Streletzky and G. D. J. Phillies. Confirmation of the erties from microrheology. Phys. Rev. Lett., 90 (2003), reality of the viscoelastic solutionlike-meltlike transition 108301 1-4. via optical probe diffusion. Macromolecules, 32 (1999), 111 A. J. Levine and T. C. Lubensky. Two-point microrheol- 145-152. ogy and the electrostatic analogy. Phys. Rev. E, 65 (2001), 94 K. A. Streletzky and G. D. J. Phillies. Coupling analysis 011501 1-13. of probe diffusion in high molecular weight hydroxypropy- 112 J. C. Crocker, M. T. Valentine, E. R. Weeks, T. Gisler, P. lcellulose. J. Phys. Chem. B, 103 (1999), 1811-1820. D. Kaplan, A. G. Yodh and D. A. Weitz. Two-point mi- 95 K. Streletzky and G. D. J. Phillies. Optical probe study crorheology of inhomogeneous soft materials. Phys. Rev. of solution-like and melt-like solutions of high molecular Lett., 85 (2000), 888-891. weight hydroxypropylcellulose. in Scattering from Poly- 113 M. A. Dichtl and E. Sackmann. Colloidal probe study mers, ed. B. S. Hsiao, (Washington, D.C.: Am. Chem. of short time local and long time reptational motion of Soc. Symp. Ser. 2000) Vol. 739, 297-316. semiflexible macromolecules in entangled networks. New 96 G. D. J. Phillies, R. O’Connell, P. Whitford and K. A. J. Physics, 1 (1999), 18.1-18.11. Streletzky. Mode structure of diffusive transport in hy- 114 M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. droxypropylcellulose:water. J. Chem. Phys., 119 (2003), Bausch and D. A. Weitz. Microrheology of entangled F- 51
actin solutions. Phys. Rev. Lett., 91 (2003), 158302 1-4. Viscoelastic properties of semiflexible filamentous bacte- 115 F. Gittes, B. Schnurr, P. D. Olmstead, F. C. MacKin- riophage fd. Phys. Rev. E, 62 (2000), 5509-5517. tosh and C. F. Schmidt. Microscopic viscoelasticity: shear 133 D. Morse. Viscoelasticity of concentrated isotropic solu- moduli of soft materials determined from thermal fluctu- tions of semiflexible polymers. 2. Linear response. Macro- ations. Phys. Rev. Lett., 79 (1997), 3286-3289. molecules, 31 (1998), 7044-7067. 116 A. Goodman, Y. Tseng and D. Wirtz. Effect of length, 134 C. Allain, M. Drifford and B. Gauthier-Manuel. Diffusion topology, and concentration on the microviscosity and of calibrated particles during the formation of a gel. Poly- microheterogeneity of DNA solutions. J. Mol. Bio., 323 mer Communications, 27 (1986), 177-180. (2002), 199-215. 135 S. Matsukawa and I. Ando. A study of self-diffusion 117 A. W. C. Lau, B. D. Hoffman, A. Davies, J. C. Crocker of molecules in polymer gel by pulsed-field-gradient 1H and T. C. Lubensky. Microrheology, stress fluctuations, NMR. Macromolecules, 29 (1996), 7136-7140. and active behavior of living cells. Phys. Rev. Lett., 91 136 I. Nishio, J. C. Reina and R. Bansil. Quasielastic light (2003), 198101 1-4. scattering study of the movement of particles in gels. 118 T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz Phys. Rev. Lett., 59 (1987), 684-687. and S. C. Kuo. Particle tracking microrheology of complex 137 I. H. Park, C. S. Johnson, Jr., and D. A. Gabriel. Probe fluids. Phys. Rev. Lett., 79 (1997), 3282-3285. diffusion in polyacrylamide gels as observed by means of 119 A. Papagiannopolis, C. M. Ferneyhough and T. A. Waigh. holographic relaxation methods: search for a universal The microrheology of polystyrene sulfonate combs in equation. Macromolecules, 23 (1990), 1548-1553. aqueous solution. J. Chem. Phys., 123 (2005), 214904 1- 138 J. C. Reina, R. Bansil, and C. Konak. Dynamics of 10. probe particles in polymer solutions and gels. Polymer, 120 B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. 31 (1990), 1038-1044. Schmidt. Determining microscopic viscoelasticity in flexi- 139 Y. Suzuki and I. Nishio. Quasielastic-light-scattering ble and semiflexible polymer networks from thermal fluc- study of the movement of particles in gels: Topological tuations. Macromolecules, 30 (1997), 7781-7792. structure of pores in gels. Phys. Rev. B, 45 (1992), 4614- 121 Y. Tseng and D. Wirtz. Mechanics and multiple- 4619. particle tracking microheterogeneity of α-actinin-cross- 140 G. C. Fadda, D. Lairez and J. Pelta. Critical behavior of linked actin filament networks. Biophys. J., 81 (2001), gelation probed by the dynamics of latex spheres. Phys. 1643-1656. Rev. E, 63 (2001), 061405 1-9. 122 M. T. Valentine, Z. E. Perlman, M. L. Gardel, J. H. Shin, 141 L. Hou, F. Lanni and K. Luby-Phelps. Tracer diffusion P. Matsudaira, T. J. Mitchison and D. A. Weitz. Colloid in F-actin and ficoll mixtures. Toward a model for cyto- surface chemistry critically affects multiple particle track- plasm. Biophys. J., 58 (1990), 31-43. ing measurements of biomaterials. Biophys. J., 86 (2004), 142 K. Luby-Phelps, D. L. Taylor and F. Lanni. Probing the 4004-4014. structure of cytoplasm. J. Cell. Biol., 102 (1986), 2015- 123 J. Xu, A. Palmer and D. Wirtz. Rheology and microrheol- 2022. ogy of semiflexible polymer solutions: actin filament net- 143 K. Luby-Phelps, P. E. Castle, D. L. Taylor and F. Lanni. works. Macromolecules, 31 (1998), 6486-6492. Hindered diffusion of inert tracer particles in the cyto- 124 J. Xu, V. Viasnoff and D. Wirtz. Compliance of actin net- plasm of mouse 3T3 cells. Proc. Natl. Acad. Sci., 84 work filaments measured by particle-tracking microrhe- (1987), 4910-4913. ology and diffusing wave spectroscopy. Rheol. Acta, 37 144 F. Madonia, P. L. San Biagio, M. U. Palma, G. Schiliro’, (1998), 387-398. S. Musumeci and G. Russo. Photon scattering as a probe 125 J. Xu, Y. Tseng, C. J. Carriere and D. Wirtz. Micro- of microviscosity and channel size in gels such as sickle heterogeneity and microrheology of wheat gliadin sus- haemoglobin. Nature, 302 (1983), 412-415. pensions studied by multiple-particle tracking. Biomacro- 145 J. Newman, N. Mroczka and K. L. Schick. Dynamic light molecules, 3 (2002), 92-99. scattering measurements of the diffusion of probes in fila- 126 F. Amblard, A. C. Maggs, B. Yurke, A. N. Pargellis and S. mentatious actin solutions. Biopolymers, 28 (1989), 655- Leibler. Subdiffusion and anomalous local viscoelasticity 666. in actin networks. Phys. Rev. Lett., 77 (1996), 4470-4473. 146 J. Newman, G. Gukelberger, K. L. Schick and K. S. Zaner. 127 A. I. Bishop, T. A. Nieminen, N. R. Heckenberg and H. Probe diffusion in solutions of filamentatious actin formed Rubinsztein-Dunlop. Optical microrheology using rotat- in the presence of gelsolin. Biopolymers, 31 (1991), 1265- ing laser-trapped particles. Phys. Rev. Lett., 92 (2004), 1271. 198104 1-4. 147 C. F. Schmidt, M. Baermann, G. Isenberg, and E. Sack- 128 L. A. Hough and H. D. Ou-Yang. A new probe for me- mann. Chain dynamics, mesh size, and diffusive transport chanical testing of nanostructures in soft materials. J. in networks of polymerized actin. A quasielastic light scat- Nanoparticle Research 1 (1999), 495-499. tering and microfluorescence study. Macromolecules, 22 129 M. Keller, J. Schilling and E. Sackmann. Oscillatory mag- (1989), 3638-3649. netic bead rheometer for complex fluid microrheometry. 148 U. A. Stewart, M. S. Bradley, C. S. Johnson, Jr., and D. A. Rev. Sci. Instr., 72 (2001), 3626-3634 . Gabriel. Transport of probe molecules through fibrin gels 130 F. G. Schmidt, B. Hinner, and E. Sackmann. Microrheom- as observed by means of holographic relaxation methods. etry underestimates the values of the viscoelastic mod- Biopolymers, 27 (1988), 173-185. uli in measurements on F-actin solutions compared to 149 I. M. Wong, M. L. Gardel, D. R. Reichman, E. R. Weeks, macrorheometry. Phys. Rev. E, 61 (2000), 5646-5653. M. T. Valentine, A. R. Bausch and D. A. Weitz. Anoma- 131 A. C. Maggs, Micro-bead mechanics with actin filaments. lous diffusion probes microstructure dynamics of entan- Phys. Rev. E, 57 (1998), 2091-2094. gled F-actin networks. Phys. Rev. Lett., 92 (2004), 178101 132 F. G. Schmidt, B. Hinner, E. Sackmann and J. X. Tang. 1-4. 52
150 M. Arrio-Dupont, G. Foucault, M. Vacher, P. F. Devaux (1998), 97-106. and S. Cribier. Translational diffusion of globular proteins 168 A. Palmer, T. G. Mason, J. Xu, S. C. Kuo and D. Wirtz. in the cytoplasm of cultured muscle cells. Biophys. J., 78 Diffusing wave spectroscopy microrheology of actin fila- (2000), 901-907. ment networks. Biophys. J., 76 (1999), 1063-1071. 151 M. Arrio-Dupont, S. Cribier, J. Foucault, P. F. De- 169 D. J. Pine, D. A. Weitz, P. M. Chaikin and E. Her- vaux and A. D’Albis. Diffusion of fluorescently labelled bolzheimer. Diffusing-wave spectroscopy. Phys. Rev. Lett., macrolecules in cultured muscle cells. Biophys. J., 70 60 (1988), 1134-1137. (1996), 2327-2332. 170 G. Maret and P. E. Wolf. Multiple light-scattering from 152 H. P. Kao, J. R. Abney and A. S. Verkman. Determi- disordered media. The effect of brownian motion of scat- nants of the translational mobility of a small solute in cell terers. Z. Phys. B, 65 (1987), 409-413. cytoplasm. J. Cell Biol., 120 (1993), 175-184. 171 G. Popescu, A. Dogariu and R. Rajagopalan. Spatially 153 K. Luby-Phelps, S. Mujumdar, R. B. Mujumdar, L. A. resolved microrheology using localized coherence volumes. Ernst, W. Galbraith and A. S. Waggoner. A novel fluo- Phys. Rev. E, 65 (2002), 041504 1-8. rescence ratiometric method confirms the low solvent vis- 172 G. Popescu and A. Dogariu. Dynamic light scattering in cosity of the cytoplasm. Biophys. J., 65 (1993), 236-242. localized coherence volumes. Optics Letters, 26 (2001), 154 O. Seksek, J. Biwersi, and A. S. Verkman. Translational 551-553. diffusion of macromolecule-sized solutes in cytoplasm and 173 I. S. Sohn, R. Rajagopalan and A. C. Dogariu. Spatially nucleus. J. Cell Biology, 138 (1997), 131-142. resolved microrheology through a liquid/liquid interface. 155 V. Shenoy and J. Rosenblatt. Diffusion of macromolecules J. Coll. Interf. Sci., 269 (2004), 503-513. in collagen and hyaluronic acid, rigid-rod–flexible poly- 174 L. F. Rojas-Ochoa, S. Romer, F. Scheffold and P. mer, composite matrices. Macromolecules, 28 (1995), Schurtenberger. Diffusing wave spectroscopy and small- 8751-8758. angle neutron scattering from concentrated colloidal sus- 156 M. Bellour, M. Skouri, J.-P. Munch and P. Hebraud. pensions. Phys. Rev. E, 65 (2002), 051403 1-8. Brownian motion of particles embedded in a solution of 175 S. Romer, F. Scheffold and P. Schurtenberger. Sol-gel giant micelles. Eur. Phys. J. E, 8 (2002), 431-436. transition of concentrated colloidal suspensions. Phys. 157 B. R. Dasgupta, S.-Y. Tee, J. C. Crocker, B. J. Frisken Rev. Lett., 85 (2000), 4980-4983. and D. A. Weitz. Microrheology of polyethylene oxide 176 S. Romer, C. Urban, H. Bissig, A. Stradner, F. Schef- using diffusing wave spectroscopy and single scattering. fold and P. Schurtenberger. Dynamics of concentrated col- Phys. Rev. E, 65 (2002), 051505 1-10. loidal suspensions: diffusion, aggregation, and gelation. 158 T. Gisler and D. A. Weitz. Scaling of the microrheology of Phil. Trans. R. Soc. London A, 359 (2001), 977-984. semidilute F-actin solutions. Phys. Rev. Lett., 82 (1999), 177 K. Rufener, A. Palmer, J. Xu and D. Wirtz. High- 1606-1609. frequency dynamics and microrheology of macromolecular 159 C. Heinemann, F. Cardinaux, F. Scheffold, P. Shurten- solutions probed by diffusing wave spectroscopy: the case berger, F. Escher and B. Conde-Petit. Tracer Microrheol- of concentrated solutions of F-actin. J. Non-Newtonian ogy of γ-dodecalactone induced gelation of aqueous starch Fluid Mech., 82 (1999), 303-314. dispersions. Carbohydrate Polymers, 55 (2004), 155-161. 178 J. van der Gucht, N. A. M. Besseling, W. Knoben, L. 160 M. H. Kao, A. G. Yodh and D. J. Pine. Observation of Bouteiller and M. A. Cohen Stuart. Brownian particles brownian motion on the time scale of hydrodynamic in- in supramolecular polymer solutions. Phys. Rev. E, 67 teractions. Phys. Rev. Lett., 70 (1993), 242-245. (2003), 051106 1-10. 161 P. D. Kaplan, A. G. Yodh and D. F. Townsend. Nonin- 179 J. H. van Zanten, S. Amin and A. A. Abdala. Brown- vasive study of gel formation in polymer-stabilized dense ian motion of colloidal spheres in aqueous PEO solutions. colloids using multiply scattered light. J. Coll. Interf. Sci., Macromolecules, 37 (2004), 3874-3880. 155 (1993), 319-324. 180 P. G. deGennes, P. Pincus, and R. M. Velasco. Cited by 162 A. Knaebel, R. Skouri, J. P. Munch and S. J. Candau. Ref. 21 as ’Personal Communication’. (1978) Structural and rheological properties of hydrophobically 181 P. G. deGennes. Scaling Concepts in Polymer Physics. modified alkali-soluble emulsion solutions. J. Polymer Sci. Cornell University Press, Ithaca (1979). B, 40 (2002), 1985-1994. 182 G. D. J. Phillies, W. Brown and P. Zhou. Chain and 163 Q. Lu and M. J. Solomon. Probe size effects on the mi- sphere diffusion in polyisobutylene-CHCl3: a reanalysis. crorheology of associating polymer solutions. Phys. Rev. Macromolecules, 25 (1992), 4948-4954. E, 66 (2002), 061504 1-11. 164 T. G. Mason and D. A. Weitz. Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids. Phys. Rev. Lett., 74 (1995), 1250-1253. 165 T. Narita, A. Knaebel, J.-P. Munch and S. J. Candau. Mi- crorheology of poly(vinyl alcohol) aqueous solutions and chemically cross-linked gels. Macromolecules, 34 (2001), 8224-8231. 166 G. Nisato, P. Hebraud, J.-P. Munch and S. J. Candau. Diffusing-wave-spectroscopy investigation of latex particle motion in polymer gels. Phys. Rev. E, 61 (2000), 2879- 2887. 167 A. Palmer, J. Xu and D. Wirtz. High-Frequency Vis- coelasticity of crosslinked actin filament networks mea- sured by diffusing wave spectroscopy. Rheol. Acta, 37 53
ν TABLE I: Fits of the concentration dependence of the probe coefficient Dp to a stretched exponential Dp0 exp(−αc ), for probes having radius R diffusing through polymers having molecular weight M. The Table gives the fitting parameters, −7 2 polymer molecular weight M, probe radius R, the polymer:solvent system, and the reference. Dp0 is in units 10 cm /s or in units of Dp(0). Concentrations are converted to g/L except g/g marks g/g as the concentration units. Polymers include BSA–bovine serum albumin, CMC–carboxymethylcellulose, dextran, ficoll, HPC–hydroxypropylcellulose, HPSS–nonneutralized polystyrene sulfonate, PAA–polyacrylic acid, PEO–polyethylene oxide, PLL–poly-L-lysine, PMMA–polymethylmethacrylate, NaPSS–sodium polystyrene sulphonate, PVME–polyvinylmethylether. Solvents include phosphate–aqueous phosphate buffer: I = nM – background ionic strength was n molar. In the Dp0 column, * indicates Dp/Dp0, f = fast mode, s = slow mode.
Dp0 α ν M(kDa) R(nm) System Refs 16 2.35 0.53 0.36 3100 100 polyacrylamide:H2O 19 1.061 0.0682 0.834 110 160 PMMA: CHCl3 19 1.082 0.1268 0.764 163 160 PMMA: CHCl3 19 1.359 0.294 0.618 268 160 PMMA: CHCl3 19 1.20 0.308 0.698 445 160 PMMA: CHCl3 −3 20 42.9 9.43 × 10 1 300 0.50 HPC:H2O −3 20 19.53 1.33 × 10 1 300 1.32 HPC:H2O −3 20 13.12 8.89 × 10 1 300 1.69 HPC:H2O −2 20 7.77 1.77 × 10 1 300 2.78 HPC:H2O −2 20 5.02 2.65 × 10 1 300 4.45 HPC:H2O −2 20 3.44 3.44 × 10 1 300 5.78 HPC:H2O −2 20 2.32 4.51 × 10 1 300 8.80 HPC:H2O −2 20 1.49 6.31 × 10 1 300 13.3 HPC:H2O 20 0.897 0.107 1 300 17.9 HPC:H2O 20 0.703 0.251 1 300 55.1 HPC:H2O −3 22 11.89 2.73 × 10 1.64 0.1 1.66 glycerol:H2O 22 12.82 0.0144 0.949 1.66 ficoll 70:H2O 22 14.56 0.222 0.49 160bp 1.66 DNA:H2O −2 23 1.007 4.21 × 10 1 65 64 polyacrylamide:H2O −2 23 0.995 2.27 × 10 1 65 65 polyacrylamide:H2O 23 1.007 0.23 1 1000 64 polyacrylamide:H2O −2 23 0.98 7.8 × 10 1 1000 65 polyacrylamide:H2O 24 6.48 0.246 g/g 1.00 2000 3.15 PEO:H2O 24 4.26 0.59 g/g 1.00 2000 5.72 PEO:H2O 24 7.72 0.388 g/g 0.633 40 3.15 PEO:H2O 24 9.82 0.34 g/g 0.73 40 2.71 PEO:H2O 24 4.19 0.661 g/g 0.83 200 5.47 PEO:H2O 24 4.15 0.428 g/g 0.80 200 5.72 PEO:H2O 101 5.73 0.574 0.73 647 TMV dextran:H2O 101 5.91 0.183 0.93 428 TMV ficoll:H2O 25 4.46 0.178 0.59 680 71kDa dextran: H2O 25 3.18 0.318 0.59 680 148kDa dextran: H2O 25 1.62 0.368 0.53 680 487kDa dextran: H2O 27 4.20 0.0636 0.616 680 19kDa dextran: H2O 27 2.33 0.0726 0.61 680 51kDa dextran: H2O 27 1.67 0.0295 0.796 680 148kDa dextran: H2O 27 1.07 0.076 0.636 680 487kDa dextran: H2O 28 1.416*f 0.247 1 700 102 CMC: H2O 28 0.449*s 1.179 0.775 700 102 CMC: H2O 28 0.844*f 0.227 1 700 47 CMC: H2O 28 0.539*s 0.676 1 700 47 CMC: H2O 28 1.011*f 0.762 1 700 14 CMC: H2O 28 0.358*s 0.150 1 700 14 CMC: H2O 29 0.229 5.04 0.37 3100 100 polyacrylamide: H2O 30 1.02* 0.0119 1.06 150 19 dextran: H2O 30 1.00* 0.027 0.965 40 19 dextran: H2O 1.034 0.0927 0.81 350 200 PMMA: thf 31 1.066 0.0328 1.07 350 200 PMMA: dmf 31 0.964 0.0644 1.05 350 200 PMMA: diox 31 54
Dp0 α ν M(kDa) R(nm) System Refs 0.87 0.209 0.76 500 19 NaPSS: 1 mM phosphate 32 0.998 0.251 0.798 500 19 NaPSS: 5 mM phosphate 32 1.014 0.488 0.624 500 19 NaPSS: 10 mM phosphate 32 0.87 0.195 0.887 500 19 NaPSS: 20 mM phosphate 32 1.028 0.48 0.679 500 19 NaPSS: 50 mM phosphate 32 1.041 0.454 0.37 500 19 HPSS: NaPSS: 20 mM phosphate 32 1.007 0.0367 0.546 70 19 NaPSS: 10 mM phosphate 32 0.985 1.42 × 10−3 1.68 500 0.5 NaPSS: 10 mM phosphate 32 1.08 0.354 0.487 500 5 NaPSS: 10 mM phosphate 32 1.015 0.493 0.614 500 19 NaPSS: 10 mM phosphate 32 105 1.05 0.48 1 4000 193 xanthan: H2O 0.998 0.0369 0.93 110 23 pS:dioxane 36 1.052 0.0858 0.803 200 23 pS:dioxane 36 39 0.85* 0.0212 0.94 50 20 PAA: H2O 39 0.995* 0.028 0.89 50 80 PAA: H2O 39 0.990* 0.0246 0.90 50 620 PAA: H2O 39 1.26* 0.030 0.91 50 1500 PAA: H2O 38 0.685* 0.14 0.75 300 20 PAA: H2O 38 0.796* 0.143 0.73 300 80 PAA: H2O 38 0.823* 0.122 0.76 300 620 PAA: H2O 38 0.99* 0.181 0.63 300 1500 PAA: H2O 39 1.09* 0.354 0.49 1000 20 PAA: H2O 39 1.14* 0.457 0.36 1000 80 PAA: H2O 39 1.12* 0.822 0.71 1000 620 PAA: H2O 42 0.20 0.149 0.73 90 80 PLL: H2O 42 0.328 0.116 1 90 49 PLL: H2O 0.154 0.192 0.78 1110 200 PS: dmf 43 0.154 0.079 0.81 215 200 PS : dmf 43 47 1.28 1.07 0.39 560 20 PAA: H2O: I = 0M 47 1.27 0.554 0.705 560 20 PAA: H2O: I = 0.01M 47 1.19 0.404 0.789 560 20 PAA: H2O: I = 0.02M 47 1.20 0.304 0.921 560 20 PAA: H2O: I = 0.1M 45 - .01 0.99 11.8 20.4 dextran: H2O 45 - 0.0171 0.90 28.5 20.4 dextran: H2O 45 - .0346 0.80 46 20.4 dextran: H2O −2 45 - 5.32×10 0.75 77.2 20.4 dextran: H2O −2 45 - 4.11×10 0.80 83.5 20.4 dextran: H2O −2 45 - 6.43×10 0.73 121 20.4 dextran: H2O 45 - 0.122 0.66 462 20.4 dextran: H2O 45 - 0.315 0.53 542 20.4 dextran: H2O 45 - .0067 1.04 11.8 20.4 dextran: H2O 45 - .0168 0.89 28.5 20.4 dextran: H2O 45 - .0232 0.89 46 20.4 dextran: H2O −2 45 - 5.06×10 0.77 77.2 20.4 dextran: H2O 45 - 0.0667 0.71 83.5 20.4 dextran: H2O 45 - 0.0549 0.77 121 20.4 dextran: H2O 45 - .185 0.60 462 20.4 dextran: H2O o 86 0.577 0.206 0.721 139 67 HPC: H2O 10 C o 86 1.044 0.107 0.90 139 67 HPC: H2O 39 C o 86 0.556 0.315 0.633 146 67 HPC: H2O 10 C 55
Dp0 α ν M(kDa) R(nm) System Refs o 86 1.05 0.174 0.793 146 67 HPC: H2O 39 C o 86 0.569 0.329 0.747 415 67 HPC: H2O 10 C o 86 0.1.159 0.274 0.789 415 67 HPC: H2O 39 C o 86 0.448 0.235 1.045 1280 67 HPC: H2O 10 C o 86 0.964 0.263 0.863 1280 67 HPC: H2O 39 C o 82 0.46 0.177 0.739 139 67 HPC: H2O 10 C o 82 0.85 0.0684 1.07 139 67 HPC: H2O 41 C 1 1.054 0.062 1 2000 93 dextran:H2O 1 1.037 0.0629 1 2000 183 dextran:H2O 1 1.027 0.0688 1 2000 246 dextran:H2O 1 1.01 0.0127 1 20 246 dextran:H2O 1 1.027 0.0246 1 70 246 dextran:H2O 1 1.027 0.0357 1 150 246 dextran:H2O 1 1.027 0.0435 1 500 246 dextran:H2O 1 1.027 0.0667 1 2000 246 dextran:H2O 57 1.03 0.121 1 300 20.8 PEO: H2O 57 1.09 0.0475 1 100 20.8 PEO: H2O 57 1.09 0.0989 0.72 18 20.8 PEO: H2O 57 1.18 0.050 0.755 7.5 20.8 PEO: H2O 57 0.462 0.139 1 300 51.7 PEO: H2O 57 0.48 0.0823 1 100 51.7 PEO: H2O 57 0.474 0.067 0.82 18 51.7 PEO: H2O 57 0.486 0.041 0.803 7.5 51.7 PEO: H2O 57 0.071 0.091 1 300 322 PEO: H2O 57 0.0733 0.0656 1 100 322 PEO: H2O 57 0.0796 0.022 1 18 322 PEO: H2O 57 0.0672 0.0148 1 7.5 322 PEO: H2O 57 0.036 0.078 1 300 655 PEO: H2O 57 0.0361 0.080 1 100 655 PEO: H2O 57 0.035 0.072 1 18 7.5 PEO: H2O −3 58 1.056 5.49 × 10 1 5 nm 5 BSA: H2O −3 58 0.99 4.3 × 10 1 5 nm 322 BSA: H2O −3 58 1.008 3.76 × 10 1 5 nm 655 BSA: H2O −2 61 0.963 1.98 × 10 1 115 5 nm DNA:H2O:0.01 M NaCl −2 61 1.007 1.51 × 10 1 115 5 nm DNA:H2O:0.1 M NaCl 0.251 0.394 0.75 1300 170 PVME: toluene 62 79 1.05 0.0736 1 110 60 HPC: H2O 79 1.06 0.090 1 140 60 HPC: H2O 79 0.986 0.104 1 450 60 HPC: H2O 79 1.043 0.234 1 850 60 HPC: H2O 79 1.049 0.072 1 110 105 HPC: H2O 79 1.024 0.089 1 140 105 HPC: H2O 79 1.016 0.189 1 450 105 HPC: H2O 79 1.011 0.30 1 850 105 HPC: H2O 79 1.029 0.0995 1 110 175 HPC: H2O 79 1.016 0.124 1 140 175 HPC: H2O 79 0.993 0.259 1 450 175 HPC: H2O 79 0.935 0.354 1 850 175 HPC: H2O 69 0.70 0.33 1 1900 159 PIB: CHCl3 69 0.59 0.0886 1 610 159 PIB: CHCl3 69 0.684 0.0584 1 182 159 PIB: CHCl3 69 0.895 0.035 1 95 159 PIB: CHCl3 69 1.036 0.0281 1 57 159 PIB: CHCl3 56
ν TABLE II: Fits of the concentration dependence of the solution viscosity η to a stretched exponential η0 exp(−αc ) for solutions studied with probe diffusion. The Table gives the fitting parameters, polymer molecular weight, polymer:solvent system, and the reference. η0 is in centipoise; concentrations are converted to g/L except g/g marks g/g as concentration units. Polymers and solvents as above.
η0 α ν M(kDa) System Refs 101 0.89 0.693 0.69 647 dextran:H2O 101 0.89 0.207 0.91 428 ficoll:H2O 30 1.045* 0.0113 0.99 150 dextran:H2O 30 1.00* 0.122 0.72 40 dextran:H2O 0.948 0.181 0.71 350 PMMA: thf 31 1.045 0.0548 0.972 350 PMMA: dmf 31 1.00 0.0554 0.979 350 PMMA: diox 31 ? 0.80 31.9g/g 0.848 70 PBLG:DMF 0.74 52.8g/g 0.669 233 PBLG:DMF ? 105 1 4.41 0.688 4000 xanthan: H2O 39 0.89 0.0268 0.91 50 PAA: H2O 39 0.89 3.13 0.803 1000 PAA: H2O 0.87 0.178 0.83 1110 PS: dmf 43 0.90 0.109 0.767 215 PS: dmf 43 45 - 0.0141 0.93 28.5 dextran: H2O 45 - .0378 0.81 46 dextran: H2O 45 - 0.0498 0.78 77.2 dextran: H2O 45 - 0.0544 0.79 121 dextran: H2O 45 - 0.164 0.62 462 dextran: H2O 86 1.26 0.189 0.837 60 HPC: H2O (10 C) 86 0.66 0.107 0.977 60 HPC: H2O (39 C) 86 1.305 0.167 1.05 100 HPC: H2O (10 C) 86 0.66 0.139 0.999 100 HPC: H2O (39 C) 86 – 0.462 0.906 300 HPC: H2O (10 C) 86 – 0.440 0.845 300 HPC: H2O (39 C) 86 – 1.21 0.962 1000 HPC: H2O (10 C) 86 – 0.931 1.04 1000 HPC: H2O (39 C) 57 0.914 0.207 1 300 PEO: H2O 57 0.908 0.0901 1 100 PEO: H2O 57 0.933 0.0064 1 18 PEO: H2O 57 0.90 0.0413 0.80 7.5 PEO: H2O 79 1.016 0.104 0.90 110 PAA: H2O 79 0.998 0.144 0.868 140 PAA: H2O 79 1.000 0.409 0.875 450 PAA: H2O 79 0.950 0.582 0.896 850 PAA: H2O 57
x TABLE III: Fits of the concentration dependence of the solution viscosity η to a power lawηc ¯ + η0, for solutions studied with probe diffusion. The Table gives the fitting parameters, lowest concentration cm included in the fit, polymer molecular weight, polymer:solvent system, and the reference.η ¯ and η0 are in centipoise; concentrations are converted to g/L except g/g marks g/g as concentration units.
η¯ x η0 cm M(kDa) System Refs −3 −3 86 1.33 × 10 2.85 3.16 × 10 20 60 HPC: H2O (10 C) −3 −4 86 1.55 × 10 2.62 −2.68 × 10 18 60 HPC: H2O (39 C) −4 −4 86 9.37 × 10 3.14 1.13 × 10 18 100 HPC: H2O (10 C) −4 −6 86 8.4 × 10 2.96 1.1 × 10 18 100 HPC: H2O (39 C) −3 −4 86 1.02 × 10 4.40 1.14 × 10 20 300 HPC: H2O (10 C) −4 −5 86 2.9 × 10 4.41 9.88 × 10 20 300 HPC: H2O (39 C) 86 0.36 3.99 0.057 6.0 1000 HPC: H2O (10 C) 86 0.092 4.01 -0.069 4.0 1000 HPC: H2O (39 C)
TABLE IV: Fits of the concentration dependence of the probe rotational diffusion coefficient Dr or rotational relaxation time ν ν τr (in µS) to a stretched exponential Dr0 exp(−αc ) or τr0 exp(αc ). The Table gives the fitting parameters, matrix molecular weight or (denoted ”*”) radius, probe radius or (denoted ”*”) major axis length, polymer:solvent system, and the reference. Dr0 ∗∗ is in /second; concentrations are converted to g/L except ( ) volume fraction . SS–silica spheres. Probes include TMV-tobacco mosaic virus, FS-fluorocarbon spheres: PBLG–polybenzyl-L-glutamate. Solvents include pyr-pyridine.
Dr0,τr0 α ν M (kDa or nm) R (nm) System Refs 101 346 0.85 0.67 647 TMV: dextran:H2O 101 336 0.183 0.95 428 TMV: ficoll:H2O 105 0.892 0.257 0.1.56 4000 93 FS: xanthan: H2O 15.4 1.19** 1 62* 68* PBLG:SS:pyr:DMF 107 32.9 13.4** 1 62* 128* PBLG:SS:pyr:DMF 107 51.6 16.1** 1 62* 170* PBLG:SS:pyr:DMF 107
TABLE V: Fits of the concentration dependence of the fast (Df ) and slow (Ds) diffusion coefficients to D0 exp(−αcν ) for probes −7 2 having bimodal spectra. ”F/S” for fast or slow. Dp0 in units 10 cm /s. R as a value for α denotes a mode showing re-entrance. Polymers include PAAM-polyacrylamide and NaPAA-fully neutralized poly-acrylic acid. Probes include PSL-polystyrene latex and hema-hematite.
F/S Dp0 α ν M(kDa) R(nm) System Refs 16 F R 3000 200 PSL: PAAM: H2O 16 S 0.235 5.3 0.36 3000 200 PSL: PAAM: H2O 16 F 0.125 3.57 0.12 large 65 NaPAM: hema: H2O 16 S 0.177 1.65 0.51 large 65 NaPAM: hema: H2O