Fluorescent Labeling Can Alter Polymer Solution Dynamics

of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, United States. Macromolecules , 2015, 48 (19), pp 7245–7255. DOI: 1...
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Fluorescent Labeling Can Alter Polymer Solution Dynamics Ryan McDonough,† Rafael Cueto,‡ George D. J. Phillies,§ Paul S. Russo,‡ Derek Dorman,‡ and Kiril A. Streletzky*,† †

Department of Physics, Cleveland State University, Cleveland, Ohio 44115, United States Department of Chemistry and Macromolecular Studies Group, Louisiana State University, Baton Rouge, Louisiana 70803, United States § Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, United States ‡

S Supporting Information *

ABSTRACT: There is a widespread assumption that modest chemical modification of a large polymer molecule, by adding a fluorescent label or the like, has no significant effect on polymer solution dynamics. We here report that light tagging of hydroxypropylcellulose (HPC) with fluorescein dye has a significant effect on the dynamic behavior of the chains, as measured using fluorescence photobleaching recovery (FPR) and dynamic light scattering (DLS). We compared unprocessed HPC, HPC processed in preparation for chemical modification, and HPC chemically modified by tagging with a fluorescein moiety. Addition of covalently bound fluorescein to the polymer eliminates the ultraslow (D ≈ 10−10 cm2/s) relaxational mode found with DLS in the unlabeled HPC samples. Our findings explain the perceived discrepancies between DLS and FPR studies of HPC solutions. The discrepancies arise from differences in sample preparation. Comparison of FPR spectra (which measure the single particle diffusion process) and DLS spectra (which measure the relative motions of pairs of particles), each with its own intrinsic time and distance scales, provides insight into the nature of diffusion and the state of dissolution in dilute and nondilute HPC solutions.

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INTRODUCTION Scattering and related solution properties techniques, including dynamic light scattering (DLS),1 optical probe diffusion,2 fluorescence photobleaching recovery (FPR),3 depolarized light scattering,4 pulsed field gradient nuclear magnetic resonance (PFGNMR),5 inelastic neutron scattering,6 fluorescence correlation spectroscopy (FCS),7 X-ray photon correlation spectroscopy (XPS),8 Mossbauer spectroscopy,9 nuclear resonant scattering,10 Fabry−Perot interferometry,11 timeresolved optical spectroscopy (TROS),12 forced Rayleigh scattering,13−15 and dielectric relaxation spectroscopy,16 have become major approaches for the study of polymer solution dynamics. On various time and distance scales, these methods reveal polymer self-diffusion and mutual diffusion coefficients, polymer whole chain rotation, segmental relaxation and side chain motion, solvent diffusion including motions in highviscosity liquids, and polymer mean end-to-end distances.17 While some of these techniques are applied to unmodified polymer molecules in solution, others (e.g., FPR, FCS, and TROS) require chemical modification of the polymer of interest before the technique can be applied. One must then be concerned that the chemical modification, however minor, has perturbed the polymer’s dynamic properties. This paper reports the application of two of these techniquesdynamic light scattering and fluorescence photobleaching recoveryto the study of diffusion in hydroxy© XXXX American Chemical Society

propylcellulose (HPC) solutions. HPC is a water-soluble synthetic derivative of cellulose. It is a neutral semiflexible chain with a persistence length of approximately 10 nm.18−20 HPC is amphiphilic; it has a dewetting phase transition at a temperature near 41 °C and a lyotropic phase transition at polymer weight fraction wHPC ≈ 0.4 in aqueous solution. HPC in water scatters enough light that it is a good target for DLS studies, even at low concentrations. Yet because of HPC’s stiffness, high viscosities can be achieved at concentrations sufficiently low that its scattering is still too weak to obstruct fluorescence photobleaching recovery or optical probe diffusion experiments. HPC solutions and gels also have commercial applications in the food, cosmetic, and pharmaceutical industries.21 The dynamic behavior of HPC solutions has been studied extensively by multiple physical techniques. Measurements of solution viscosity as functions of polymer concentration and molecular weight have been reported by Russo et al.,22 Yang and Jamieson,23 and Phillies and Quinlan.24 Viscosity, η, increases markedly with increasing concentration, c, and molecular weight, M. The extensive viscosity measurements of Phillies and Quinlan24 show that at smaller concentrations Received: July 27, 2015 Revised: September 5, 2015

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In contrast to the DLS measurements, FPR studies find no sign of a new slow mode that appears only at larger polymer concentrations.36 A long list of possible reasons for the difference between the FPR and DLS measurements can be generated; the simplest is that differences in sample preparation, concentration, or polymer sample were responsible. Here all studies were done on the same samples. DLS is a fluctuation spectroscopy. It studies evanescent variations in the intensity I(t) of monochromatic light scattered by a nominally homogeneous equilibrium solution. The scattered intensity is time-dependent because it is determined by the relative positions of the molecules in solution, those positions changing as the molecules diffuse. A DLS system generates a statistical characterization, the time-dependent intensity−intensity correlation function S(q,t) = ⟨I(τ)I(τ + t)⟩, of those evanescent intensity fluctuations. FPR operates on a solution that initially contains a uniform equilibrium concentration of fluorescently tagged molecules and perhaps a matrix of untagged molecules. A strong pulse of light is then used to photobleach some of the tagged molecules in solution; once bleached, the molecules do not fluoresce. The bleaching process creates a macroscopic nonequilibrium variation in the relative concentrations of bleached and unbleached fluorescently tagged molecules. The total of the concentrations of the tagged and untagged molecules is very nearly the same everywhere, but the relative concentrations of bleached and unbleached molecules depend on the location in the sample cell. The relaxation of the relative concentration profile is monitored via the time-dependent FPR decay profile AC(t). DLS and FPR both measure diffusive properties of a polymer solution. The methods differ in the diffusive property to which each responds, the time and distance scales to which each is sensitive, and in their modes of operation. The diffusive processes measured by DLS and by FPR are not the same. DLS measures the relaxation of concentration fluctuations, a process described on a macroscopic level by the mutual diffusion coefficient Dm.44,45 FPR measures the motion of labeled molecules through a solution in which the total concentration of labeled and unlabeled molecules is the same everywhere, a process described on a macroscopic level by the self-diffusion coefficient Ds. DLS can also be sensitive to rotational and segmental motion. As practiced here, FPR responds to processes on long time scales on which rotational and segmental motions are too rapid to contribute to the FPR relaxation function, except as rotation and segmental motion influence the long-time average relaxation. Dm and Ds in general depend on the solute concentration. The concentration dependences are created by direct and hydrodynamic interactions between the solute molecules. Mutual diffusion may properly be said to be driven by the solute’s osmotic stiffness, as modulated by hydrodynamic interactions and more subtle correlations. Implicit in uses of FPR to measure self-diffusion is a central physical assumption, namely that tagging a molecule with a fluorescent group, and then photobleaching the fluorescent group, has no significant effect on the molecule’s diffusive properties. DLS is sensitive to distance scales as determined by the scattering vector

the viscosity η(c) is a stretched exponential in c, but at some critical concentration c+ has a sharp, analytic (first derivative continuous) crossover to a power-law concentration dependence at larger c. This “solution-like−melt-like” crossover was a primary motive for many of our later studies of this system by other methods.22,23,25−34 Scattering experiments include (a) static light scattering studies,23,25 (b) DLS measurements of polymer mutual diffusion in HPC:water,26 and (c) optical probe diffusion studies using DLS of polystyrene spheres of many sizes in HPC:water.22,23,27−34 Extending the probe diffusion studies to smaller probe sizes, FPR was used to measure Ds of dextrans and sodium fluorescein through HPC solutions.35,36 Systematic reanalysis of these studies leads to a coherent phenomenological image of relaxational modes, dynamic transitions, and fundamental length scales in HPC solutions.25,26,37 DLS spectra are multimodal, containing at least two stretched-exponential relaxations. The critical concentration c+ was determined. For our 1MDa HPC, c+ ≈ 0.6 wt % was found. Above this concentration, DLS spectra of some diffusing probes and of the polymer itself show a third, very slow mode. The concentrations at which the third optical mode appears and at which the viscosity has its dynamic transition are the same. The very slow mode appears to result from long-lived dynamic structures and local vitrified regions.37 Static light scattering measurements find that the normalized scattered intensity I/c has a slow decrease with increasing c, with no discontinuity at c+, proving that the vitrified regions cannot be aggregates, that is, local regions of elevated polymer concentration, unless they are so large that measurements performed so far missed them by not extending to sufficiently low q. Multiple lines of analysis converge on the conclusion that HPC solutions have a longest dynamic length scale ξ. Optical probe spectra of polystyrenes spheres in HPC:water show distinct small-sphere and large-sphere behavior, the dividing line between small and large spheres being approximately Rg at all c.25,31,37 DLS probe spectra only show the very slow mode for smaller probes. From the DLS studies, ξ is thus approximately independent of concentration and approximately equal to the polymer radius of gyration Rg. In contrast, FPR results on probes in HPC solutions show no signs of the very slow mode, even though the FPR probes are smaller than the polystyrene sphere probes.36 Comparison35 of the concentration dependence of Ds of fluorescein with simple obstruction models,38−43 which treat the HPC chains as stationary objects, was unsatisfactory. Ds decreases far more sharply with increasing HPC concentration than would be expected from any of the obstruction models. It could be proposed that the effective diameter of the HPC chains is increased by bound water, but the required amount of bound water is considerably larger than the amount of water binding inferred from calorimetry, indicating that in this system obstruction models for the retardation of diffusion do not appear to be satisfactory as explanations. Comparison22,23 of viscosity measurements with probe diffusion of polystyrene latex spheres found (after addition of traces of the surfactant TX-100 to prevent probe aggregation) that probes in lowermolecular-weight (110, 140, and 292 kDa) HPC solutions diffuse at the rate expected from η, but diffuse more rapidly than expected from η in large-M (450 and 850 kDa) HPC solutions.

q = 4πn sin(θ /2)/λ

(1)

λ being the wavelength of the scattered light in vacuo, θ being the scattering angle, and n being the solution index of refraction. For our 632.8 nm laser, q is ca. 2 × 105 cm−1 B

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particle and as part of an aggregate or vitrified zone, so that only a single averaged diffusive mode might be apparent. In this paper, we have several major objectives. First, we explore whether or not fluorescent labeling perturbs the polymer’s diffusion. Second, by combining DLS and FPR measurements, we seek to obtain a fuller image for P(Δx,t) and P(2)(Δx,t,R12). Third, we seek to understand the slow mode found in DLS studies of HPC:water and in some optical probe spectra of the same solutions. The next section briefly outlines our experimental methods. The third section describes our spectral analysis techniques, namely time moment analysis25 and inverse Laplace transforms via the CONTIN algorithm.49 The fourth section presents our findings based on FPR and DLS. Finally, in the fifth section we discuss our results and present an interpretation.

depending on the scattering angle. Routine DLS measurements are sensitive to time scales from a few microseconds to several seconds; by using a two-detector system, DLS spectra have been observed on time scales as short as 50 ns.46 FPR as practiced here uses optical methods to create repetitive bleaching patterns that gradually fade as bleached and unbleached molecules interdiffuse; our optical system gave a spatial frequency K ≈ 583 cm−1. FPR relaxations are readily observed on times scales from 0.1 to ca. 103 s, the lower time limit being fixed by the duration of the photobleach laser pulse. Our DLS and FPR measurements thus observe diffusion on thousand-fold different distance scales and up to million-fold different time scales. On a molecular level, FPR and DLS correspond to displacement distribution functions.47,48 FPR corresponds to the single-particle distribution function P(Δx,t), which describes the likelihood that thermal motion will displace a given molecule through a distance Δx during a time interval t. The DLS spectrum receives contributions from the spatial Fourier transforms of two displacement distribution functions, namely from P(Δx,t) and from the two-particle displacement distribution function P(2)(Δx,t,R12). Here R12 is a particular component at time 0 of the vector from particle 1 to a second particle 2, so that P(2)(Δx,t,R12) gives the likelihood that particle 1 will have a displacement Δx during the time interval (0, t) given that R12 is specified at time 0. In dilute macromolecular solutions, Ds and Dm are very nearly equal, P(Δx,t) is a Gaussian in Δx, ⟨(Δx)2⟩ increases linearly with t, the effect of P(2)(Δx,t,R12) vanishes, and S(q,t) and AC(t) decrease exponentially with t. In nondilute macromolecule solutions, Ds and Dm differ substantially, P(Δx,t) is not in general a Gaussian in Δx, the effect of P(2)(Δx,t,R12) is significantly nonzero, and the relaxations of S(q,t) and AC(t) typically gain multiple, nonexponential decay modes. The single particle motions described by P(Δx,t) that contribute to the FPR decay profile AC(t) also contribute to the DLS spectrum S(q,t), the two techniques being sensitive to P(Δx,t) on quite different time and distance scales. However, the inverse is not true; P(2)(Δx,t,R12) contributes to S(q,t) but not to AC(t). The physical reason that P(2)(Δx,t,R12) does not contribute to AC(t) is that fluorescent scattering loses information on the phase of the incident light, so that FPR is sensitive to the motion of fluorescently tagged particles but not to the relative motion of pairs of tagged particles. In contrast, the quasielastic scattering process observed by DLS retains information on the phase of the incident light, so that DLS is sensitive both to the motion of single scattering particles and also to the relative motion of pairs of scattering particles, the latter being the motions that P(2)(Δx,t,R12) describes. As an additional complication, in complex fluids P(Δx,t) and P(2)(Δx,t,R12) have relaxations that act on particular time and distance scales. Because FPR and DLS are sensitive to molecular motions on disparate time and distance scales, features of P(Δx,t) that are apparent in FPR measurements may not be apparent in DLS measurements, and vice versa. Thus, for example, if diffusing particles sometimes formed short-lived aggregates, DLS might report single-particle and aggregate diffusion as two distinct modes. Because K ≪ q, FPR is sensitive to particle motions on much longer distance and hence time scales. On long time scales, each diffusing particle could have alternated repeatedly between diffusing as a single

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MATERIALS AND METHODS

Because one of our interests was in determining whether or not the labeling process modifies the polymer’s solution dynamics, an elaborate solution preparation protocol was invoked. Changes in the polymer’s solution properties might arise either from (i) fractionation or other outcomes attendant to various dissolution and drying steps or from (ii) the chemical modification step of tagging the individual polymer chains with our fluorescent label. We therefore had three polymer samples, namely unlabeled-unprocessed polymer (UL-UPHPC) made by dissolving HPC in water, unlabeled-processed polymer (UL-P-HPC) made by passing the polymer through all processing steps except chemical modification, and finally the labeled fluoresceintagged polymer (FITC-HPC). In more detail: HPC was obtained from Scientific Polymer Products, Inc., with a nominal vendor-supplied molecular weight of 1 MDa. Gel permeation chromatography with multiple-angle light scattering detection (GPC/ MALS) data given in detail below instead found 0.3−0.4 MDa for the molecular weight. The HPC polymer was dissolved in deionized water (Millipore Academic) to a concentration of 1.0 wt % and equilibrated for several days with stirring. The solution was then filtered through a 0.45 μm cellulose acetate membrane (GE Water & Process Technologies). The filtered solution was lyophilized via the shelf freezing technique to obtain a solid of low water content. Approximately one-third of the new solid was redissolved in deionized water to make a 1.0 wt % aqueous solution. This solution was used to make the unlabeled-unprocessed (UL-UP-HPC) samples. The remaining solid was dissolved in acetone. Half of the HPC/acetone solution was used to prepare the unlabeled-processed (UL-P-HPC) samples; the other half of the HPC/acetone solution was mixed with a saturated FITC/acetone solution. To tag the HPC molecules and form labeled FITC-HPC, the preparation protocol of Mustafa et al.35 was followed. The FITC-HPC solution was allowed to react at room temperature for several days in a dark in order for the hydroxyl− isothiocyanate reaction to reach equilibrium. A label content of approximately one FITC group per HPC molecule was obtained. The fluorescent intensity was measured during dialysis until the intensity level became stable. The fluorescence intensity was calibrated using unattached dye. n-heptane was added to the FITC-HPC/acetone and HPC/acetone solutions to precipitate the polymer. The precipitates were dried in a vacuum oven at 40 °C for several hours, then redissolved in water, and dialyzed using membrane tubing (Spectrum, MWCO 3500). The deionized water in the outer membrane was frequently replaced until no trace of yellow-green color due to the dye could be detected. This process of dialysis took several days. After being exhaustively dialyzed, the UL-P-HPC and FITC-HPC solutions were lyophilized and then redissolved in deionized water. Solutions of all three materials, UL-UP-HPC, UL-P-HPC, and FITC-HPC, were prepared to final concentrations of 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9, and 1.0 wt %. Figure 1 presents a schematic diagram comparing the three materials. GPC/MALS was used to obtain average molecular weights for all three materials, using samples from several solutions of the final C

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3000 Ar+ operating at 488 nm with power of about 1.0 W provided illumination for the photobleaching event and the post-photobleach recovery period. The laser intensity was modified by acousto-optic modulation (Newport, EOS-N35085-9) for photobleaching and recovery. The bleaching light was about 2000 times brighter than the reading light. The signal was detected with an RCA 7265 photomultiplier tube, protected from the photobleaching event by an electronically synchronized shutter. Several grating constants K = 2π/L were available. The optimum choices of Ronchi ruling and microscope objective for our FITC-HPC samples were found to be 50 lines/in. and 18×, respectively, leading to K = 583 cm−1. A typical FPR run lasted from 5 to 30 min, depending on the sample viscosity, with at least three runs for each sample and optical setting. No significant recovery of the bleached FITC dye was observed in any experiment. For our DLS studies, a HeNe laser operating at λ = 632.8 nm with a power of 35 mW illuminated the polymer solution. This weak red laser was used in order not to destroy the dye, which has an absorption maximum near 490 nm. The sample was housed in a Brookhaven goniometer (BI-200M); light was detected at a 90° scattering angle with a BI-DS2 photomultiplier. The intensity autocorrelation function S(q,t) was obtained with a Brookhaven 9000 computer board. A typical DLS measurement extended for 1 h in order to determine slow modes accurately. To ensure reproducibility, at least three DLS spectra were obtained for each sample. The system used square quartz sample cells having a 1 cm optical path length. When no dust or other intense scattering was observed, there was a high degree of reproducibility between DLS spectra on a given sample. The samples examined by DLS and FPR were precisely the same; solutions were drawn into FPR microcapillary tubes directly from DLS sample cells.

Figure 1. Schematic diagram of HPC sample comparison. Three types of stock solutions were made from the precleaned HPC powder: unlabeled and unprocessed solution (UL-UP-HPC), processed for labeling but unlabeled solution (UL-P-HPC), and FITC labeled solution (FITC-HPC). concentrations. The starting concentrations were 0.1−1.0 wt %. Concentrations in the GPC/MALS system are much lower (typically 10−2−10−5 wt % at peak) due to dilution as the sample elutes. The virial coefficient A2 was assumed to be zero, based on the absence of any consistent trend in molecular weight with concentration. For the unlabeled-unprocessed HPC (UL-UP-HPC), measurements on the 0.1, 0.3, and 0.5 wt % samples averaged to Mn = 228 300 ± 25 000 Da and Mw = 290 100 ± 44 000 Da. For the unlabeled-processed HPC (UL-P-HPC), measurements on the 0.1, 0.3, 0.5, and 1.0 wt % samples averaged to Mn = 272 900 ± 23 000 Da and Mw = 350 300 ± 31 000 Da. For the labeled HPC (FITC-HPC), three measurements on the 0.5 wt % sample averaged to Mn = 307 600 ± 7000 Da and Mw = 379 000 ± 6000 Da. The unlabeled-unprocessed HPC had Mw/Mn ≈ 1.2, while for the unlabeled-processed and labeled samples the polydispersity was slightly larger, Mw/Mn ≈ 1.3. The ≈20% difference of both Mn and Mw between the two unlabeled samples suggests that a lower-molecular-weight fraction of the HPC is lost in processing. The only difference between the dilute-solution conformation plots (Rg vs M, see Supporting Information, Figures X1a and X1b) of the three materials is a slight nonlinearity at low M for the unlabeledunprocessed HPC, in a range of M in which the other two samples have no material. The overall consistency of molecular weight measurements across the concentration range of 0.1−1.0 wt % suggests that irreversible noncovalent binding is not responsible for the differences in Mn and Mw between samples. GPC/MALS delivers not just averages but the full molecular weight distribution of each sample, as seen in Figures X2a and X2b. The differences between the molecular weight distributions of the unlabeled-processed HPC and the FITC-labeled HPC samples are relatively small, while the molecular weight distribution of the unlabeled-unprocessed HPC is quite different from the other two distributions. Indeed, GPC/MALS measurements on the unlabeledprocessed HPC and the FITC-labeled HPC agree with each other to within experimental error, indicating that the labeling process does not create covalent cross-links between polymer chains. All FPR and DLS experiments were performed at 22 °C. The FPR system used rectangular Precision Ronchi Ruling Glass Slides (Edmund Scientific), which created a striped photobleaching pattern during the photobleaching event. The Ronchi ruling was mounted on a cylinder connecting two opposing loudspeakers driven by an amplifier. During the recovery period, this arrangement gave a periodic sidewise motion of the Ronchi ruling with the width of one stripe, at a frequency of 16 Hz. Using a tuned amplifier, the fluorescence recovery is observed as a difference signal created by alternative illumination of bleached and unbleached areas. This modulation detection scheme greatly increases instrumental sensitivity, thereby decreasing the fraction of dye groups that must be bleached, excluding noise outside the detection bandwidth, and ensuring that each diffusive mode corresponds to a single-exponential mode.3 The modulation detection typically required only shallow (5−20%) photobleaches; even shallower photobleaches were not possible to observe because of limits of instrument stability. A modified Olympus BH-2 epifluorescence microscope36 was used to illuminate the Ronchi ruling from the rear focal plane. Samples were loaded into flat glass vessels of optical path length ≈100 μm. A Lexel

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DATA ANALYSIS We have two major sets of spectra. DLS yields the intensity− intensity time correlation function S(q,t). Because we measure alternatively the intensity from bleached and unbleached regions of the sample, our FPR signal (after passing through the tuned amplifier to suppress higher harmonics) consists a low-frequency (typically 16−32 Hz), nearly sinusoidal oscillation. The experimental signal of interest in these FPR measurements is the decay profile of the FPR envelope AC(t). The functions S(q,t) and AC(t) each have complex nonexponential forms. We used two very different schemes to extract meaningful numerical parameters from each of these functions. Spectral time moment analysis25 decomposes S(q,t) and AC(t) into a sum of stretched exponential modes, whose shape parameters are then combined into a single characteristic relaxation time for each mode. The CONTIN algorithm49 provides a smoothed inverse Laplace transform of S(q,t) and AC(t). To analyze DLS measurements, S(q,t) was decomposed into the field correlation function g(1)(q,t) using the relation50 S(q , t ) = A(g(1)(q , t ))2 + B

(2)

which reflects the fact that the volume from which light scattering is observed is much larger than the volumes over which particle positions or displacements are correlated. Here the amplitude A is an instrumental parameter, B is the baseline to which S(q,t) relaxes at long times, and q is defined above in eq 1. We have previously shown that g(1)(q,t) of HPC:water solutions, and of spherical probes in HPC:water,37 is accurately described as one or a sum of a few stretched exponentials in time. We therefore here used the same expression for fitting g(1)(q,t) D

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Figure 2. Normalized, baseline-subtracted DLS spectra of (◇) unlabeled-processed (UL-P-HPC), (○) unlabeled-unprocessed (UL-UP-HPC), and (△) labeled HPC (FITC-HPC), at concentrations of (a) 0.2 wt % and (b) 0.9 wt %. Spectra of the unlabeled-unprocessed and unlabeled-processed materials are similar but not the same. Labeling the HPC with fluorescein has a substantial effect on the polymer’s DLS spectrum.

Figure 3. Normalized DLS (red circles) and FPR (black squares) relaxations S(q,t) and AC(t) for FITC-labeled HPC at concentrations (a) 0.2, (b) 0.3, (c) 0.5, (d) 0.7, (e) 0.9, and (f) 1.0 wt % and outputs of CONTIN/ANSCAN analyses for the DLS (blue) and FPR (green) spectra. N

g(1)(q , t ) =

∑ Ai exp(−θit β )

magnitude in t and several orders of magnitude in S(q,t)/ S(q,0), while the characteristic relaxation times of the three modes spanned 3 orders of magnitude in time. Even so, nine fitting parameters represent an extreme outer limit to the number of parameters that can be extracted from a DLS spectrum.51 The FPR signals span narrower ranges in time and AC(t)/AC(0). To clarify our results, the method of spectral time moments25 was used, which combines the fitting parameters θi and βi for each mode into a single characteristic relaxation time. The rationale for using time moments is that the errors in determining θi and βi are significantly cross-correlated. The errors tend to cancel, so that the scatter in the time moments is much less than the scatter in θi or βi viewed separately.

i

i=1

(3)

The above is a sum over N spectral modes with index i labeling the mode. In each term of the sum, Ai is the mode amplitude, θi is the relaxation pseudorate, and βi is the mode’s stretching parameter. θi is a pseudorate, not a true rate, because its dimensions in general are not time−1. There is no implication in this equation that the separation into modes is unique or that the separate modes correspond one-to-one to distinct physical relaxations. Ai, θi, and βi were obtained from the measured spectra by using nonlinear least-squares functional minimization based on the simplex algorithm. Two or three modes were used in the analysis. The use of so many modes is only possible because we are able to track S(q,t) over 4−6 orders of E

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Figure 4. Nominal diffusion coefficients and normalized mode amplitudes for the DLS spectra (a and b, respectively) and for the FPR spectra (c and d, respectively) of labeled HPC. Representative spectra are seen in Figure 3. Diffusion coefficients and amplitudes were obtained from fits to sums of two stretched exponentials and interpreted using time moment analysis.

Spectral time moments arise from the exponential moment integral Mn ≡

∫0

∞

dt t n exp( −Γt ) =

γ(1 + n) Γ1 + n

M 0i =

∫0

∞

dΓ A(Γ) exp( −Γt )

(5)

where A(Γ), the decay amplitude at decay rate Γ, is normalized as ∫ ∞ 0 A(Γ) dΓ = 1. The nth moment integral of g(t) is then Mn ≡

∫0

∞

dt t ng(1)(t ) =

∫0

∞

dΓ A(Γ)

γ(1 + n) Γ1 + n

∞

1/ βi

dt exp( −θit β) = γ(1 + 1/βi )/θi

(7)

as the spectral time moment. M0i is the mean relaxation ⟨Γ−1⟩. Each mode has its own M0i. We identify the averaged diffusion coefficient ⟨Di−1⟩−1 = 1/(M0iq2) corresponding to M0i as the nominal mean diffusion coefficient. The method of time moment analysis can equally be applied to the FPR amplitude profile AC(t). In contrast to the spectral time moment method, which decomposes S(q,t) and AC(t) into a sum of stretched exponential functions on the way to identifying a characteristic time for each stretched exponential mode, the CONTIN method is based on a smoothed inverse Laplace transform of S(q,t) or AC(t). To cope with the physical constraint on the number of free parameters implicitly present in S(q,t) or AC(t), CONTIN actually presents a smoothed approximation to A(Γ). The smoothing constraint in CONTIN means that the reported amplitudes, each with their corresponding Γ, are not simply a discrete representation of the distribution function A(Γ). The CONTIN algorithm was applied to our FPR and DLS spectral data using the ANSCAN software developed in one of our groups.52

(4)

Here Mn (not to be confused with number-average molecular weight) is a moment of order n, Γ is the decay rate of a simple exponential, and γ is the Gamma function (not to be confused with the spectral decay rate Γ). For a simple exponential, M0 = 1/Γ and M1 = 1/Γ2. g(1)(t) can be written as a Laplace transform, a sum of exponentials, namely g(1)(t ) =

∫0

(6)

Just as determining spectral cumulants accurately requires measuring g(1)(t) accurately at short times, so also does determining time moments accurately require measuring g(1)(t) accurately at long times. To solve noise problems, we fit our spectra to algebraic forms that reflect a small number of spectral modes and then perform analytic integration of the functional forms to compute the moment integrals. For a stretched exponential relaxation, integration gives

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EXPERIMENTAL RESULTS

Figure 2 shows the normalized DLS spectra of the three polymers at dilute and nondilute concentrations. These are log−log plots, so S(q,t) is clearly visible through a 3 orders of magnitude change in S(q,t) and F

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Figure 5. Nominal diffusion coefficients for the DLS spectra of (a) unlabeled-unprocessed and (b) unlabeled-processed HPC. Representative spectra are seen in Figure 2. Diffusion coefficients were obtained from fits to sums of three stretched exponentials and then interpreted using time moment analysis. a 5 orders of magnitude change in t. As made visible by Figure 3, this choice of plot axes reveals some spectral features but masks others. Figure 2a gives the normalized DLS spectra of the three materials in relatively dilute 0.2% solutions. Spectra of the two unlabeled polymers are both visibly polymodal; namely, there are local regions in which the second time derivativethe curvature of S(q,t) vs log(t)is positive. The fast mode of the UL-P-HPC is visibly weaker than the fast mode of the UL-UP-HPC, and correspondingly the slow mode is more intense for UL-P-HPC than for UL-UP-HPC. This weakening of the fast mode corresponds to the reduction by the processing steps in amount of low-molecular-weight HPC, as discussed above. The initial decay of S(q,t) is slower for the labeled than for the unlabeled polymer. The effect of processing is also apparent for times 300 μS−20 ms, for which S(q,t)/S(q,0) of the unlabeled-processed material is visibly larger than S(q,t)/S(q,0) of the unlabeledunprocessed material. For the two unlabeled polymers, S(q,t) shows a longest mode that persists until times >100 ms. This mode is absent from S(q,t) of the labeled polymer; S(q,t) of the labeled polymer simply decays into the background noise for times ≈20 ms. Figure 2b shows spectra of the nondilute 0.9 wt % solutions. The qualitative mode structures of the unlabeled and labeled materials do not change when the polymer concentration is increased. As also seen in dilute solution, spectra of the unlabeled polymers are visibly polymodal. The spectrum of the labeled polymer does not have the visibly separated very slow mode of the unlabeled polymers. Increasing c from 0.2 to 0.9 wt % slows the relaxations by more than 10-fold; the effect of increasing concentration is much larger for the labeled than for the unlabeled HPC. At 0.2%, the three spectra overlap; at 0.9%, S(q,t) of the labeled polymer lasts several-fold longer than does S(q,t) of either of the unlabeled polymers. Our qualitative results thus clearly demonstrate that labeling has a dramatic effect on the DLS spectrum of HPC:water. Processing has a much more modest effect. The implication is clear: comparisons are appropriate between FPR spectra and DLS spectra of labeled polymer. Such comparisons may illustrate the above-mentioned differences between what the two methods see. The labeled and unlabeled materials are quite different in their dynamics, so comparisons between FPR spectra (necessarily of labeled polymer) and DLS spectra of unlabeled polymers are less likely to be revealing about the differences between the techniques but instead hint at the nature of aqueous HPC. Figure 3 shows DLS and FPR spectra of the labeled polymer at concentrations 0.2, 0.3, 0.5, 0.7, 0.9, and 1.0 wt % of FITC-HPC and the corresponding CONTIN/ANSCAN mode decompositions. Our DLS and FPR experiments examine relaxations on very different time and distance scales, here approximately unified by normalizing the initial amplitudes of S(q,t) and AC(t), and by plotting the DLS and FPR relaxations against the time-distance variables tq2 and tK2, respectively. This unification reflects the issue that FPR is observing

particle motions over hundreds of times the distances to which DLS is sensitive. The unification masks the issue that the DLS spectra almost completely relax at times less than 1 s, this being about the time at which the relaxation in the FPR spectrum first becomes visible. Figure 3 shows semilog plots, the abscissa being the logarithmic axis. The use of a linear ordinate reveals the two modes, hard to see in Figure 2, that are needed to fit each DLS spectrum of the labeled polymer. These modes are a weak relaxation at early times and a longer-lived relaxation that slows dramatically with increasing polymer concentration. This longer-lived relaxation is part of the spectrum seen in Figure 2 for the labeled polymers and is not the very long-lived relaxation seen as the tail of spectra of unlabeled polymers. The FPR spectra, on the other hand, have approximately sigmoidal shapes that change in time scale as the polymer concentration is increased. In all cases the DLS relaxations extend farther as functions of tq2 than the FPR spectra extend as functions of tK2, the time−distance values at which the two spectra reach zero differing by an order of magnitude. Figure 4 represents the decomposition of the spectra of labeled HPC into sums of stretched exponential relaxations. Nonlinear leastsquares fits followed by time moment analyses provide quantitative values for the nominal diffusion coefficient and amplitude of each mode. As seen in Figure 4, the mode parameters depend on polymer concentration. DLS spectra of the labeled HPC require only two modes to be fit well. Efforts to fit the DLS spectra to a sum of three stretched exponentials gave scattered results; the compliance of a nineparameter, three-stretched-exponential fitting function is too soft to be supported by the measured spectra. As seen in Figures 4a and 4c, each set of spectra has a fast mode with D ≈ 1 × 10−7 cm2/s and a slow mode with D in the range 4 × 10−8−4 × 10−10 cm2/s. As shown by Figures 4b and 4d, the slow modes are dominant. The concentration dependences seen for the DLS and FPR spectra are similar but not the same. The fast mode for the DLS spectra becomes slightly more rapid with increasing c; for the FPR spectra, the fast mode slows 4-fold over our concentration range. The slow modes depend much more strongly on concentration than the fast modes do, in each case decreasing more than 10-fold over the observed concentration range. However, D for the slow mode of the DLS spectra is nearly 10-fold slower than D for the slow mode of the FPR spectra. For DLS, the amplitude of the slow mode increases with increasing concentration, starting from ca. 2/3 of the total amplitude at small c and increasing to 90% of the total amplitude at the largest concentration studied. Because the FPR spectra relax over a relatively narrow dynamic range, determinations of the mode amplitudes for FPR spectra are somewhat scattered; however, for the FPR spectra the slow mode has about 3/4 of the total amplitude, with no apparent concentration dependence. Figures 5a and 5b show the nominal diffusion coefficients for DLS spectra of unlabeled HPC before and after processing, respectively. These spectra were fit to sums of three stretched exponentials, finding G

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Macromolecules in addition to the fast and slow modes an additional ultraslow (D ∼ 10−10 cm2/s) mode. D of the fast and slow modes of both unlabeled HPC types of samples are qualitatively similar in their concentration dependences to the D of the labeled polymer. D of the ultraslow mode decreases with increasing c. The combined amplitude of the two slow modes (slow and ultraslow) depends strongly (Figures X3a and X3b) on c, increasing from about 55% at c < 0.5 wt % to about 70% at c ≥ 0.5 wt %. Processing the polymer removes some low-molecular-weight material but only has a very modest effect on the DLS spectra. What is the justification for fitting to three rather than two modes? Just as with the labeled polymer, we fit spectra both to sums of two and to sums of three stretched exponentials. Between labeled and unlabeled polymers, the outcomes were entirely different. For the labeled polymers, three-mode fits were unstable; there is no third mode for the fitting process to find. For unlabeled polymers at larger (c ≥ 0.5 wt %) concentrations, a third mode is clearly visible in spectra; the improvement in the RMS error of the fit on moving from two to three modes is a factor of 2 to 4. For the same polymers at lower concentrations (c ≤ 0.4 wt %) the addition of a third ultraslow mode improves the RMS error, but only by 10% or 30%. At lower concentrations, there is an ultraslow mode, but it is so weak as almost to be lost in the noise at the tail of the spectra. We also analyzed our spectra using the CONTIN/ANSCAN software. ANSCAN combines a graphical user interface to CONTIN with nonlinear least-squares multiple exponential fitting and cumulants. For example, the ANSCAN operator selects the minimum and maximum decay rates used by CONTIN, along with other parameters. The full set of CONTIN control parameters appears in the Supporting Information (Table X1) for the benefit of readers who may wish to reproduce our analysis. Most of the operating parameters (quadrature, regularization)49 were not varied during the present study, and the CONTIN spectra shown are the ones chosen automatically after it tries several different regularizers. The mode decompositions appear in Figure 3, being seen as the curves and dots along the bottom of each subfigure. CONTIN decomposes DLS spectra into a mode that remains near 107 s/cm2 and another more intense mode whose location moves from 108 out beyond 109 s/cm2 as concentration is increased over the observed range. Except perhaps at the highest concentration studied, there is no indication in the CONTIN analysis of a very long-lived, concentrationindependent peak in the DLS spectra. CONTIN analyses of the FPR spectra find at low concentration a single sharp peak. As the polymer concentration is increased, the FPR spectrum becomes progressively broader. CONTIN decomposes these broad relaxations into a sum of two or three relatively sharp peaks. At all concentrations, the DLS spectrum is seen to relax approximately 1 order of magnitude later in scaled times tq2 than the scaled times tK2 at which the FPR spectra relaxes. Correspondingly, the CONTIN decomposition of DLS spectra always has a substantial peak at larger scaled times than the scaled times at which the longest-lived FPR peak appears. Finally, we made several exploratory experiments to investigate possible causes for the effect of dye labeling. Measurements were made on systems to which 0.05 wt % of Triton X-100 had been added. Triton X-100 is a neutral amphiphile; in the presence of HPC, no micelles would be expected to be observable with DLS at this Triton X-100 concentration. The rationale for these experiments is that the surfactant might bind to HPC chains and reduce the incidence of interchain interactions. By reducing these interactions, whatever effects underlie the slowing of the slow mode of the labeled chains, relative to the same mode in the unlabeled chains, might be diminished, thus modifying the mode structure of the labeled chains. If true, these effects should be apparent in a change of the mode structure. Reducing the strength of interactions might also disperse hypothesized temporal associations responsible for the ultraslow mode in unlabeled polymers. For both unlabeled and labeled chains addition of Triton-100 reduces the amplitude of the slow mode and displaces its relaxation to shorter times but does not eliminate it (Figures X4 and X5). Careful examination of DLS spectra at low HPC concentration also shows (Figure X4) that the spectra of the labeled chains are somewhat more

sensitive to the presence of Triton X-100 than are the spectra of the unlabeled chains. Mixtures containing 10% of the labeled polymer and 90% of the unlabeled-processed polymer were prepared to total HPC concentrations of 0.1 and 1.0 wt % HPC. The experiment examined whether the presence of a small fraction of labeled polymer was sufficient to abolish the ultraslow mode. The ultraslow mode remained, although it was diminished in amplitude; the mixtures had spectra very similar to spectra of the unlabeled-processed polymer and significantly different from the spectra of the labeled polymers. The presence of Triton X100 affects the dynamics of unlabeled polymer much more strongly than the presence of 10% of the labeled polymer (see Figure X5a). Measurements were also made at 10 and 25 °C, on both labeled and unlabeled polymers. Reducing the temperature slows all diffusive processes, but moving the system farther from or closer to the 41 °C dewetting transition had no other effect. The differences between the labeled and unlabeled polymers are the same at all three temperatures studied. Finally, we made limited studies of the q dependence of the DLS spectral mode amplitudes, including decomposing the total scattering intensity into the intensities to be attributed to each scattering mode.53 The results for unprocessed, unlabeled HPC suggest that both slow and fast decay modes are associated with correlation lengths that well exceed the molecular size as determined from our GPC/MALS measurements of Rg (approximately 50 nm). These samples were prepared using very slightly different preparation and protocols compared to the rest of the study (different filter, different correlator). A more detailed study along these lines, taking into consideration the subtle effects of processing and labeling, is warranted.

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DISCUSSION What, then, do we learn from our measurements? Our first major finding is that minor chemical modification of our 0.3 MDa HPC, by attaching to it an average of one fluorescein residue (molecular weight 330) per polymer chain, has a substantial effect on the molecule’s diffusive properties. The DLS spectrum of unmodified HPC has three modes, nominally fast, slow, and extremely slow. On modification with fluorescein, the extremely slow mode is abolished, and the slow mode’s relaxation rate is slowed by 2-fold or more. Modification does not, however, substantially affect the relaxation rate of the fast mode. It follows that comparisons of DLS, FPR, and if available FCS spectra must be made on exactly the same materials, rather than making the fluorescence studies on labeled material and the DLS studies on unlabeled materials, at least until there is adequate evidence that dye labeling does not affect polymer dynamics in the system of interest at the concentrations of interest. We also compared the effects of polymer labeling and mild polymer processing. The effects of mild processing, preparatory to chemical modification of the polymer chains, are shown to be much less significant than the effects of chemical modification. It is not plausible that the difference between labeled and unlabeled DLS spectra of HPC could be caused by the diffusion in solution of free dye. The free dye is so low in molecular weight that as a highly dilute solute component it would be almost impossible to observe using DLS. Also, D of the free dye has been measured experimentally in pure water and HPC solutions.35 D is 5.5 × 10−6 cm2/s in pure water and only slightly less in 1.0 wt % HPC.35 Addition of a very small amount of a rapidly diffusing component can have no effect on the detectability of a moderately intense mode whose diffusion coefficient is 4 or 5 orders of magnitude slower. Motion of free dye might be slowed greatly if it adsorbed to the polymer H

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differences in the probability distributions sensed by DLS and FPR. Our fast mode does not appear to correspond to the constrained lateral diffusive mode successfully identified by Li et al.56 in some other systems. In “tube”-type models of polymer dynamics, as treated by Li et al.,56 polymer chains are said to have rapid lateral diffusion within their tubes. That lateral diffusion is confined by polymer entanglement points to very small distances ξ, distances much less than the polymer radius of gyration. Tube models restrict confined diffusion to semidilute and concentrated solutions, but our fast mode is more intense in dilute than in nondilute solutions. Ignoring the question of whether or not our solutions, which reach viscosities η(c) ≈ 1 × 103 times the viscosity of water, are entangled, a central finding of this paper is that the fast mode is equally observed by DLS and FPR, with about the same relaxation rate, even though these methods function on greatly different time and distance scales. Over a 400-fold variation in distance and 106 in time, the fast mode has a characteristic diffusion coefficient near 1 × 10−7 cm2/s, which harshly constrains physical interpretations of this mode. For DLS, (qξ)2 < 1, so the potential amplitude of a fast mode due to tubeconstrained lateral motion is quite limited. For FPR, Kξ is infinitesimal; FPR is no more sensitive to tube-constrained lateral motion than it is to polymer rotation. For our experiments, KRg ≪ 1 and qRg ≈ 1; it follows that our FPR relaxation modes both overwhelmingly correspond to wholechain diffusion, while the DLS relaxational modes mostly share this characteristic but do weakly reflect other contributions. Finally, our measurements on the q dependence of the DLS mode amplitudes imply that both modes correspond to chain motions correlated over distances much larger than a single chain, while constrained lateral diffusion occurs over distances much smaller than a single chain. Indeed, our angular dependence studies indicate that the two DLS relaxations both correspond to the correlated motions of several neighboring polymer chains. In our systems, the fractional amplitudes of the fast and slow modes are not the same in the DLS and FPR spectra. The fast mode is more intense in the FPR spectra than in the DLS spectra. Setting aside differences in sensitivity of the two methods to polydispersity, if one supposed that the FPR slow mode was faster than the DLS slow mode due to averaging of some of the fast mode into the slow mode, which is a mechanism described by Li et al.,56 then one might have expected intensity to transfer from the fast to the slow mode as one moved from short to long time scales, i.e., as one moved from the DLS to the FPR spectra. That expectation is not met. Our slow mode is actually more intense in FPR, on its longer distance and time scales. The differences between DLS and FPR spectra should not be surprising. DLS and FPR operate on very different distance scales, q−1 and K−1 being ≈54 nm and ≈19 μm, respectively. The techniques also observe very different time scales, namely 5 μs−1 s and 0.1 s−103 s, respectively. In addition, the two techniques measure different correlation functions. FPR observes only the single-particle displacement distribution function P(Δx,t), which corresponds to self-diffusion, while DLS measures a combination of P(Δx,t) and the two-particle displacement distribution function P(2)(Δx,t,R12). Our measurements compare one correlation function at one time and distance scale with a weighted average of two correlation functions on completely different time and distance scales, so

chains, but such dye molecules would not reasonably be expected to be visible in DLS studies. Our second major finding is that comparison of DLS and FPR on the very same labeled samples gives new information on the physical interpretation of the observed mode structure. Prior studies based on DLS, static light scattering, and optical probe diffusion proposed that each spectral mode corresponds to a single physical process.25,26,37 In this model, the fast mode was attributed to the consequences of fundamentally local motions, such as bending and structural relaxation of polymer chain segments, as they relaxed overall concentration fluctuations. The slow mode was attributed to the center of mass displacement of chains in response to their direct and hydrodynamic interactions. The ultraslow mode was believed to arise from multiple-chain, loosely structured, temporal aggregates or perhaps polydispersity fluctuations, subject to the constraint given by static light scattering26 that the local concentration of polymers within the temporal aggregates could not be significantly different from the average concentration of polymers in the solution. Such vitrified regions might readily be more apparent to DLS than to FPR because scattering methods are exquisitely sensitive to large objects but also because those objects do not need to last very long to contribute to the relaxation of the DLS signal. However, in the solutions studied with FPR there is no DLS ultraslow mode at all. Multiple mode behavior is not unique to aqueous hydroxypropylcellulose. For example, FPR and DLS measurements on ternary systems forming transient networks find that FPR and DLS both reveal the presence of distinct fast and slow modes.54,55 Li et al.56 present an extremely systematic review of observations of slow modes (and also some systems in which slow modes are absent), including neutral polymers, polyelectrolytes, and chemically cross-linked chains. Unfortunately, few of these studies have applied DLS and FPR to the same samples. An exception is formed by studies of low-salt polyelectrolyte systems.57−60 In these studies, DLS spectra exhibited distinct fast and slow relaxational modes. In contrast, FPR studies on the same systems found only one mode, which relaxed on a time scale intermediate between the DLS fast and slow modes. Special care was taken to keep experimental samples as similar as possible, as by using a small dye:polymer labeling ratio, removing free dye in the FPR studies, and using in situ dialysis to change the pH of the DLS samples. A critical assumption was that attaching a very small amount of dye to a large and flexible polymer chain does not change the physical properties of the polymer chain. For these polyelectrolyte systems, confirming evidence for this assumption was given by gel permeation chromatography measurements. Differences between the DLS and FPR spectra were explained as arising from the different time scales studied by DLS and FPR, but without benefit of the probability distribution considerations mentioned above. It was proposed that DLS sees rapid movements of unattached chains, and slow motions of chains in temporal clusters, as distinct diffusive modes, while on the much longer FPR time scale all diffusing molecules have spent some time being unattached and some time in a temporal cluster, so for FPR only an average D was observed. Similar reasoning could be applied herefor example, to explain why the FPR decays in Figure 3 are generally narrower and quicker (after adjusting for spatial frequency) than their DLS counterpartsbut this simple picture is clouded by the I

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to a single-particle displacement distribution function. Optical probe diffusion in unlabeled HPC has been studied extensively.25,36 Optical probes as studied with DLS have polymodal spectra, with a fast mode similar to the fast mode of HPC. The decay rates of the slow modes seen with DLS, FPR, and optical probe diffusion all decrease markedly with increasing c. However, optical probes are rigid bodies. Any sort of diffusive mode that requires the diffusing body to rearrange its shape, the way a long snake might slither through a bamboo grove, cannot be performed by a rigid optical probe such as a polystyrene sphere. As the DLS slow modes for spherical probes and for the polymer itself are qualitatively similar, it is reasonable to infer that polymer chains in our systems move via the sorts of motion that are accessible to rigid spheres. It is sometimes proposed that the slow mode observed in DLS experiments arises from polymer self-diffusion. For the system considered here, our measurements (cf. Figure 3) suffice to reject this interpretation. The self-diffusive modes observed by FPR are perhaps 10-fold faster than the DLS slow mode. Finally, proposals that the ultraslow mode represents HPC aggregates that are removed or dispersed by our processing steps and not by chain labeling are multiply rejected by experiment. On one hand, the processed but unlabeled material shows the ultraslow mode. On the other hand, measurements of the static light scattering intensity as a function of concentration find that I/c decreases slowly with increasing c, with no sign of the jump that would be expected near 0.4 wt % if one were seeing concentration-induced aggregation that sets in near that concentration.25

effects of time scale, distance scale, and correlation functions are challenging to resolve. How is it possible for minor chemical modification to abolish the ultraslow mode? From the overwhelming size ratio between the polymer and the fluorophore, one reasonably infers that the unlabeled and labeled chains should have very nearly the same hydrodynamic properties. One possibility is that the dye somehow accelerates the transfer of polymer chains into and out of the vitrified regions, so that even on the shorter DLS time scales chain and vitrified region modes average together. If this were to occur, the DLS slow mode would be slower for the labeled HPC than for the unlabeled HPC, exactly as observed. Alternatively, the hypothesized temporal aggregates that lead to the ultraslow mode are extremely fragile, so that the labels somehow disrupt the temporal aggregates. D of the free dye differs from D of the slow mode by close to 4 orders of magnitude, so there is no obvious mechanism whereby remnant free label could directly alter the visibility of the slow mode. As a third possibility, our chemical reaction attaches to each HPC molecule an average of one fluorescein residue. However, there are multiple binding sites on each polymer molecule. Statistical fluctuations guarantee that some polymer molecules will have multiple bound dye molecules. Interactions between multiple dye molecules bound to a single HPC molecule could then form transient rings or transient, very long end-to-end-linked chains, but it is unclear how such structures could abolish the ultraslow mode. It might be proposed, correctly, that during a photobleaching session in FPR there are two concentration gradients, one each for the bleached and unbleached polymer molecules, so that therefore FPR is a diffusion measurement in which the excess chemical potential gradients drive diffusion, exactly in a classical mutual diffusion experiment. However, in an FPR experiment the total concentration of bleached and unbleached polymer molecules is everywhere the same. So long as the bleached and unbleached polymer molecules are physically very nearly the same, they all make very nearly the same contributions to the excess chemical potential, and to the gradients in the excess chemical potential, of each polymer molecule. At each point the gradients in the concentrations of the bleached and unbleached polymer molecules are necessarily equal in magnitude and opposite in direction, so their resultant gradients in the excess chemical potential of a given polymer molecule sum to zero and cancel. Only self-diffusion contributes to diffusion of bleached or unbleached molecules in an FPR measurement.44,45,59 A plausible interpretation of the observed mode structure may be given in terms of the correlation functions to which FPR and DLS are sensitive. First, FPR is entirely determined by the single particle displacement distribution function P(Δx,t), so the modes in the FPR spectrum necessarily show aspects of single-scatterer self-diffusion. P(Δx,t) has a component that relaxes rapidly and (especially at higher concentrations) another component that relaxes much more slowly. Second, DLS spectra are relaxed by a weighted average of P(Δx,t) and the two-particle displacement distribution function distribution function P(2)(Δx,t,R1,2). The fast mode in the DLS spectrum, being the same as in the FPR spectrum, corresponds to P(Δx,t). The slow mode of the DLS spectrum, which appears to be rather different than the slow mode of the FPR spectrum, must then correspond to the P(2)(Δx,t,R1,2). FPR measurements are rationally compared with DLS studies of optical probe diffusion. Both methods are sensitive

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CONCLUSION

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ASSOCIATED CONTENT

The desirable features of FPRamong them insensitivity to dust, small sample size, selectivity that allows even small probe polymers to be seen against an invisible background of large matrix polymers, relative insensitivity to the thermodynamic driving forces that govern DLS, and the focus on the single particle distribution functionare purchased at the price of a fluorescent label. The same is true for FCS. It is not known whether other polymers will be affected by dye, but care is required when labeling polymers intended for polymer dynamics investigations. DLS can be a sensitive way to test whether the system is altered by the label. Even when DLS and FPR are used to study the very same labeled samples, differences may be encountered. In addition to the effects of very different distance and time scales, the methods fundamentally probe different displacement distribution functions. In very dilute solutions, this difference is small, but that still leaves the different response of the methods to polydispersity, which has been ignored in the above analysis but depends on labeling scheme.

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01669. Figures X1−X5 and Table X1 (PDF) J

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (K.A.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the support of NSF-DMR 1005707 (P.R.) and The Donors of the Petroleum Research Fund (P.R.); NSF-REU CHE-0648841 and CSU’s Engaged Learning Undergraduate Research Program (R.M.); CSU’s 2005 Faculty Research Development Award and 2006 COTTRELL Science Award CC6821 (K.A.S.); NSF-IMR 0526949 (R.C.).

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DOI: 10.1021/acs.macromol.5b01669 Macromolecules XXXX, XXX, XXX−XXX