4196
J. Phys. Chem. 1992, 96, 4196-4200
Line Shape and Line Width Effects in Optical Probe Studies of Glass-Formlng Liquids George D.J. Phillies* and Delphine Clomenil Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 (Received: November 11, 1991; I n Final Form: February 7, 1992)
Quasi-elastic light scattering spectroscopy was used to observe optical probe particles diffusing through water:erythritol, water:glycerol, and neat glycerol. High-precision measurements of the spectral line width (as characterized by the first cumulant Kl of the spectrum S(q,t)) and of the spectral line shape (characterized by the second spectral cumulant K2)were made. The objective was to resolve discrepancies between previous studies of Kl for probes in water:glycerol and to search for the appearance of memory function effects in probe diffusion through viscous fluids, as revealed by changes in line shape. K2 is virtually independent of T and solution composition, ruling out memory-function/line shape changes as an explanation for disagreements between prior studies of K I .
Introduction Transport in viscous and glass-forming liquids has become a significant topic in modern chemistry, with extensive studies of glasses being made via experiment, analytic theory, and computer simulation. For a wide variety of fluid substances, viscosity increases rapidly with decreasing temperature, often diverging quasi-exponentially as a glass temperature Tg is approached.] Under some conditions anomalies are apparently observed in translational transport through viscous liquids. For example, Heber-Greene,2 who observed small molecules moving through viscous solutions of larger molecules, found that the drag coefficientf of the small molecules scaled with viscosity as f qa for a in the range 0.67 f 0.03, rather than the a = 1 observed for probes in low-viscosity, small-molecule liquids. Nonlinear relationships between f and q have also been reported for diffusion by n-hexane and naphthalene in compound oils3 and for the electrophoretic mobility of simple electrolytes in sucrose4 and mannitolS solutions. On the other hand, Berner and Kivelsod used NMR spectroscopy with paramagnetic enhancement to measure D of di-tert-butyl nitroxide in a variety of solvents, finding f7-I varied by no more than a factor of 4 when q was varied over 4 orders of magnitude. Translational diffusion of large probes often follows the Stokes-Einstein equation
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chenko and Litvinov proposed that their data may be explained by writing
in which 5 is a new length scale-a dynamic correlation range-in the fluid. Physically, one could describe 5: as a distance over which fluid motions are correlated, so that a moving particle of radius R drags with it a fluid skin having an effective thickness 4. To explain their data, Kiyachenko and Litvinov estimated 5 0 at high temperature but 5 = 140 A at low temperatures. One notes that data based on a single sphere size does not provide direct evidence that eq 2 is preferable to eq 3 for K > 1.
(3)
Wiltzius and van Saarloos9duplicated aspects of Kiyachenko et al.3 experiment, measuring D of three different polystyrene latices, all in neat glycerol. Reference 9 reports that D of all three species scaled with temperature in exactly the same way, so that, e.g., D I / D zis independent of T. The constancy of this ratio is inconsistent with eq 2 but is consistent with eqs 3 or 1. The possible utility of still further experiments on probes in water:glycerol and other glass-forming liquids-experiments that are reported below-is suggested by findings on diffusion by D = - kBT mesoscopic probe particles in polymer solutions. In polymer (1) 6a9R systems (which all include several length scales) the StokesEinstein equation sometimes fails badly.1° However, the degree. Here kB is Boltzmann's constant, Tis the absolute temperature, of retardation of probe particles by polymer solutions is largely R is the physical radius of the particle, and q is the solution independent of probe radius, except for very small probe particles, viscosity. For example, this authors used QELSS to measure D so eq 2 does not describe non-Stokes-Einsteinian behavior in of bovine serum albumin and 0.091-pm polystyrene spheres in polymer solutions. A demonstration that &R/T is independent water:glycerol and water:sorbitol mixtures. 9 of the solutions was of R, as so clearly provided by ref 9, thus does not entirely exclude measured with Ubbelohtie and Cannon-Fenske (capillary) visthe possibility that interesting non-Stokes-Einsteinian effects OCCUT cometers. The experiments covered a wide range of temperatures in water:glycerol systems. and solution compositions, T/q being varied by more than 3 orders Experiments on probe diffusion in glycer01*~~ in some ways of magnitude. D for either probe obeyed eq 1 to good accuracy. resemble work on probes in hydroxypropylcellulose (HPC): On the other hand, Kiyachenko and Litvinov' measured q of water."-15 In both systems, independent studies failed to agree neat glycerol and D of a dilute latex suspended in the same liquid, on the validity of eq 1. For HPC:water, a possible reconciliation finding that eq 1 fails by up to 60% at low temperatures. In these of results was proposed by Russo and collaborators,12who found experiments, in viscous liquids probes were found to move more conditions under which light scattering spectra of HPC:water slowly than expected from the macroscopic viscosity. This reduction in D below D T/q is contrary in sense to above-n~ted~-~ contain a fast-decaying mode not seen in dilute solution. Russo et a1.12 show that the apparent degree of failure of the Stokeshistorically-known anomalies in viscous-liquid transport. KiyaEinstein equation can than depend on one's data analysis method, because different methods of interpreting QELSS spectra respond
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( 1 ) Angel, C. A. In Relaxations in Complex Systems; Ngai, K. L., Wright, G. B., Eds.; National Technical Information Service: Springfield, VA, 1985. (2) Heber-Greene, W. J . Chem. SOC.1910, 98, 2023. (3) Hiss, T. G.; Cussler, E. L. AIChE J . 1973, 14, 698. (4) Stokes, J. M.; Stokes, R. J . Phys. Chem. 1956,60, 217. ( 5 ) Stokes, J. M.; Stokes, R. J . Phys. Chem. 1958, 62, 497. (6) Berner, B.; Kivelson, D. J . Chem. Phys. 1979,83, 1401. (7) Kiyachenko, Y. F.; Litvinov, Y. I . Pis'ma Zh. Eksp. Teor. Fiz. 1985, 42, 215. (8) Phillies, G. D. J. J . Phys. Chem. 1981, 85, 2838.
0022-36S4/92/2096-4196$03.00/0
(9) Wiltzius, P.; van Saarloos, W . J . Chem. Phys. 1991, 94, 5061. (10) Lin, T.-H.;Phillies, G. D. J. J . Colloid Interface Sci. 1984, 100, 82. (1 1) Mustafa, M.; Russo, P. S. J . Colloid Interface Sci. 1989, 129, 240. (1 2) Russo, P. A.; Mustafa, M.; Stephens, L. K.; Cao, T. J . Colloid Interface Sci. 1988, 122, 120. (13) Yang, T.; Jamieson, A. M. J . Colloid Interface Sci. 1988, 126, 220. (14) Brown, W.; Rymden, R. Macromolecules 1986, 19, 2942. (15) Brown, W.; Rymden, R. Macromolecules 1987, 20, 2861.
0 1992 American Chemical Society
Optical Probe Studies of Glass-Forming Liquids differently to weak, rapidly-decaying spectral modes. An objective of this paper is to see whether the anomalies in water:glycerol data have the same source as the anomalies in the water:HPC data. Following the leads suggested by Russo et a1.,I2 we reexamined light scattering spectra of water:glycerol:probe mixtures, searching for temperature-dependent fast or slow decay modes. An second decay mode, combined with fortuitous choices of photon correlator operating conditions in different laboratories, might explain the discrepancies between refs 8,9, and 7. Extra decay modes, if they could be characterized reliably, might also give information about glass transitions in solution. Experimental Methods Quasi-elastic light scattering spectroscopy (QELSS)was used to observe the diffusion of latices through several viscous and glass-forming liquids, including glycerol, water:glycerol, and water:erythritol. The probe particles were carboxylate-modified polystyrene latex spheres (Seradyne, Inc.) with nominal diameters of 38 and 67 nm. The actual hydrodynamic diameter (as inferred from light scattering spectra) of the spheres in a given sample may differ significantly from the nominal diameter, but the actual diameter is highly reproducible from measurement to measurement. Solutions were prepared with 18-MQ conductivity water, itself clarified by passage through a 0.45-pm pore size filter, and were clarified by passage through Nuclepore filters having pore sizes in the range 0.1-1.0 pm. Polystyrene spheres were transferred from stock to the clarified solutions. The light scattering spectrometer employed a 20-mW He-Ne laser and a 144-channel Langley-Ford digital correlator. The effort here is to search for multiple time scale effects in spectra taken at a single scattering angle. Since the solvent molecules are all much smaller than X/200, spatial distance scales are not expected; the scattering angle was left fixed at 90°. Cells were held in a copper block provided with slits for the passage of the incident and scattered beams. Temperatures were scanned over the range 10-60 OC using an automatic computer-controlled circulating bath (stability f50 mK). Viscosities of water:erythritol mixtures were determined by means of capillary viscometers mounted in a stirred, regulated bath. Data analysis employed the method of cumulants, in which light scattering spectra S(q,t) are expanded as a power series N Ki(-t)' -21 log (S(q,t) - B ) = 1-0 ET 1. (4) Here B is the baseline, determined either (i) from correlator delay channels placed ca. 1000 correlator channel widths beyond the final channel or (ii) from a theoretical estimate P2/n based on the total number of received photocounts P and the duration n of the experiment in units of the correlator channel width. The Ki are the cumulants. Equation 4 is linear in the Ki, so the Ki were obtained by a weighted linear least-squares fit to measured spectra. N is the truncation value of the expansion. Each spectrum was fit to eq 4 for N = 1,2, 3, .... With increasing N , goodness-of-fit parameters (such as the RMS error in the fit) improve and then approach plateau values, sometimes asymptotically. We consistently selected, as the best fit, the smallest N such that the plateau in the goodness-of-fit parameters had nearly been reached. With 144 channels, channel widths arranged so that most channels fell in the first decade of decay, and a signal-to-noise ratio of 1000-2000, most of our spectra were fit adequately with N = 2; in a few cases N = 3 or rarely N = 4 was required. The major interest here was in determining reproducible values for the first cumulant K1 (reported herein as a translational diffusion coefficient D) and the second cumulant K2 (reported as the spectral variance V = 100dK2/K1). Vprovides a description of the spectral line shape. V is zero for a simple exponential spectrum; for more complex spectra, Vprovides an estimator for the degree of nonexponentiality. Specific precautions were taken to avoid artifacts that might lead to spurious values of K2. To avoid spectral distortion caused by the finite time width of the correlator channels, the one-half
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4197
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Figure 1. D as a function of T / s for 0.038-pm polystyrene latex spheres in pure water, showing Walden's rule (D T / T )behavior ( D in ficks; 1 fick = 1 X lo-' cm2/s).
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Figure 2. Variance V against temperature (plotted as T / q ) for 38-nm polystyrene spheres in pure water, showing that the spectral line shape is virtually temperature-independent. Straight line is a linear leastsquares fit.
decay time of S(q,t) was required to be a t least 15 correlator channels. Fluctuations in B (no matter how B is determined) lead to correlated fluctuations in the amplitudes of all correlator channels and thence to artifactual values of K These fluctuations are particularly significant ifS(q,r) is followed out to small values (S(q,t)/S(q,O)
