A Surrogate for Debye-Waller Factors from Dynamic Stokes Shifts.

We show that the short-time behavior of time-resolved fluorescence Stokes shifts (TRSS) are similar to that of the intermediate scattering function obtained from neutron scattering at q near the peak in the static structure factor for glycerol. This allows us to extract a Debye-Waller (DW) factor analog from TRSS data at times as short as 1 ps in a relatively simple way. Using the time-domain relaxation data obtained by this method we show that DW factors evaluated at times ≥ 40 ps can be directly influenced by α relaxation and thus should be used with caution when evaluating relationships between fast and slow dynamics in glassforming systems. Keywords: Time Resolved Stokes Shifts, Solvation, Debye-Waller factor, Mean-squared-displacement, Alpha relaxation The Van Hove single particle correlation function, Gs(r,t) gives the probability of finding a particle at a position r at time t relative to the position of that particle at time t=0. Thus, this function and its Fourier space analog, the incoherent intermediate scattering function, Fs(q,t), contain significant information about dynamics and thermodynamics of solids and gasses1. A time-weighed average of Fs(q,t), the Debye-Waller (DW) factor is commonly used for characterization of condensed matter systems. The concept of a Debye-Waller (DW) factor, originally defined as the mean-squared displacement 〈u2〉 of atoms around equilibrium positions in a crystal2, has been extended to a harmonic oscillator approximation of motion in amorphous materials, and has emerged as an important parameter in theories of and experiments on supercooled liquids and glass3–4. DW factors and changes in their behavior have been connected in one way or another to virtually all of the classical characteristics of these systems, including the Kauzmann temperature (TK), crossover temperature (Tc), glass transition temperature (Tg), viscosity and fragility6. This striking relationship between long-time and short-time dynamics seems also to be manifest in the relationship of DW factors to the dynamic coupling between proteins and solvent7, between protein function and dynamics8 and between host dynamics and stability of proteins encapsulated in sugar-based glass.9–10 This latter context is of considerable practical importance, as protein instability accounts for significant losses in the biopharmaceutical industry11. Both gaining a better fundamental understanding of this striking relationship and exploiting it for screening protein-stabilizing glasses requires the ability to routinely and reliably measure DW factors without access to a nuclear reactor or need for Mossbauer-active elements in the sample. Until now, this has not been possible. DW factors can be obtained from x-ray11 and neutron12 scattering, as well as from Mossbauer spectroscopy.7 Below we show that one can also obtain a surrogate for DW factors in glasses through time resolved fluorescence Stokes shifts (TRSS). Unlike x-ray, neutron, and Mossbauer scattering, Stokes shifts are not sensitive to oscillatory motion per se, but respond to the same molecular rearrangements that cause decay of the correlation functions from which 〈u2〉 values are derived. The TRSS of a probe molecule senses changes in position and orientation of host molecules through their influence on the electric field at the probe. Upon S1 ← S0 photoexcitation of a dye molecule there can be a significant change in dipole moment. Concomitant with this change, nearby host molecules begin to reorganize to accommodate the new local electric field. As this occurs the energy gap between the S1 and S0 state of the probe becomes smaller, resulting in a time-dependent red shift in the emitted fluorescence. The magnitude and time dependence of the red shift depends on the solvents relative permittivity (er) and the timescale for relaxation in the material respectively.14 TRSS decays presented here were measured in the time-domain subsequent to a sub-ps excitation pulse by spectrally dispersing fluorescence from rhodamine 6G (R6G) onto an 8-channel avalanche photodiode array (id150 Quantique)*. Detector output was digitized using a Becker and Hickl SPC 830 module. The convolved time resolution of the detector and digitizer was 54 ps. 515 nm excitation light was generated by pumping a 10 cm length of photonic crystal fiber (Crystal-Fiber) with 10 nJ pulses of ~790 nm, 30 fs from a cavity dumped Ti: Sapphire oscillator (Kapteyn-Murnane Laboratories Inc.). Fluorescence was collected in a 180° geometry, and dispersed onto the detector as eight equally spaced emission bands in the range (533 to 600) nm, covering the major fluorescence peak of R6G. The transmission efficiency of each spectral channel was calibrated against steady-state fluorescence of R6G in glycerol at 300 K. The instantaneous fluorescence energy was calculated as the first moment average of the signal from each detector channel; v(t)=∑i vini(t)/∑i ni(t), where n is the number of detector counts in time bin t, v is the average energy of the light detected in each channel, and i runs from 1 to 8. Assuming a linear relationship between molecular relaxation and a change in fluorescence energy, a decay function, Φ(t), was defined in the usual way as15: Φ(t)=v(t)−v(∞)v(0)−v(∞) (1) where v(∞) is obtained from the slightly temperature-dependent long-time values of v(t). In this work, v(∞) obtained at the lowest reliable temperature is used at all lower temperatures. The value of v (0) = 18100 cm−1 is obtained from steady-state Stokes shifts in a series of solvents of decreasing polarity, as described in Ref 16.