TDFRS

GPC/SEC, MALDI-MS, membrane osmometry, vapour pressure osmometry, viscometry, light scattering, TDFRS, SAXS, SANS, SEC-HPLC, SEC-MS, SEC-IR, FFF, ultracentrifugation, MALDI-TOF-MS, NMR, capillary electrophoresis... 

TDFRS Thermal diffusion forced Rayleigh scattering... 

Figure 17 Molar mass distributions of polystyrene in ethyl acetate obtained by dynamic light scattering (photon correlation spectroscopy, PCS) and TDFRS with short and long exposure time tp. The dashed curves represent the distribution as determined by SEC. Reproduced with permission from Rossmanith and Kohler [107]. Copyright 1996 American Chemical Society.

Table 4 Molar mass averages (in kg/mol) calculated from the molar mass distributions obtained by SEC, short and long exposure TDFRS, and DLS...

Total dietary fiber (TDF) content of potato dry matter is determined according to the AACC (2000) method 32-05 following the total dietary fiber assay procedure (Megazyme k-TDFR 01/05). This is a gravimetric method that is simpler and faster than other analysis methods. In addition to total dietary fiber content, both soluble and insoluble dietary fiber content can be determined by this method. 

TDFRS allows for experiments on a micro- to mesoscopic length scale with short subsecond diffusion time constants, which eliminate almost all convection problems. There is no permanent bleaching of the dye as in related forced Rayleigh scattering experiments with photochromic markers [29, 30] and no chemical modification of the polymer. Furthermore, the perturbations are extremely weak, and the solution stays close to thermal equilibrium. 

In this contribution, the experimental concept and a phenomenological description of signal generation in TDFRS will first be developed. Then, some experiments on simple liquids will be discussed. After the extension of the model to polydisperse solutes, TDFRS will be applied to polymer analysis, where the quantities of interest are diffusion coefficients, molar mass distributions and molar mass averages. In the last chapter of this article, it will be shown how pseudostochastic noise-like excitation patterns can be employed in TDFRS for the direct measurement of the linear response function and for the selective excitation of certain frequency ranges of interest by means of tailored pseudostochastic binary sequences. 

The main parts of the TDFRS setup and some experimental aspects will be discussed in this section without going into details of data evaluation and signal modelling. A derivation of the equations for a phenomenological description will be deferred until later. 

A TDFRS setup is, in its main components, almost identical to standard forced Rayleigh scattering (FRS) setups as described in many publications [28, 31, 32, 33, 34]. The symmetric arrangement as employed for the more recent work in our group is sketched in Fig. 1. 

A distinction between homodyne and heterodyne detection must be made in optical scattering and diffraction experiments. Without careful treatment of the background, there is always the risk of mixed or unknown coherence conditions, and the diffusion coefficient determined from such data may be off by a factor of two. At least for the signal and background levels present in TDFRS, heterodyne detection is always superior to homodyne, especially since the heterodyne signal, contrary to the homodyne one, turns out to be very stable against perturbations and systematic errors. Even under nearly perfect homodyne conditions the tail of the decay curve is almost unavoidably heterodyne [34]. 

Ideally, there is no phase shift between the reference and the diffracted beam (0=0), and, since TDFRS is completely nondestructive without dye bleaching, the signal can be accumulated over almost arbitrary times. In order to maximize the heterodyne signal amplitude, some means for phase adjustment and stabilization are needed. Without such active phase-tracking, 0 would have some arbitrary value and would slowly drift away due to the almost unavoidable slow thermal drift of the whole setup. 

An elegant phase-tracking and stabilization procedure has been developed, which utilizes the fast signal contribution from the heat mode with a relaxation time of the order of 10 /as, which is always present in TDFRS measurements [34]. 

Some constraints apply for the measurement of mass and thermal diffusion by TDFRS, which originate from excessive sample heating at high laser powers, the resulting onset of convection, and the need to avoid boundary effects at the cuvette windows when the grating constant becomes comparable to the sample thickness. The problem of optimization of the experimental boundary condi-... 

The phenomenological description of signal generation in TDFRS is, in principle, straightforward [27,28,33,44,45]. The primary source of excitation is the optical interference grating formed by the two writing beams, which intersect under an angel 6 inside the sample ... 

Fig. 7. Sketch of the geometry of the TDFRS-setup with the two writing beams intersecting under an external angle 6. Also shown is a cross section through the sample cell...

Equation (26) can directly be employed for the description of TDFRS experiments on simple liquid mixtures, which are characterized by a single diffusion time r. 

In this section it will be outlined how the different molar masses contribute to the TDFRS signal. Of especial interest is the possibility of selective excitation and the preparation of different nonequilibrium states, which allows for a tuning of the relative statistical weights in the way a TDFRS experiment is conducted. Especially when compared to PCS, whose electric field autocorrelation function g t) strongly overestimates high molar mass contributions, a much more uniform contribution of the different molar masses to the heterodyne TDFRS diffraction efficiency t) is found. This will allow for the measurement of small... 

Fig. 11. Relative statistical weights for polydisperse polymers as a function ofM. The line with slope 1 corresponds to PCS, the shaded area to TDFRS. The long exposure limit corresponds to the random coil value >=1/2...

From Eq. (39), the amplitude factors in Eq. (33) can be expressed as a function of molar mass and concentration, and the normalized decay function for a dilute polymer solution, which is et(t) in case of TDFRS andgj (t) in case of PCS, can be written as... 

The exponent a depends on the type of experiment performed, and it has been introduced to allow for a consistent description of the different measurements. a=l for PCS, a=0 for short exposure TDFRS, which guarantees strictly concentration proportional amplitudes, and a=b for long exposure TDFRS. a = 1 leads directly to the z-average diffusion coefficient (D)cM characteristic for... 

This situation is illustrated in Fig. 11, where the relative statistical weights and, hence, the amplitudes contributing to the multiexponential decays, are plotted as a function of M. The curves are arbitrarily normalized to unity at M = 105 g/mol. The strong underestimation of components with low molar mass in PCS is obvious. The shaded area is accessible to TDFRS, and the two limiting cases correspond to short and long exposure times. 

Fig. 12. Accessible q - t- range for typical PCS and TDFRS experiments. The lines correspond to diffusion coefficients D = (q2t) 1 of 10 3,10 7,and 10 11 cm2/s...

Figure 12 gives an overview over the accessible q- and f-range for typical experiments. The numbers used are 0.5° < 6 < 15°, X = 488 nm (TDFRS) and 15° < Q < 150°, X = 647 nm, n=1.5 (PCS). The wavelengths are the ones routinely employed in out laboratory. A change of wavelength does not significantly change Fig. 18. Note that...

The boundaries in Fig. 12 are soft, and PCS can be extended to lower and TDFRS to higher q, however at the expense of increased experimental problems, like stray light from the cuvette in case of PCS and a loss of signal amplitude, which is proportional q 2, in the case of TDFRS. 

This section is concerned with the information that can be obtained by TDFRS about the molar mass and size distribution and the various molar mass averages. 

Once the scaling relation of Eq. (39) is known, the molar mass distribution can, at least in principle, be obtained from a Laplace inversion of the multi-exponential decay function as defined in Eq. (40). At this point, the differences between PCS and TDFRS stem mainly from the different statistical weights and from the uniform noise level in heterodyne TDFRS, which does not suffer from the diverging baseline noise of homodyne PCS caused by the square root in Eq. (38). 

Again, a = 1 for PCS, a = 0 for short exposure TDFRS, and a = b for long exposure TDFRS, and b is the scaling exponent from Eq. (39). For TDFRS with arbitrary exposure time, c(M) is given by... 

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