Contrast Factors

The contrast factor k may be written in a slightly different form  

Techniques using the contrast factor have been applied to test the theoretical prediction of the behavior of a polymer in solutions (see Chapter 5)  

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Small angle neutron scattering measurements were carried out with the PACE diffractometer at the Laboratoire Leon Brillouin, (CE Saclay, France). The q range observed was 3.4 lO" to 0.2 A l. Samples were prepared in deuterated instead of ordinary water to achieve a suitable value for the neutron contrast factor. 

I(q) is the intensity at wave vector q, (bjjr-bp) is a contrast factor arising from the difference in scattering lengths of deuterated and protonated species, M is molecular weight of the deuterated polymer, c is concentration in gm/ml, S(q) is a particle scattering factor, and A contains machine constants, detector efficiency, and other fixed quantities. For the purpose of the current study, S(q) is the quantity of significance, and it is given by... 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

According to [291] the intensity scaled to the scattering contrast factor Ap may be written as ... 

Figures 6.2 and 6.3 compare the SANS scattering responses, for RH=0.11 and RH=0.87, of the dry carbon, the carbon+TD, carbon+0 0, and carbon + D O+TD. The loss of intensity due to the reduced contrast factor is evident in the region 0.2A

In the opposite sitnation, when the carbon is exposed to D O at RH=0.87 in association with a 1 1 (v/v) mixtnre of TH and TD (THD), a large difference is observed with respect to the case where the tolnene is fnlly deuterated (Fig. 6.4, curve 4). I. shows a substantial increase, to 0.11 cm , and the contrast factor also increases as a result of the replacement of deuterium atoms in the toluene (fep=H-0.66 X10 cm) by protons (b -0.37 x 10 cm). It is also noticeable that the accumulated intensity at the peak around 1.7 A is much weaker owing to the reduced scattering power of the TH molecules. 

The fundamental idea is to determine the solubility parameter of the polymer, and then to use tabulated results to identify a number of solvents that have solubility parameters close to this value. The list of potential solvents is then narrowed to two or three candidates. Solvents that are too volatile, too toxic, too flammable, too expensive, and so on can be removed from the list. Other criteria would depend on the nature of the studies to be pursued. If the objective is to carry out light-scattering measurements, the need for maximizing the contrast factor would make the index of refraction of the solvent an additional important consideration. 

The contrast factors (dn / dT)cp and (An / dc)Tp, taken at the wavelength of the readout laser, are the only quantities which must be measured in separate experiments. For this purpose, a Michelson interferometer has been developed, whose mirrors can be scanned over X/2 [43] (Fig. 6). 

The 5-function accounts for the fast contribution from the temperature grating. The normalization to the amplitude of the temperature grating in Eq. (24) takes away the need for difficult absolute intensity measurements for the determination of Dt and S7-, which otherwise would be necessary. The only quantities that must be obtained from separate measurements are the two contrast factors (d n/d T)cp and (d n/d c)Tp. All the transport coefficients Dlh,D,Dv and ST can be extracted from the sample response to suitably chosen excitation patterns. 

Obviously, there is a strong isotope effect, and the Soret coefficient is reduced in the case of (C6D6), which has exactly the same mass as the other component, the cyclohexane. The measurements were conducted at T=21°C, q = 8030 cm1, and the contrast factors are almost identical for both mixtures. The sign of ST is such that benzene migrates towards the warmer regions [52]. 

When such different techniques as in Table 1 are compared, there is always the problem of different sensitivities for different aspects of the distribution. If, for example, information about the high molar mass tail is of importance, PCS may be the method of choice. It may also be incorrect to regard the SEC distribution as the true molar mass distribution as it may suffer from calibration problems, solute-column interactions, peak broadening, and a molar mass dependence of the contrast factor d n/dc, and hence the detector sensitivity. 

The contrast factors have been measured interferometrically [87] and with an Abbe refractometer, respectively. The sample is contained in a fused silica spectroscopic cell with 200 pm thickness (Hellma). The sample holder is thermostated with a circulating water thermostat and the temperature is measured close to the sample with a PtlOO resistor. The amplitude of the temperature modulation of the grating is well below 100 pK and the overall temperature increase within the sample is limited to approximately 70 mK in a typical experiment [91], which is sufficiently small to allow for measurements close to the critical point. 

For zinc pyridine reference complexes in toluene AS values are about 50 J K 1 mol-1, and hence the upper limit for EM is in the region of 400 mol dm-3. For the trimer 10.7, the experimental value (in CH2CI2) of 100 mol dm-3 is close to the theoretical maximum, which suggests that this system is nearly geometrically optimal (i.e. the monomer units are highly complementary). In contrast, factors such as ring strain may contribute to the lower value for the dimer 10.6. 

Here also, the various self and cross correlations have to be weighed by the contrast factors in order to obtain the scattered intensity. 

Elimination of intense scattering contributions requires high-quality optical components. The finite contrast factor of polarization optics (106-107) and possible higher-order processes (e.g., six-wave mixing) to the nonlinear scattering limit the suppression potential (39). 

The intensity of the scattered neutron beam is related to numerous parameters, but a full discussion of these is not appropriate here. There are several texts that give a more detailed account of these variables and of SANS in general (e.g., Richards, 1989). However, one of the most important parameters in the determination of the pattern and intensity of scattered radiation is the contrast factor, as defined in Eq. 2,... 

If the experimental data are now plotted In the form of ln[l(q)] vs. q, a straight line Is expected, with the slope being a third of -. Note that l(q) Is proportional to the contrast factor (Ap ). This observation leads to the Bablnet principle which states... 

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