The Contrast Factor

The choice of the most appropriate scattering technique depends upon two main requirements the contrast factor and/or the need for time-resolved experiments. Presently, time-resolved experiments can be essentially carried out with X-rays thanks to the extremely high flux of photons delivered by synchrotron radiation facilities. In some high flux neutron facilities, such as ILL in Grenoble, some apparatuses allow one to do near-time-resolved experiments under certain experimental setup conditions (low sample-detector distances). 

The contrast factor is by far the characteristic difference between neutrons and X-rays. As was said in Section 4.1, these particles do not see matter the same way. 

Another item of interest is the low variability in scattering amplitudes of most of the atoms. These amplitudes do not depend on the atomic number and are rather random. Values are, in most cases, somewhere between 0.2 a 1 (10 cm), with a few negative values often found for isotopes (Table 4.2 shows some of the values for atoms often found with polymers). Unlike X-rays, heavy atoms do not scatter very differently from lighter atoms. Note that scattering lengths are only determined from experiments. 

The calculation of the contrast factor must take into account molar volumes of each species, as the scattered intensity is always by unit of volume. The contrast factor Kf  

TABLE 4.2 Values of Scattering Lengths and Incoherent Scattering Cross Sections = 4ir i ) for Various Atoms 

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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-... 

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). 

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. 

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

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... 

C 7jfc/ is the average contrast factor, which accounts for the main effect of anisotropy." It can be shown that the contrast factor can be written in terms of the fourth order crystallographic invariant for the Laue group of the studied phase. In other words, can be written for any lattice symmetry, for instance using the invariants given by Popa. ... 

The experimental form factor P( ) shown in Fig. 12a can be expressed as P q) = [bcFc q,rc) + bsFs q,rc,R )], where bc,bs are the contrast factors for the core (c) and shell (5) with core radius and overall micelle radius R, whereas Fc q), F q) aiQ the scattering amplitudes of the core and shell, respectively. Under core contrast conditions (Z s 0), the expected first minimum for the compact sphere at high q values falls outside the ("/-range, whereas under shell contrast conditions the power-law behavior arising from blob (swollen PEO shell) scattering is observed. Hence, the dual colloid-polymer character of the particle is clearly reflected in Fig. 12a. 

We now consider the behavior of the confined iBA-rich liquid mixture (mean mass fraction of iBA wa = 0.54) along path II in Fig. 4.18. Density profiles from the model calculation for four different temperature are shown in Fig. 4.22. At this composition (mean mass fraction ica = 0.54) the mixture does not undergo phase. separation (see path II in Fig. 4.19). As water is the minor component of the mixture and is strongly preferred by the substrate, almost all water accmnulates in the first and second layer at low temperatures (T < 30 C), whereas the core liquid is almost depleted of water (see Figs. 4.22(c) and 4.22(d)]. For this reason the contrast factor kp attains a high and nearly constant value at low temperatures, as can be seen in... 

Wilkens [WIL 70] followed by Khmanek and Knzel [KLI 88] estabhshed the expression of the contrast factor based on diagrams similar to the one found in Figure 5.5. Generally, this factor % is expressed as follows ... 

This is due to the fact that the contrast factor in the underlying Rayleigh equation [10] depends on the density difference and compressibility difference of a dispersed matter in the surrounding media. In particular, if a liquid is dispersed in another liquid, the value of the contrast factor is about 0 (i.e., there is almost no density difference and, thus, no compressibility difference between liquids) and a solid dispersed in a Hquid increases the value just slightly to about 1-2. However, in the case of air in water (or blood) this contrast factor raises to 10 ", which makes dispersed air bubbles a perfect contrast agent for ultrasound. 

Figure 6.8 Points are the values of app) determined with a diblock copolymer of deuterated polystyrene and hydrogenous polyisoprene, dissolved in a solvent that is a mixture of hydrogenous and deuterated cyclohexane. The contrast factor Y, defined by (6.71), depends on the proportions of the two isotopic species in the solvent. The curve is the theoretical parabola calculated according to Equation (6.70) with best-fitting val-...

A systematic SANS study of the PS-PA block copolymer has been achieved. The PS-PA co-polymers were obtained through thermal annealing of the diblock precursor PS-PPVS (polyphenylsulfoxide), a detailed description of the synthesis and preparation of the PA sequence free from defect can be found in references [54-55], To get a perfectly dispersed micelle structure in solution, the PA must be short and so the study was devoted to the diblock PS64-PA1 [56-58], The contrast factor of the PS unit is about 95 x 10 cm , while the contrast of the PA chain is negligible. As given in Section 1.2.2 the structure factor of a particle varies as a function of its geometrical shape. For instance, a... 

The contrast factor, je, is defined as the slope of the natural logorithm vs. % resist thickness remaining curve measured at the resist/substrate interface after the convention of W. Arden et.al., in Reference 1 above. 

The contrast factor, which gives the ratio of maximum to minimum transmission ... 

A higher finesse F caused larger reflectivities of the reflecting films not only decreases the bandwidth but also increases the contrast factor. With R — 0.98 F = 4R/(l — R) = 9.8 x 10, which means that the intensity at the transmission minimum is only about 10 of the peak transmission. 

While the expression (2.21) might be more familiar to the particle sizing community, Eq. (2.22) is customary in colloid and polymer science. The contrast factor H contains the refractive index increment dmsuJQcm, which is independent of size and concentration for very fine particles. 

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