X ray scattering

Scattering from structures of any size is regular, i.e. takes place at well defined angles, only when the structures are periodic. The scattering is then usually called diffraction. The most important periodic structures suitable for WAXS investigation are crystals, which are periodic in three dimensions. 

X-ray scattering can be meastued by the classical Kratky camera or more modem synchrotron techniques. Technical details can be formd in a munber of books, e.g., those by Guinier and Fouret, Chen and Yip, or Glatter and Kratky.  

second, third. osmotic virial coefficients Mj molar mass of the polymer 

According to the scattering theory of polydisperse polymers (please see, for example, the book written by Strobl), the molar mass of the polymer, Mj, is equal to the mass average, Mv of polydisperse polymers. The exact application of Equation [4.4.47] is to be made again at the scattering angle 0=0. The constant K is now given by  

The second exponential term on the right-hand side is the familiar Debye-Waller factor exp(— W). Now 

in contrast to smectic A and the two-dimensional lattice, for which the displacement-displacement correlation is of logarithmic form, the columnar liquid crystal gives the usual Bragg reflexions. As noted in 6.1, this is borne out by high resolution X-ray experiments on well oriented monodomain discotic strands. 

So far it has been assumed that the length (L ) of the liquid columns is much larger than L. One may similarly consider the opposite situation U L. In this case, it turns out that for the bounded sample 

If the surfaces are free m will have additional terms, which may be analogous to those for a two-dimensional lattice. In any case, it is clear that the mean square fluctuations of the lattice as well as the Debye-Waller factor may be expected to show a certain dependence on the linear dimensions of the sample. 

The mean square amplitude of the orientational fluctuations of the director and the corresponding intensity of light scattering may be worked out in a similar fashion. For example, when the incident and scattered beams are both polarized in the plane perpendicular to the optic axis (or column axis) the scattered intensity is given by 

There is some evidence for the existence of polymorphic forms in PE. Semi-crystalline polymers exhibit amorphous and powder features simultaneously. The amorphous halo is seen when the alkane is raised above its melting temperature. Semi-crystalline PE exhibits the amorphous halo, with sharp lines (but perhaps broader than for the -alkane) superimposed. 

The results of X-ray scattering thus on the one hand show that around 180 K, the Peierls transition to the CDW ground state occurs, and on the other, that above the Peierls transition over a large temperature range, fluctuations between the metallic and the semiconducting states are present. 

One of the earlier blend characterization studies employing SAXS investigated PVC/PCL blends [ 145]. The lamellar structure of crystalline PCL was volume filling up to 50% PVC. The analysis indicated that the amorphous phase was partially phase separated. Further studies using purified materials showed that PVC is incorporated between the PCL lamellae and the amorphous phase is homogeneous down to molecular dimensions [146]. More recent SAXS studies of PVC/PCL blends investigated the effects of molecular weight of both components 

SAXS measurements have also been useful in studies of the morphology and phase behavior of block copolymer/homopolymer blends. SAXS data on PS blends with SB diblock copolymers showed the presence of microdomains, which disappeared as the order-disorder transition was approached with increased temperature [162]. The order-disorder transition and the order-order transition (change in gyroid to cylinder morphology) was investigated with SAXS for PS/SI diblock copolymer blends [163]. The interfacial thickness of PS/PMMA samples with 

Reviews of light and x-ray scattering applied to polymer bends can be found in [167]. 

All these topics will be gathered under the heading X-ray Scattering . 

The second part of this illustration will concern the X-ray Absorption Spectroscopies , mainly XANES and EXAFS the later becoming a routine structural investigation technique of the short range order. 

Nowadays, however, synchrotrons are available that provide million times higher intensity and wide spectrum of the polarized emission. One can use different wavelength ranges and short expositions when studying dynamic processes. Of course, there are not so many synchrotron accelerators all over the world but they have many output beams, as shown in Fig. 5.2, and attached are many experimental stations. Such a work is usually organized at the international level. 

What does an X-ray diffraction experiment bring about In fact, a lot  

Number of diffraction peaks on a diffractogram, their precise positions and the symmetry of the pattern 

The peak amplitudes I and areas A under peaks as functions of temperature, pressure, external fields, etc. 

The peak profile that is the profile of the diffraction intensity I(q) within a particular diffraction spot, which is a functirm of the diffraction angle or scattering wavevector q. The key problem of X-ray analysis is how to relate I(q) to the electron density function or density correlation function that takes into account thermal fluctuations. 

4 Observable implications of the pair correlation function 5.4.1 X-ray scattering 

For normal liquids the characteristic structural distance is of order 1 A. A probe of liquid structure should therefore have a characteristic wavelength X in this range. This calls for using X-rays or light atomic particles as probes. In liquids we are interested in the short-range structure, on the scale of intermolecular distances. This implies the need to apply short range interactions and therefore the use of particles should be limited to neutral ones, such as neutrons. 

Problem 5.1. Discuss the approximation made in Eq. (5.28). (a) Under what conditions can we replace fa by Rc in the denominator, as done (b) Why is it impossible to make this substitution in the phase factor a  

Because kjn = kout I, the scattering angle 9 and the modulus of the scattered wavevector are related to each other by 

The total scattered intensity is the absolute-value square of the scattered amplitude, which is in turn a combination of scattered waves like Eq. (5.28) summed over all scattering centers. The signal at the detector is therefore 

Diffraction is a process composed of elastic or coherent scattering of the x-ray radiation with the dispersion centers of the material, and thereafter these scattered rays interfering between them. 

To understand the scattering process, we will distinguish it from other processes, starting with electron scattering, which was studied in 1906 by J J. Thomson. He found that the intensity scattered by an electron interacting with an x-ray radiation is given by following equation [20,26]  

The Physical Chemistry of Materials Energy and Environmental Applications 

The expression given by Equation 1.51 is normally named the polarization factor. 

The next step is the scattering by an atom. This effect is basically the addition of the scattering of the electron cloud around the nucleus, since each electron in the atom scatters part of the incident radiation in a coherent form in agreement with the Thomson equation. Owing to the fact that the electrons in an atom are located at different points within the atom, and the fact that the x-ray wavelength is of the same order as the atomic dimensions, there will be path differences between waves scattered by different electrons if these path differences are less than one wavelength, then the interference will be partially destructive [20,22,26], To describe this effect, the parameter/is defined, also called the atomic scattering factor, which is the ratio of the amplitude scattered by an atom, Aa, to the amplitude scattered by an electron, Ae, that is [21] 

In contrast to low-energy electrons, X-rays are very weakly scattered by atoms, a property which leads to the success of X-ray diffraction as a means of determining the structure of bulk solids through scattering from atoms over a large depth into the 

The radiation damage problems with these incident X-ray methods are similar to those described in the previous section for photoelectron diffraction and SEXAFS, namely that there are potential problems, but they can mostly be overcome by appropriate precautions. 

To find a more useful form for 5 (k) we first separate it to its diagonal (/ = y) and non-diagonal parts. The diagonal part yields unity. In a homogeneous isotropic system all N(N — 1) non-diagonal terms are identical. We get 

The quantum mechanical treatment of the electron distribution about an isolated atom nucleus gives an electron density, p(r), which is peaked at the nuclear position and falls off smoothly as a function of the distance from the nucleus. Each unit of volume, dv, around this center can scatter X-rays and those that are scattered coherently will interfere with those scattered from other unit volumes near this atom, depending on the scattering angle. From Chapter 1 we know that the interference occurs as a phase shift, 4 , between the scattered waves parallel to the vector, S, from two unit volumes separated by r by  

The incoherent scattering is then found by difference via the Thompson formula as  

---

Other important characterization techniques include electrophoresis measurements of droplets [11, 12] (see Section XIV-3C), infrared absorption of the constituent species [13], and light or x-ray scattering. NMR self-diffusion measurements can be used to determine droplet sizes in W/0 emulsions [14]. 

The structure of microemulsions have been studied by a variety of experimental means. Scattering experiments yield the droplet size or persistence length (3-6 nm) for nonspherical phases. Small-angle neutron scattering (SANS) [123] and x-ray scattering [124] experiments are appropriate however, the isotopic substitution of D2O for H2O... 

Phospholipid molecules form bilayer films or membranes about 5 nm in thickness as illustrated in Fig. XV-10. Vesicles or liposomes are closed bilayer shells in the 100-1000-nm size range formed on sonication of bilayer forming amphiphiles. Vesicles find use as controlled release and delivery vehicles in cosmetic lotions, agrochemicals, and, potentially, drugs. The advances in cryoelec-tron microscopy (see Section VIII-2A) in recent years have aided their characterization [70-72]. Additional light and x-ray scattering measurements reveal bilayer thickness and phase transitions [70, 71]. Differential thermal analysis... 

X-ray scattering arises from fluctuations in electron density. The general expression of the absolute scattered intensity (simplified as I(q) from now on) from the tliree-dimensional objects iimnersed in a different... 

Brumberger H (ed) 1967 Small-Angle X-ray Scattering (New York Gordon and Breach)... 

Balta-Calleja F J and Vonk G G 1989 X-ray Scattering of Synthetic Polymers (New York Elsevier)... 

Cakmak M, Teitge A, Zachman FI G and White J L 1993 On-line small-angle and wide-angle x-ray scattering studies on melt-spinning poly(vinylidene fluoride) tape using synchrotron radiation J. Polym. Sc/. 31 371- 81... 

Toney M F, Floward J N, Richter J, Borges G L, Gordon J G, Meiroy O R, Wiesier D G, Yee D and Sorensen L B 1995 Distribution of water moiecuies at Ag(111 )/eiectroiyte interface studied with surface x-ray scattering Surf. Sc/. 335 326-32... 

Ocko B M, Wang X J, Adzic R and Wandiowski Th 1998 Surface x-ray scattering studies of Eiectrosorption Synchrotron Radiat. News 11 23-30... 

Tidsweii i M, Lucas C A, Markovic N M and Ross P N 1995 Surface structure determination using anomaious x-ray scattering Underpotentiai deposition of copper on Pt(111) Phys. Rev. B 51 10 205-8... 

Ocko B M, Magnussen O M, Wang J, Adzic R R, Shi Z and Lipkowski J 1994 A criticai comparison of eiectrochemicai and surface x-ray scattering resuits at the Au(111) eiectrode in KBr soiutions Electroanal. Chem. 376 35-9... 

Melroy O R, Toney M F, Borges G L, Samant M G, Kortright J B, Ross P N and Blum L 1989 An In situ grazing incidence x-ray scattering study of the initial stages of electrochemical growth of lead on silver(111) J. Electroanal. Chem. 258 403-14... 

Figure C2.17.12. Exciton energy shift witli particle size. The lowest exciton energy is measured by optical absorjDtion for a number of different CdSe nanocrystal samples, and plotted against tire mean nanocrystal radius. The mean particle radii have been detennined using eitlier small-angle x-ray scattering (open circles) or TEM (squares). The solid curve is tire predicted exciton energy from tire Bms fonnula.

Mattoussi H efa/1996 Characterization of CdSe nanocrystalline dispersions by small angle x-ray scattering J. Chem. Phys. 105 9890... 

Schdmaiic illustration of an X-ray scattering experiment. The X-ray beam travels a different distance when ered by an electron at the origin compared to an electron situated at r... 

This is the result we have sought, although it needs a bit of additional manipulation to make its usefulness evident. The derivation we have followed in this section was developed by Debye in the context of x-ray scattering by the individual atoms of small molecules. Since s o , this function again empha-... 

O. Kiathy, in O. Glatter, O. Kiathy, eds., Small-Angle X-Ray Scattering, Academic Press, Inc., New York, 1982. 

---