Size matters!

What kinds of things can we observe with x-rays?  If we make a few simple calculations using Braggs' law

lambda = 2d sin theta

we can get some ideas.  Suppose we use CuKalpha radiation (lambda = 1.5418 Å), and calculate 2theta for various values of "d", where "d" is now considered an 'interaction distance'.

d                                                    2theta                       q

10 Å (0.001 micron)                       8.84°                   0.628 Å^-1

50 Å                                              1.77°

100 Å (0.01 micron)                       0.88°                   0.0628 Å^-1

300 Å                                             0.29°

600 Å                                             0.15°

1000 Å (0.1 micron)                       0.09°                   0.00628 Å^-1

10,000 Å (1 micron)                       0.009°                 0.000628 Å^-1

q = (4/lambda) sin theta = 2/d

From the results above, we can see that high-angle x-ray scattering, usually about 2°2theta to 160°2theta gives information about the structure on an atomic scale.  We may even be able to extract atomic scale information from scattering data for crystalline substances with large interaction distances at angles smaller than 2°.  As we go to lower scattering angles, we can measure larger "things".

What are these larger "things"?

Examples

1.  WAXS and SAXS study of (m)TMXDI-PDMS siloxane-urethaneureas

FIBRES & TEXTILES in Eastern Europe (January/December 2003), 11, No. 5, 44

http://www.fibtex.lodz.pl/44_23_107.pdf

The (m)TMXDI-PDMS polymer molecule consists of soft and hard chain segments as shown below.

The hard segments form regions with crystal-like order, phase regions surrounded by the amorphous siloxane segment chains.

Below are shown saxs measurements for these polymers with different compositions.  From the scattering curves, it is seen that, as the NCO/OH ratio (a = 1.5/1; e = 4.5/1) increases, the position of the scattering maximum moves closer to q = 0.  Thus, the size of the hard segment "crystalline" regions increases as the fraction of the diisocyanate increases.

2. Microstructure orientation and nanoporous gas transport in semicrystalline block copolymer membranes

Polymer (2000), 41,  46714677

Channel die processing (see figure below) of semicrystalline ethylene (E)/ethylenepropylene (EP) diblock E/EP and triblock E/EP/E copolymers results in orientation of the block copolymer lamellar microstructure both parallel and perpendicular to the direction of shear. Gas permeability in these oriented block copolymer systems varies: parallel --> lower permeability, and perpendicular --> higher permeability (see 2^nd figure below).

saxs was used to obtain the lamellar periodicity.

This saxs photo was for the perpendicular texture type. (Ignore grid instrument artifact)

3. Nanometer to Micrometer Void Microstructure Characterization of SOFC Layers and Interfaces by Small Angle Scattering (SAXS) and Computed X-ray Microtomography(XMT)

http://www.netl.doe.gov/publications/proceedings/03/seca core/Andrew Allen NIST.pdf

The fuel cell configuration is shown below.

This cross-section micrograph shows the porosity.

4.  Critical Dimension Metrology of Nanoscale Structures with Small Angle X-ray Scattering

http://polymers.msel.nist.gov/highlights/Critical-Dimension-Metrology-Nanoscale-Structures-Small-Angle-X-ray-Scattering.html

NIST is developing a transmission x-ray scattering based method capable of angstrom level precision in critical dimension evaluation over large, (50 x 50) mm, arrays of nanoscale periodic structures. With a wave-length more than an order of magnitude smaller than the pattern size, patterns can be analyzed using methods traditionally employed in crystallographic diffraction.

SAXS geometry, showing incident and scattered beams (red lines), sample with pattern oriented at rotation angle w, and 2-D detector (right).

SEM image (top left) of a photoresist grating on a silicon wafer compared to the resulting 2-D SAXS image (top right). The diffraction peaks were fit with a simple model (solid blue line).

Streaks of intensity emanating from diffraction peaks are indicative of deviations from the ideal grating and may provide information about defects such as long wavelength line edge roughness.

 (a) blue rectangles represent etched regions in a film, and (b) the resulting SAXS detector image.

5.  The measurement of the micro-fibril angle in soft-wood

JOURNAL OF MATERIALS SCIENCE LETTERS 20, 2001, 2245 2247

The major portion of the wall of a wood cell consists of bundles of a crystalline arrangement of cellulose chains (microfibrils).  The microfibrils align quite parallel and are arranged in a spiral around the cell wall, with the axis of the spiral along the vertical or long cell direction.

Above are shown typical scattering patterns from a Norway spruce sample with a mean MFA of 20°. The longitudinal cell axis was vertical. (c) SAXS pattern recorded at alpha = 0°. The two-dimensional image shows three streaks of high intensity. Integration over the scattering vector q gave a curve with three peaks that were fitted with three Gaussians, giving a MFA of 20.0±0.6°. (d) SAXS pattern recorded at alpha = 45°. In the scattering pattern one can see two streaks of high intensity. The curve resulting from integration was fitted with two Gaussians. The MFA was determined to be 20.4±0.6°.