·Table of Contents
·Materials Characterization and testing
X-Ray Refraction Topography for Non-Destructive Evaluation of Advanced Materials
Bernd R. Müller, Manfred P. Hentschel,
Karl-Wolfram Harbich, Axel Lange, Jörg Schors
Federal Institute for Materials Research and Testing (BAM)
D-12200 Berlin, Germany
The non-destructive characterization of high performance composites, ceramics and other advanced materials can be difficult. Anisotropy, heterogeneity and complex shapes reduce the performance of traditional non-destructive techniques, which have been optimized for isotropic single phase materials, preferably for metals.
The effect of X-ray refraction provides unconventional small angle X-ray scattering (SAXS) techniques which have been developed and applied in the last decade at our laboratory to meet the actual demand for improved non-destructive characterization of advanced materials. X-ray refraction reveals the inner surface and interface concentrations of nanometer dimensions due to the short X-ray wavelength near 0.1 nm. Sub-micron particle, crack and pore sizes are easily determined by "X-ray refractometry" without destroying the structure by cutting or polishing for microscopic techniques.
Beyond this analytical potential for (integral) analysis, spatial resolution can be achieved, when the sample is scanned across a narrow X-ray beam. In this case we call it "X-ray refraction topography". The X-ray refraction topogram of a probe can be measured within relatively short time, as the scattered intensity at very small angles of few minutes of arc is much higher than in conventional wide angle X-ray scattering (WAXS).
PHYSICS AND INSTRUMENTAL
The physics of X-ray refraction is quite similar to the well known refraction of the visible light by optical lenses and prisms, which is governed by Snell's law. However a major difference to the visible optics is, that the refractive index n of X-rays in matter is nearly one. This causes deflections at very small angles in the order of a few minutes of arc.
e is the real part of the complex index of refraction, r the electron density and l
the X-ray wavelength. In case of X-rays, where n < 1 the converging effect of convex lenses changes to divergence. Fig. 1 demonstrates the effect of small angle scattering by refraction of cylindrical lenses: A bundle of glass fibres with a diameter of 15 µm each as used for glass fibre reinforced plastics (GFRP) deflects collimated parallel X-rays within several minutes of arc. The oriented intensity distribution is collected by an X-ray film or a CCD camera while the straight (primary) beam is omitted by a beam stop. Monochromatic radiation below 20 keV is applied like in crystallography, which is relatively soft for NDT purposes.
The shape of the intensity distribution of cylindrical objects is always the same even for very different materials, if the scattering angle is normalized to the "critical angle" q
C of total reflection (see Fig. 2). This parameter depends only on the refractive index: q
2C = 2e
Fig 1: Effect of oriented small angle scattering by refraction of glass fibres.
Fig 2: The normalized shape of the angular intensity distribution of cylindrical objects.
Fig. 3 demonstrates the X-ray deflection at circular objects (sections of fibres or spheres) by refraction and (very few) total reflection. In fibres and spherical particles the deflection of X-rays occurs twice, when entering and when leaving the object (Fig. 3, left). The intensity of the deflected X-rays falls down to nearly zero at the critical angle of total reflection (see Fig. 2 and Fig. 3, right). A cross section of 10-3 of the fibre diameter contributes to the detectable intensity above typically 2 minutes of arc. The effect of total reflection of X-rays occurs as well at the angle
Fig 3: X-ray deflection at circular objects by refraction and total reflection (left); angular intensity distribution for fibres and spheres (right)|
of grazing incidence but only 10-6 of the diameter is involved and therefore negligible. But well oriented planar surfaces can produce strong reflections. Based on Snell's Law the angular intensity distribution has been calculated and approximated for cylindrical fibre and spheres, as illustrated by Fig. 3. The refracted intensity of a cylinder without absorption effects can be expressed as
R is the radius of the fibre, q
is the scattering angle, I0 the intensity of the incident and I the intensity of the transmitted X-ray beam.
The SAXS instrumentation is relatively simple (see Fig. 4) but sometimes delicate in terms of its (thermo-) mechanical stability. It requires a fine structure X-ray generator, a (commercial) small angle X-ray camera (collimator), an X-ray refraction detector and a sample manipulator. A reference detector looks at a scattering foil in order to monitor sample absorption and beam stability.
Fig 4: SAXS instrumentation with primary X-ray beam, collimator, sample, sample manipulator, scattering foil, a refraction detector with a slit unit, which measures the refracted intensity of the sample (IR) or if the sample is not in place (IR0) and a reference detector, which measures the intensity I '0 or I ', which is proportional to I0 (without sample) or I (with sample), respectively.|
The refraction intensity can be measured according to Eq. (3). IR* depends on the transmitted intensity I, the thickness d and the inner surface density x
= N ×
R (N is the amount of fibres) of the sample, respectively. The proportional factor k is a specific constant of the used apparatus and can be determined by measuring a probe with a known inner surface density.
The proportional factor k and the inner surface density x
define the refraction value C = k×
, which is a relative measure of the surface density of the sample. For practical measurements the refraction slit remains at a fixed scattering angle 2q
, so that the surface density of the sample can be measured according to:
The conventional understanding of "continuous" small angle X-ray scattering (SAXS) is governed by the interpretation of diffraction effects. Apart from Guinier's theory for separated particles Porod explains diffraction of densely packed colloids similar to Eq. (3). However both deal with particles two orders of magnitude smaller. A simple proof for the refraction effect at large objects can be found by scanning a fibre through a narrow X-ray beam and collecting the intensity at each fibre position (Fig. 5). Even focussing by pores is possible. The behaviour is exactly the same as in an experiment with visible light. Any diffraction effect would result in a symmetric intensity above background level.
Fig 5: X-ray refraction by a 125 mm polymer fibre is demonstrated by scanning a fibre through a narrow X-ray beam and collecting the intensity at each fibre position.
Fig 6: Comparison of inner surface densities of selected non-metallic materials
Average specific surfaces can be determined by "stationary" X-ray refractometry. Such kind of analytical investigations can be useful in the field of new materials, when cutting or polishing has to be omitted. In Fig. 6 the specific surface densities of selected non-metallic materials are compared. The values are relatively small in case of very "porous" materials as the pores are very large, which reduces the surface to volume ratio. The plotted surface values are taken from the refraction factors C, corrected for the different densities of the materials: C/r
² (see Eqs. (1) and (2)). In case of composites like paper and carbon fibre reinforced plastics (CFRP) the refraction value C is a composition of the refraction at inner surfaces of each component and at the interfaces. In case of the CFRP the porosity is very low, therefor only the interface contributes to the signal. The measuring time is usually a few seconds or less, if ±
1% error is accepted. The short measurement time allows scanning for spatial resolution or statistical evaluations.
The determination of the pore size distribution in glass ceramics by X-ray refraction results in diameters which correspond to the chordlength distribution in microscopic analysis. The mean values of the diameters are identical within ±
3%. The measurements are performed with Mo-k-a
-radiation at different positions of a 1.4 mm ceramics plate, sintered at 850°C (Fig. 7). Further pore size measurements on SiC and Al2O3 ceramics by X-ray refractometry reveal good agreement with other techniques, especially with high pressure mercury intrusion.
Fig 7: Pores in glass ceramics: micrograph, optical chordlength analysis of pores and pore diameter probability by X-ray Refraction
The measurement of the crack density in light weight materials can be performed by X-ray refractometry as well. The knowledge of the crack development is believed to play the key role in all long-term material behaviour. Fig. 8 correlates the residual shear strength of CFRP to the average inner surface of cracks created by ageing treatment at 150°C, 180°C and 200°C up to 10,000 h. The investigation compares epoxy and BMI matrix systems for high temperature applications in supersonic aviation. Although BMI has a high strength at the beginning, it falls below epoxy at the end of the ageing treatment. The results explain clearly the dependence of the shear strength on the crack density. The slope defines an ageing module which can be regarded as a new materials parameter.
Fig 8: Correlation of the residual shear strength of CFRP to the average inner surface of cracks created by ageing treatment at 150 °C, 180 °C, 200 °C for 1,000, 3,000, 5,000, 10,000 h.
Fig 9: Model of X-ray refraction at interfaces of bonded and debonded fibres of a composite, X-ray topography of model, investigation of single fibre debonding at different fibre volume ratios.
Single fibre debonding in composites is not measurable - except by X-ray refractometry (although some attempts of pulling off individual fibres under the microscope have high artistic value). It is a central parameter of composites characterization. The basic principle can be understood by the optical analogue: compare the focussing properties of a lens in air (fibre in air) and in a liquid (fibre in matrix)! The refraction effect is lower in the second case. The density difference between fibre and matrix determines the X-ray scattering effect as well.
A model composite has been made in order to demonstrate the refraction behaviour of a debonded and a bonded 140 µm sapphire fibre in wax matrix (Fig. 9, left). The upper ray crosses the bonded fibre-matrix interface causing a small amount of deflected intensity. At the debonded fibre and at the matrix surfaces (lower ray) much more X-rays are deflected, as the larger density difference between the materials and air corresponds to a higher index of refraction.
The middle of Fig. 9 shows the resulting intensity distribution of a refraction scan of the model composite. The wax channel is clearly separated from the fibre surface. The bonded fibre is less contrasted. A practical measurement of the fraction of debonded fibres in a real thermoplastic C-fibre composite is given on the right of Fig. 9. There is a non-linear dependence of debonding on the fibre volume fraction. This can be explained by the very viscous thermoplastic matrix, which is hindered to penetrate between densely packed fibres during melt impregnation processing. Formulas for the calculation of individual or collective fibre debonding have been given.
REFRACTOGRAPHY (REFRACTION TOPOGRAPHY)
Scanning X-ray refraction localises the projection of inner surface concentrations or individual edges of surfaces and interfaces such as sub-micrometer pores or cracks. The spatial resolution can be better than 10 µm, although this is not the main advantage of refraction techniques, as the signal level itself contains the information about inner surfaces.
An example of refraction topography is given in Fig. 10. It images the crack pattern in unidirectional CFRP after ageing of 10,000 h at 180°C (epoxy) and 200°C (BMI). Although the average inner surface densities are the same the additional spatial information shows a difference in the type of crack distribution. The differences in shear strength can be understood by the different fractal behaviour of cracks. The type of cracks is of course a mixture of cracks at the fibre-matrix interface (single fibre debonding) and matrix cracks. Fig. 10 shows as well the directions of the cracks.
Fig 10: Model of X-ray refraction at interfaces of bonded and debonded fibres of a composite, X-ray topography of model, investigation of single fibre debonding at different fibre volume ratios.
Fig 11: X-ray refraction topographs of crack patterns in CFRP after ageing of 10,000 h at 180°C (epoxy) and 200°C (BMI).
Another problem of CFRP characterization relates to impact damages. Ultrasound C-scans resolve delaminations created by impact very well, but the single fibre debonding area, which develops at lower loads, is only detectable by X-ray refraction topography. In Fig. 11 seven impact areas are imaged at 1 mm resolution. The reduction of details compared to Fig. 10 is compensated by 100 times faster measurements (10 mm²/s). (The three bright capitals are not impacted, simply pencil written [graphite scattering].)
X-RAY REFRACTION COMPUTER TOMOGRAPHY
Although two-dimensional refraction topography provides an effective new probe for analysing meso-structures of all kind of heterogeneous materials, it is sometimes interesting to have section images of transversal resolution as known from X-ray computer tomography in order to overcome the overlap of details by projection effects. Fig. 12 demonstrates the feasibility of X-ray refraction computer tomography: The sample micro-graph shows a 3 mm by 3 mm bar of phenolic resin CFRP laminate, which is a standard precursor in C/C and C/SiC CMC processing.
Fig 12: X-ray refraction Computer-Tomography of CFRP laminate: micrograph, left; conventional absorption tomography, middle; interface tomography, right.|
The computer tomography experiment is carried out by 18 keV single beam scanning in a Kratky camera according to Fig. 4. Linear scans are performed for 360 angular positions, Fourier filtered for linear smearing on a PC and added up in an image file (filtered back projection). The reconstruction of detector signals I¢
shows a quite homogeneous density of the conventional (absorption) computer tomographic image (Fig. 12, centre). The final refraction image reveals the spatial interface/inner surface distribution free of absorption effects. The typical layer and crack structure of the micrograph can be recognised by a non-destructive technique.
X-ray refraction techniques combine analytical capabilities of sub-micrometer structure detection with the requirements of non-destructive full volume characterization. X-ray refraction therefore might help faster materials development, better understanding of meso-structures and partly replace micro analysis and mechanical testing in advanced materials science.
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