Some background for the advanced

Some information for the curious

The interface distribution function (IDF) comprises lamellar thickness distributions. The IDF is computed from the scattering curve.

Even nowadays scattering data from polymer materials frequently are evaluated inadequately (reflex positions, Bragg's law). An improved method was proposes in 1967 by Vonk ("linear correlation function"). But it took another ten years until Ruland developed the notion of an interface distribution function. The IDF constitutes the relation between the scattering curve on one hand and the structural features of the layer thickness distributions on the other. Considering poorly ordered systems like polymer materials the older methods are insufficient. Their results are incomplete or even incorrect.
Macromolecules (1991) 24:5980-5990

Polymer Materials.

Polymer materials are plastics. Fibres, casings of devices, suture materials, gas pipelines, shoe soles, printing plates, membranes, rubber, styrofoam, soft foams, gaskets, spouts and isolating materials in the automotive industry, CDs, bulletproof glass, ...

Mellin convolution of two Gaussians

The nanostructure is composed from domains. It defines the properties of the material. The domains are not placed at random inside the part. Even in distorted structure there is a certain short-range order. Electron microscope and X-ray scattering exhibit the amalgamation of domain size statistics and interdomain distance distributions. Thus the information content is rather low. Several models are discussed that describe the arrangement of domains. We have demonstrated how to unify these models. The result has 7 structural parameters only. Applied to scattering data the resulting parameter values are highly significant.

Nanostructure in Polymer Materials.

Inside a crystal we observe order from a periodic arrangement of uniform molecules. In polymers crystallites are small. Their size is subject to considerable variation. Amorphous regions outside the crystallites occupy a considerable fraction of the volume. A typical feature of semicrystalline polymers and other kinds of plastics is a multi-phase structure: There is a hard (crystalline) and a soft (amorphous) phase. Crystallites are embedded in amorphous regions; hard domains are placed in a soft matrix (or the other way round, as in high impact plastics that are used to manufacture casings of household devices). Because polymers are chains of concatenated monomers, the crystallites prefer to form lamellae.

Layer systems are forming a considerable fraction of the technical polymer materials. At very small scattering angles the scattering corresponding to a lamellar system may be superimposed by the effect of pores (voids) in the material. What does the scattering of the pure lamellar system look like? A useful statement for application is given by mathematics only for the slit-smeared, one-dimensional scattering curve. Experiment and theory show that in the corresponding region the shape of the favourable curve is a straight line. Mathematical deduction yields both axis intercept and slope of this line after expressing the slit-smearing operator in terms of a Mellin convolution. The bottom line tells that the line is only changed if long period (i.e. the lamellar repeat) or material density are changing - or, of course, if voids are emerging.

Void scattering.

Most of the technical plastics (Polyethylene [PE], polyvinylidenefluoride [PVDF, PVF2] are semicrystalline comprising layer-shaped crystals ("lamellae"), stacked on top of each other. In between there are amorphous layers. Imagine a pipeline made from plastics is getting porous. That's bad. Favourable it is that we can detect the pores in the beginning of the scattering curve. The more voids there are, the more the intensity does increase. And the scattering method is very sensitive. Void fractions ranging in parts per thousand give a considerable effect. But what is it good for if we do not know how the curve is affected by other effects? What does the curve look like if there are no voids? How fast does it increase only from the fact that there are lamellae? It would be reassuring to know the shape of the curve for common materials - moreover, it would become possible to tell, how many per-mille of void fraction would cause a bubbling pipeline. Let's do the mathematics. After the deduction we have learned what kind of X-ray camera to choose, how to carry out the experiment and how to do the analysis in order to be able to report figures instead of cloudy notions.

PVDF linings were studied. Material was taken from the inner, outer and the central part of the sheath. Measurements of the absolute SAXS were carried out using a Kratky camera. Pore fraction is increasing with increasing age of the material. Porosity is increasing from the inner to the outer surface of the part. Voids are generated inside the amorphous layers because of escaping plasticizer. The amorphous layers collapse and leave behind some voids.

Ultra small-angle X-ray scattering of oriented polyethylene at 128°C

Frequently the scattering pattern of oriented material shows several reflections. In 1999 methodical work at the synchrotron beamlines resulted in a quantum leap concerning data quality. How to tackle the promising data analysis now? There were no multi-dimensional methods available. It appeared reasonable to prepare an interesting partial aspect from the nanostructure by putting it into a scattering curve and then to analyse it using methods established for the treatment of curves. We employed the projection operations proposed by Bonart in 1960 and developed two methods - one for each of the main axes of the fibre scattering. Trial evaluations in the sequel exhibited that many materials show a complex nanostructure. We felt that the required complex modeling would not be appreciated by the community and only published the results from the simple materials. Then we started development of a multidimensional method.

Anisotropy of Fibres.

Fibres are spun from dyes under draw. Thus polymer chains and the domains of the nanostructure are oriented. Propagating through the material after this process, the sequence of segments crossed in hard or soft material is a function of the chosen direction (with respect to the fibre axis). The material exhibits anisotropy. Complex arrangements of differently shaped domains can be imaginated - and nature shows that complexity is not unusual in anisotropic materials. Is this finding related to complexity structure evolution? What are the processes governing the evolution? Will we be able to identify and to control them?

Poly(ether ester).

Poly(ether ester) (PEE) are quite common. They seal the tap, help to keep dirt from the drive shafts of our cars (because of their look the PEE spouts are named "christmas trees") and are sympathetic to us in the shape of "Sympatex" fibres. In this material the components of the chain molecules are not all the same, but there are "ether"-units (monomers) and "ester"-units. ethers are forming a soft phase, esters a hard one. This separation into two phases is only working, because the chemist has concatenated the monomers in a special way. Several monomers of the one kind are coupled together, until several monomers of the other kind are added to the end of the growing chain. A group of monomers of the same kind is called "a segment". Thus, soft and hard segments alternate several times when propagating along a chain. We call the material a "multi-block copolymer". The monomers of the one kind do not like the monomers of the other and try to form domains of their own kind. Concerning the hard phase it is very important that at ambient temperature the hard domains can only be disintegrated by application of brute force. On the other hand, the chain segments of the soft domains are behaving quite flexible as they cross space, until there is a hard segment fixed in a hard domain. Thus, at ambient temperature, the chains are cross-linked by hard domain that form the nodes of the network. Deformable but strong the material, more or less, exhibits the elastic properties of a rubber.

A narrow subset of digital image processing (DI) methods has been utilized in the analysis of X-ray scattering images. Nevertheless, before a quantitative analysis can be started flaw detection and repair has to be performed. In order to do so automatically, we introduced additional operators from DI (Sobel, erode, closure, ...) in the field of scattering pattern analysis. Multi-dimensional extrapolation into remnant blind areas can sufficiently be accomplished by the concept of radial basis function interpolation. Thus termination effects can be minimized.

Removing ticks.

We intend to retouch the scattering image. Doing the job of replenishing old grammophone recordings, is quite similar. Ticks are searched and the wave form is restored. Advantageous in the field of X-ray analysis is the symmetry of scattering patterns. Thus, quite frequently, the flaw can be restored from flawless scattering intensity taken "from one of the other quadrants" of the picture. Whenever this method is impossible, extrapolation has to be carried out.

In the field of small-angle X-ray scattering the information on nanostructure is sitting on top of a background that has to be subtracted. Several deviations of the real material structure from the ideal notion of nanostructure compile to form the background. For the isotropic case there are established but cumbersome methods to model the background that cannot be transferred to the anisotropic case. We propose to determine the background from the scattering pattern itself. Because of the fact that it cannot be "patterned considerably", we construct a suitable background from the low spatial frequencies of the scattering pattern. Moderate corruption of the structural information related to small domains is accepted.

Air is scattering light.

"It's simply air", we state when we look up to the sky. We should know precisely where the air stops. Otherwise we risk to run into a lamppost. But we don't care about peanuts. We imagine something like ideal air. Transparent, as to say. Really transparent? The "peanuts", i.e. the short-range random fluctuations of the air density, limit the transparency of the air at daytime: Sunlight is scattered. We observe a blue sky instead of the darkness of the outer space. This blue is not "considerably patterned", isn't it? Rather a simple model should work and describe its features.

Revisiting the background subtraction method developed for the anisotropic case, it can be considered the introduction of signal processing method. After the filter characteristics of the spatial frequency filter has been determined, all the scattering patterns from a time-resolved study can be processed automatically. After back-transformation of the principle to the isotropic case we gain automation for the study of isotropic materials. Manual effort and biased evaluation are avoided. Productivity is increased.

Signal processing.

We are searching for the information describing the ideal nanostructure in the scattering pattern. Doing the job of replenishing old grammophone recordings, is almost the same. Rumble and hiss have to be filtered - and the result is quite close to the original - at least what the essentials are concerned.

The proposed chord distribution is visualising the distances between the surfaces of the domains in 2 or 3 dimensions in space. We do not extend the corresponding definition of the isotropic case as a "2nd derivative of the correlation function with respect to its radial component", but choose the Laplacian of the multidimensional correlation function. Doing so we introduce a dependency of the function's value from the angle between the normals of the correlated surfaces.

The figure shows the chord distribution z(r) of a PEE sample during a cyclic straining experiment in the state "relaxed after straining to 88% elongation". View is from the lattice face. Thus the peaks demonstrate the distances between domains in the distorted lattice with only short-range order. In this view the domain peaks are observed below the surface pointing downwards.

Multi-dimensional Structure Information.

Studying fibres and other anisotropic materials we would like to learn how the domains are oriented (with respect to the fibre axis), how they are arranged and shaped (layers, rods, spheres). How do we extract this information? Let us consider the case of the robot that is supposed to handle a brick. First "edge amplification" of the viewed picture is carried out. The human eye is known to do the same. Then the contour of the brick (domain) is emerging in the electronic brain and the arm of the robot can execute a targeted move. Edge detection is carried out by a searc for steep changes in brightness or colour. In mathematical language the image must be differentiated. Sometimes a double differentiation (2nd derivative) is favourable. This is one of these cases. Scattering theory gives the proof. On the other hand, our image is multi-dimensional. And in this case we have to choose from a bunch of differential operators. Quite favourable appears to choose the Laplacian. It is implemented in scattering theory ("Fourier transform") with ease, the result of this kind of "edge detection" can be defined concisely with respect to physics and it is quite common in the field of digital image processing. Last but not least it is possible to transform the raw data early in a manner that produces the requested edge detection in the course of the following evaluation steps automatically. Thus error and computational effort are kept limited.

In the edge-enhanced image the complexity of nanostructure is revealed. We recognise layer stacks, row structures, sattelite domains, systems of cylinders. Because of these rich structures a quantitative analysis appears to be quite a challenge. It will require new strategies, complex models and some computing power.

Chord distribution at 128°C. Injection moulded PE. Triangular peaks show lamellae grown on row structures.

Oriented injection moulded rods from PE were heated in the synchrotron beam. We observe the disappearing of nanostructure in a sequence of chord distributions. We invert the time and interpret the sequence beginning from its end. Then we first see a regular arrangement of tiny crystal nuclei on rows. From this structure crystal lamellae are growing. Continuing in the direction toward lower temperature new, less perfect lamallae are inserted in the amorphous zones. By a second step of insertion the initial long period appears to be reduced to a quarter in the ambient temperature sample. Nevertheless, the inserted layers are distorted to such an extent that the strong and constant fingerprint of the primary lamellae is the most prominent feature of almost all of the X-ray patterns. At ambient temperature there are three structural components: the row structure, Lamellae grown on the rows and inserted lamellae. In the X-ray pattern they are represented by layer lines, 2 point-shaped reflections and an intensity ridge stretching along the meridian, respectively.

Ultra-high molecular weight polyethylene (UHMWPE) is of some technical importance. E.g. it is used to manufacture artificial prosthesis in medical application. Here it shall show low wear. High crystallinity and big crystals are sought. After injecting the melt into a mould under very high pressure, a very peculiar material is obtained. When heated in the differential scanning calorimeter (DSC), initially id does not show indications of melting or crystallisation. Late at very high temperature a strong melting peak is observed that is related to very big crystals. X-ray scattering contains information on the crystallite thickness distribution as a whole at every temperature. We find that the peculiar material initially comprises many small crystals. They are melting during heating of the material while thick crystals are formed at the same time. The net heat flow of this coupled process is low and is not considered in DSC analysis. Thus the sought morphology is only established by an annealing process.

Time-resolved Materials Research.

Left: Crystal thickness distributions at room temperature. Right: Change of the distributions between room temperature and 120°C. 9 materials with molecular masses of 4, 5, and 6 106 g/mol; produced at low, medium and high pressure.

Time-resolved measurements of the small-angle X-ray scattering (SAXS) have some tradition at synchrotron sources. Nevertheless, the method must still be considered developing. Only in 1998 we received sufficiently stable beam conditions on an international level that is sufficient to proceed beyond test experiments. Data quality is intriguing. Now it is the time to proceed to a new level of development in evaluation method as well.

We show what can be reached after methodical development. In the presented study an isotropic material was investigated. The quantitative evaluation for is well-established for such materials. Nevertheless, the problem was the big amount (500) of scattering patterns that were to be evaluated. After back-transformation of the signal processing method from the anisotropic to the isotropic case, the evaluation could be done automated. By doing so no "key-press method" is established. Before running automatically it will have to be adjusted to the individual experimental conditions. Time-consuming will always be the fitting of structural models to the measured data, error propagation analysis, and the interpretation of the results. On the other hand there is a reward, since mother nature is generously telling surprising stories to an open ear.

Nafion is a membrane material invented by DuPont. Because of its complex nanostructure the membrane exhibits a complex X-ray pattern in the middle- and small-angle regions. The pattern has frequently been interpreted. Using our CDF method the middle- and nanostructure can be separated from each other and visualized in physical space. Thus, for the first time, rich quantitative data concerning domain extensions and distances are acquired. Our data fit well in the qualitative models discussed in the literature.

Additionally we have strained the membranes and obtained information concerning the change of the inner structure upon mechanical load. The astonishing extensibility of the commercial film is a result of a peculiar distribution of the ionomer channels, which tear apart when the material is elongated. Such behavior is supposed not to be in aid of the separation properties - especially because the favorable initial structure does recover upon release of the membrane.

Thanks to GKSS we received some funding for this study. A post-doctorate scientist was employed, and the funding of material, books, and travel expenses is gratefully appreciated.

The long period face of the CDF clearly exhibits the lattice with short-range order that was formed during earlier straining.

No funding of this study

Results in a nutshell

Both granular (blocks) and lamellar crystallites in ethylene-octene copolymers react to drawing and form microfibrils. Only the microfibrils formed from lamellae stabilize the relaxed stucture and a permanently oriented macrolattice is generated. The chord distribution analysis reveals it nanostructure and how it was formed.

Poly(ether esters) (PEE) are polymer materials that are elastic and harden during elongation because of crystallization. Thus it is difficult to tear them apart. Moreover, they can stand low and high temperatures rather well. But after they have been strained far enough, they do not shrink back to their original length. The material is used in gaskets, as a dirt barrier in automotive, and as textile fiber (Sympatex). We have studied Arnitel E1500/50, a material produced by DSM (Geleen, Netherlands).

No funding of this study

Lamellae sintered from identical blocks are torn apart

How blocks are extracted from lamellae during elongation. It is interesting to note that they do not move straight downward. The reason is that the loose blocks are connected to their neighbors by polyether chain segments of identical length.

If the crystallization temperature is high, then crystalline blocks are observed for quite a while - layers are only formed later on.

We cause the polymers to crystallize from the melt with the crystallites oriented in one preferrential direction. During this process we record the X-ray scattering (SAXS, WAXS) using 2D detectors with high frequency. We vary the temperature programs (isothermal crystallization at various temperature, non-isothermal crystallization according to various temperature profiles). Nanostructure data are displayed as a CDF in real space - and there the nanostructure evolution performs in great detail before our eyes. Here we report on even some unpublished results based on measurements that were performed in February 2004.

• During melting prior to the crystallization experiment the structure is clarifying. We begin to see lamellae and small blocks. But the blocks are melting first, and the lamellae do not disintegrate into floes before melting.
• During crystallization the arrangement of the crystallites is purely random. Initially there is no lattice-generating force that is perturbed. A primacy of the "distorted lattice" is not found.
• When there is little space on the "parking ground" for crystals, searching and thus ordering starts.
• At high temperature crystallization is slow. In this case there are blocks built in the beginning and it takes quite a while, before the gaps between the floes freeze and lamellae are formed. Every lamella is ripening individually. Its relation to its neighbors is nothing but the required packing correlation.
• Although a lamella is the preferred shape of a crystal, it is not the final real structure. When the space on the parking ground and the time is vanishing, only frustrated lamellae (e.g. blocks) are formed, and these secondary blocks undergo short-range arrangement with neighboring lamellae and each other. With this network of harmony they, finally, dominate the observed scattering pattern albeit their individual imperfection.

More information on a separate web page

No funding of this study

Polymers crystallize

When a polymer melt is injected into a mold and cooled down, it frequently undergoes crystallization. The crystallites are layers with a thickness of several nanometers. They can be observed in the electron microscope and deduced from small-angle X-ray scattering patterns. These layers are stacked, and the resulting nanostructure of soft and hard domains determines the materials properties.

Until now there was no method capable to visualize the nanostructure during processing directly without interference with the materials processing. Though time-resolved 2D data can be recorded for more than 10 years now, nobody had undertaken the effort to compute images of the nanostructure from it.

Nevertheless, this is not a problem too complicated. Each medical CT or MRI-device demonstrates the feasibility: The body is irradiated and projections of the density function are obtained. Data are transferred to the reciprocal space by Fourier transformation - and have become slices (Fourier-slice-theorem). From the slices an reciprocal space image is interpolated and the data are transformed back. The image is there. In our case it is both a little bit easier and a little bit more complicated: Our data are already from reciprocal space, so we only require a one-way ticket. But our device has swallowed part of the information. Thus we are unable to reconstruct a common picture. Instead, the CDF shows the distances between domain surfaces. Therefore a lamella becomes a smooth triangle, and a trapezoid with dents is an ensemble of blocks instead. The future lamella is divided into several floes.

Mesostructure at the beginning of crystallization (a) entanglement strands (b) nuclei (c) pairs of crystalline lamellae

We study the initial phase of oriented, isothermal crystallization of polyethylene (PE) and measure SAXS and WAXS with high time-resolution. Before the first nuclei are observed there is already a mesostructure visible in the SAXS. The melt is separated into entangled and disentangled regions of diferent density (a). The SAXS cannot recognize, if the entangled regions form a network - but the fact that the strands are inclined with respect to the fiber direction are a strong argument. Row nuclei (red squares) are a dynamical structure that is observed before crystallisation starts. Thereafter crystalline lamellae are formed at the ends of the entangled strands (b). The only regularity is the formation of pairs of lamellae (c). Whereas the lamellae are highly oriented, the first crystallites are not. Only as thickness grows of the crystalline layers is observed, orientation of crystallites in fiber direction is observed.

This study is supported by the DFG

Crystallization from a quiescent melt

The simultaneous X-ray studies show: The formation of crystalline lamellae from the quiescent melt is a rather complex process in which various states (entanglement network, row structure, blocks, lamellae) may be encountered, but never does the quiescent melt generate layer stacks with long-range order. There is only random population of the volume and all order is restricted to arrangement of a domain with its adjacent neighbors.

Evaluation methods for structure investigations of synthetic soft matter can only be advanced, if the predominant aspects of structure are sufficiently considereed in the theory. This paper is closes a gap. Moreover, it is intended to be the fundament of advanced data evaluation.

Steps towards a general mathematical theory of polydispersity.

Except for some macromolecules in living organism, soft matter is ruled by polydispersity. Crystallite thicknesses vary, as do the lengths of molecules. Structure of soft matter is governed not by regularity of a lattice, but by irregular arrangement of polydisperse domains. Nevertheless, the nanostructure of soft matter is still discussed using notions from crystallography. Reason may be the missing general theory. After I have proposed a theoretical approach for 1D structures in 1993, I now proceed and discuss 3D structures by means of the Mellin convolution concept. As a result, statements on the distortion of polydisperse structure in experimental scattering data are obtained, and a method of correction is obvious. There are some historical papers dealing with aspects of polydispersity. The respective results are confirmed by application of the new formalism.

Visualization of zones of different nanostructure in a PE rod

Bone is hard outside, but flexible inside. A synthetic part can be made to possess high crystallinity outside and lower crystallinity inside. Size and arrangement of the crystallites characterize the nanostructure of the gradient material. The discrete SAXS is generated from this nanostructure. A wealth of scattering patterns is recorded, if a part is scanned by means of an X-ray microbeam. Mathematics of tomography permits to slice the material virtually into small cubes (voxels). the nanostructure in each voxel can be addressed without the need to slice the material. The spatial resolution is restricted by the cross-section of the microbeam. If the SAXS signal is used for imaging, it becomes obvious that the fiber is a gradient material, indeed.

We analyze the nanostructure.

Variation of nanostructure inside a PE rod. The reconstructed SAXS evaluated

There is more than imaging that can be done using SAXS patterns. They can be evaluated and the nanostructure of the part can be quantitatively described and spatially resolved. This can be carried out using the CDF method, because the fibers are highly oriented. Thus we find that the material is containing both lamellae and microfibrils. The lateral extensions of the lamellae and the long periods are determined as a function of the position in the fiber cross section. The method is time consuming. Recording of the data from one fiber cross section takes 12 hours. Nevertheless, mechanical cutting is more involved, and a spatial resolution of 80µm cannot be achieved. With improved microbeam technology spatial resolutions of 1 µm should soon become available.

Nanostructure at beginning crystallization. The CDF (topI shows reflections of a lattice-like nanostructure that can be interpreted in 2 ways.

Just after the start of crystallization the fiber-symmetric CDF exhibits a lattice-like nanostructure. From the positions of the peaks the lattice can be constructed. Nevertheless, there is ambiguity from Babinet's theorem. Both Strobl's block structure and Keller's cross-hatched structure explain the scattering data. Nevertheless, the evolution in time shows that the initial structure grows into lamellae. Because crystallinity is growing and there is no mechanical force, the observed initial structure must be blocks.

Discussion

Unfortunately we were unable to reduce the cycle time below 30 s. If an intense X-ray source is used to record good SAXS patterns at low exposure, then the beam additionally heats the sample and melts a hole into it.

The further evolution of the SAXS at different crystallization temperatures support the concept of crystallization from a mesophase that forms blocks which merge into lamellae. On the other hand, the monitoring of the crystallization by WAXD (Macromolecules (2009), in print) permits a different explanation at least for the regime in which crystallinity is growing fast, and in which block-merging does not occur (according to Keller's notion): The initial blocks are made from crystallites in two different orientations. They start to grow after different latency periods - with different crystallization velocities. Taking these observations into consideration, the classical crystallization mechanism explains both SAXS and WAXS results: blocks (nuclei) are continuously growing to form lamellae. Normally oriented crystallites always grow faster than the cross-lamellae, but the latency period is differently varying with crystallization temperature. At 150°C first the normal lamellae start growing. At 140°C the cross-lamellae are the first.

Fiber tomography. After a simple microbeam scan of the scattering the structure along the fiber radius is reconstructed

As a fiber is irradiated by an X-ray microbeam, the SAXS pattern stays constant when the fiber is rotated about its axis. Change is only observed when the fiber is translated. Skipping the unnecessary rotation speeds up the tomography experiment (20 min) and it becomes interesting for practical application (study of industrial fibers). Even the reconstruction is easier and faster (8 min) as in the general case! Her the solution of the reconstruction problem is Abel inversion. After the common regularization this inversion returns low-noise "desmeared" scattering patterns that describe the variation of the nanostructure along the fiber radius.

In the example we irradiate a composite made from an isotropic polymer in which needle-shaped domains of an oriented polymer are distributed. The needles are indicated in the scattering pattern by the horizontal streak, crystals in the isotropic polymer are indicated by the reflection ring. Where are the needles? Where is the isotropic material highly crystalline?

The Abel inversion

Fiber tomography. The recorded scattering pattern is smeared (projection). The desmeared image (section) is sought.

The mathematical problem of this one-dimensional tomography and its principal reconstruction is known for decades in scattering theory. It has been applied for "slit desmearing" (after Guinier and DuMond) of scattering curves measured with the Kratky camera (slit focus). Scatterers have not managed to suppress the emerging noise during desmearing. This problem has been solved in tomography by regularisation. If the measured data are rather smooth, we use the BASEX algorithm:
Rev. Sci. Instr. (2002), 73(7), 2634-2642
 
Isn't there anything special in the series of reconstructed images? - The central pattern does not fit in the series! This is not an artifact, but explained by the geometric peculiarity of tomography, as not only numbers (gray values), but matrices (complete scattering patterns) are reconstructed:
Macromolecules (2008), 41(20), 7637-7647

Perfect reconstruction occurs only, if the structure inside the fiber shows local fiber symmetry (LFS). This is only the case for voxels with axial grain (contrast undulation in fiber axis direction). Structural entities with radial or tangential grain, respectively, are characterized by restricted visibility in microbeam scanning experiments. This is the reason that the reconstruction of the central voxel is different: It has accumulated the structure information of all the structural entities with tangential grain.

Geometry of fiber mapping (Illustration with Polanyi sphere). Variation of the pattern on the detector upon tilting the fiber (in the center of the Ewald sphere)

Materials scientists frequently analyze shape and intensity of wide-angle X-ray reflections in diffraction. For this purpose they utilize mapping algorithms that have been devised by crystallographers for the analysis of the structure of the crystal lattice. These methods are slow, because they start from an approximated set of parameters that is refined iteratively and manually. But why should such a simple geometry as fiber mapping not give analytical equations? Admittedly, the refinement is necessary for crystallographers in order to determine high-fidelity lattice parameters but this is "only" a result of inherent inaccuracy of the measured pattern. As materials scientists we can tolerate such inaccuracy. In particular, as the crystallographer has already provided us with exact lattice parameters of our material. For example, if we register a WAXD pattern every 4 s during an hour, it becomes almost impossible to refine every image manually.

Critical review of literature shows from 1950 an awkward demonstration of fiber mapping geometry, from which an errative tilt-angle relation has been deduced. Moreover, some of the general statements on inaccuracy are indefensible. In summary, the puzzle can be solved. A simple automatic algorithm for the mapping of the detector pattern into reciprocal space is presented. It can be applied, as long as a reference reflection is at hand that is neither on the meridian nor on the equator of the fiber pattern. Nevertheless, A peculiar unclear aspect of the intensity norm is not mathematically resolved by us. Here we follow the equation used in crystallography that appears to be plausible from physics.

The fiber mapping

As the polypropylene fiber becomes hot, it becomes floppy in the X-ray beam. The tilt angle ß must be determined and considered for analysis.

After Polanyi the diffraction image of the fiber (i.e. the cylinder in the center of the Ewald sphere) in reciprocal space (black co-ordinate system) is considered to be spread on spherical surfaces around the origin of reciprocal space. One of these Polanyi spheres is shown (reddish glass). Images of a reflection are fixed to its Polanyi sphere. Because of fiber symmetry the reflection images degenerate into rings about the meridian (fiber axis). Detectable is the section between Polanyi sphere and Ewald sphere. This section is the reflection circle. Even if the fiber is tilted, the reflection circle does not change. What is changing upon fiber tilt is the information about reciprocal space along the reflection circle. The reflection circle is mapped on the detector by central projection. Its image is the detector circle. As the fiber is tilted, the images of the reflections are moving on the detector circle. From the positions of the reflection images the center of the detector pattern, the rotation and the tilt of the fiber can be determined with geometrical accuracy. Thus, the mapping problem is solved with sufficient accuracy for application in materials science.

In practice, the positions of the 4 reflection images of a known reflection are determined from the intensity maxima on the detector. Their center is the focal spot. From the positions the rotation angle and the tilt angle ß are determined with high accuracy. Using the crystallographical parameters of the material and of the reference reflection the measured pattern is mapped into reciprocal space. In doing so, the scattering intensity must be corrected. As proposed by Fraser et al., we only consider the radial component of the fiber mapping. The azimuthal component is neglected. We understand this from physics, but not from mathematics.