Data analysis of 2D-SAXS Patterns with Fibre Symmetry from some Elastomers

Universität Hamburg, Institut für Technische und Makromolekulare Chemie, Bundesstr. 45, D-20146 Hamburg, Germany

Using a special programming tool for scientific use (pv-wave), a program is developed in order to analyse 2D SAXS patterns with fibre symmetry quantitatively. After that the first detectors for synchrotron radiation become available which allow to record high resolution patterns with a good S/N-ratio in short time, there is urgent need for such evaluation tools. Using data recorded during the straining of two thermoplastic elastomers, the program and the underlying concepts are presented.

1. Experimental

2. Choice of the programming tool

Presently my program is simply a collection of procedures and functions, which are called from the interpreter prompt. The advantages of pv-wave are in particular short development cycles, the extensive library and the close relation to the programming language of "IDL?", another well-known programming tool for the scientist. Before one will be able to write effective programs for pv-wave, one will have to understand the paradigms of a programming language, which is designed for the evaluation of vectors and matrices.

3. Data structure and file format

Each of the evaluated patterns has to be archived. I store each pattern in a file of its own and map the complete data structure (pixel values and pattern parameters) without any conversion into the file. Thus even large images can be read and stored fast. Representations of floating point numbers and even of integers (big-endian vs. little-endian) vary from platform to platform (e. g. UNIX on an Intel-based machine and UNIX on a Silicon Graphics workstation). In order to read files across platforms using pv-wave, the interpreter offers a special binary format named "exchange data format?. This format enables any local pv-wave interpreter to convert foreign binary formats transparently into the local binary representations.

In fact, the achieved portability is not really far-reaching, since my algorithms rely on a particular data structure. Thus a new island solution is created. A major reason for this presentation was my interest in a discussion which may lead to a redesign according to a common proposal of an interchange data format for 2D scattering patterns.

Generation and definition of my image data structure can be extracted from the following example.

; img=sf_structure( $
; 'Sample', '19961001', $
; 500, 450 ); Generates a
; sf_structure with
; img.Width  = 500
; img.Height = 450
; img.map  (all zero
;  floating point matrix)
; img.BoxLen = (0, 0)
;  (floating point vector)
; img.Center = (0, 0)
;  (integer vector)
; img.Title = 'Sample'
; img.Date = '19961001'
;
FUNCTION sf_structure, $
 name, stamp, x, y; Generate structure and
; fill in parameter values
  RETURN, $  { , Width:x, $
       Height:y, $
       map:FLTARR(x,y), $
       BoxLen:FLTARR(2), $
       Center:INTARR(2),$
       Title:name, $
       Date:stamp }
END ; Func sf_structure

4. Raw data features

Figure 1: Raw SAXS data of a PBT/PEG, 1 min after recovering from an elongation epsilon=1.48. Image plate exposure: 2 min.

4.1 Shade of vacuum tube and background fog

The sharp shadow boundary can be used to align sample scattering and empty scattering before correction. This is necessary, since the positions of the patterns vary from gel-file to gel-file. By extrapolating a surface into the blind outer region it may be possible to extend information on the shape of the scattering.

4.2 Pattern tilt

4.3 Area shaded by beam stop

5. Concept of masks

Fig. 2 shows the mask which can be generated by first copying the image (m = r480) to a mask image and then setting all intensities to "true?, which are greater than 300 counts (m.map = r480.map GT 300). To see the result as presented in Fig. 2 the operator types tvscl, m.map. One observes that this mask is good for the determination of the position of the scattering pattern on the image plate. Moreover it can be used to set the noisy blind area to zero intensity (r480.map = r480.map * m.map).

Before the program can start to fill information into zero intensity pixels, the image must be centred and aligned. Masking m.map = r480.map GT 8000 yields the mask shown in Fig. 3. After manual erasure of the three fragments near the beam stop this mask can be used by an alignment procedure, which starts with computing centres of gravity from both the contiguous regions in the mask.

6. Filling in missing information

Figure 4: Examples of extrapolated aprons using 2D-SAXS data from different polyetheresters. Left: Arnitel A2000/20. Here the sample-detector distance is too long to give a reasonable 2D-extrapolation. Right: PBT/PEG. In this series all the extrapolated aprons show the expected shape of a SAXS fluctuation background.

In particular the 2D-extrapolation of a surface into the outer blind area shall be demonstrated. It is well-known from the quantitative analysis of 1D-SAXS that one should gather information on the shape of the SAXS curve over a wide angular range including the beginning of the WAXS curve [1]. This ensures that the determination of the electron density fluctuation background is possible. If the measured data do not extend far enough into the region of Porod's law, one will have to guess the fluctuation background and information on the size distribution of domains is inaccessible.

An extrapolation may be superfluous, if the vacuum tube can be widened without risking failure of the vacuum window, if the distance between sample and detector can be shortened without loss of information in the central part or if it is possible to measure the same state of the sample with two different instrument adjustments. The special benefit of a 2D-extrapolation may result from the fact that the fitting of a surface to given points is stiffer than the 1D-fitting of a curve. Thus it might follow the expected shape of a 2D-SAXS pattern, if only the measured data are sufficient for a prediction. Since we generally record patterns from a series of states, we can compare the extrapolated aprons and thus obtain some heuristic measure on the validity of the extrapolations. Examples are shown in Fig. 4.

7. Evaluation of the patterns

Modelling of the 2D-scattering pattern as a whole is a challenging task for future quantitative evaluation methods. But in analogy to the one-dimensional case a quantitative evaluation can be performed on curves, which can be extracted from the scattering patterns by projection operations. Thus the "longitudinal structure? and the "transverse structure? of the fibre may be studied [3] by application models which describe the hard domains and the soft domains in the elastomer by 1D- or 2D-models in real space. The scattering intensity of the longitudinal structure I₁(s₃) is given by the projection

of the pattern intensity I(s₁₂,s₃) and the scattering intensity of the transversal structure I₂(s₁₂) is defined by the projection integral

The latter can be treated like any SAXS curve measured with the Kratky camera. Each of the two projections from the same scattering pattern has its own Porod law, and if the range of the measured data is insufficient for a complete evaluation of I₁(s₃), the data in I₂(s₁₂) may still suffice for a quantitative evaluation of the transversal structure.

Figure 5: Interference function G₁(s₃) computed from vertical projection I₁(s₃) after correction for the scattering effects of non-ideal multiphase systems. Data from sample PBT/PEG at an elongation epsilon=1.48.

The effect of an insufficient measuring area can easily be demonstrated by comparing different evaluations of the same sample. One of the evaluations relies on the measured pattern, the second is based on the same data after extrapolation of an apron. In order to find Porod's law in the projection I₁(s₃) obtained from the narrow area data, one has to assume a negative fluctuation background, which is of no physical sense. The corresponding projection from the wide area data shows a well-expressed Porod law with a positive fluctuation background, if the shape of the apron was of physical sense (as found for patterns from the sample PBT/PEG). Quantitative evaluation of the longitudinal structure is based on the analysis of an interference function G₁(s₃)[4], in which the heights of the soft domains and the heights of the hard domains are given by attenuated cosine-functions. The frequency of the cosine is related to the average domain height; the attenuation factor is related to the width of the domain height distribution. An example is shown in Fig. 5. The position of the peak and the general shape of the interference function are not altered by missing data and thus the average domain heights should be extractable even if the measured area is rather narrow. On the other hand, the attenuation is considerably higher in the curve obtained from the narrow scattering pattern. Thus the determination of the width of domain height distributions becomes wrong. Model fitting on several series of straining data from thermoplastic elastomers confirm this assessment. They show that determined distribution widths as a function of elongation can only be connected by a smooth curve, if the previous analysis of the Porod region has been performed in a sufficiently long angular interval.

References

[2] Stribeck, N., Sapoundjieva, D., Denchev, Z., Apostolov, A.A., Zachmann, H.G., Stamm, M., Fakirov, S. (1997) Macromolecules 30, 1329-1339

[3] Bonart, R. (1966) Colloid Polym. Sci., 211, 14-33

[4] Ruland, W. (1977) Colloid Polym. Sci. 255, 417-427