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Michelle A.
Calabrese
,
Norman J.
Wagner
and
Simon A.
Rogers†
*

University of Delaware Department of Chemical and Biomolecular Engineering, Center for Neutron Science, 150 Academy St., Newark, DE, USA. E-mail: srogers@udel.edu; Fax: +1 302-831-1048; Tel: +1 302-831-8079

Received
15th December 2015
, Accepted 4th January 2016

First published on 12th January 2016

A deconvolution protocol is developed for obtaining material responses from time-resolved small-angle scattering data from light (SALS), X-rays (SAXS), or neutrons (SANS). Previously used methods convolve material responses with information from the procedure used to group data into discrete time intervals, known as binning. We demonstrate that enhanced signal resolution can be obtained by using methods of signal processing to analyze time-resolved scattering data. The method is illustrated for a time-resolved rheo-SANS measurement of a complex, structured surfactant solution under oscillatory shear flow. We show how the underlying material response can be clearly decoupled from the binning procedure. This method greatly reduces the experimental acquisition time, by approximately one-third for the aforementioned rheo-SANS experiment.

Time-dependent scattering is often analyzed by an averaging procedure known as ‘binning.’ The binning procedure groups the scattering information into fixed intervals (bins) of duration t_{w}, referred to as the bin width. Here, the material response is examined on a timescale much shorter than that of the full experiment, t_{w} ≪ T. By dividing the scattered particles into discrete time bins, the material properties are averaged over t_{w} and are assumed to be relatively constant within the bin. For example, in a scattering experiment of 30 minute duration (T = 30 min), researchers may choose to examine the structural changes after every minute (t_{w} = 1 min). In this case, the average material structure per one-minute time bin is analyzed.

In order for such analysis to be performed, the detector must time-stamp each detected particle in addition to recording its spatial position. Fig. 1 shows an example data set, where the X and Y position on the detector and the time of detection for each scattering event are shown in three dimensions for a typical small angle neutron scattering (SANS) experiment. The red data points in Fig. 1 represent neutrons scattered during a cycle of an oscillatory shear experiment of period T, so that the time axis is normalized as t/T. The spatial and temporal dependence of the scattering events leading to this detector response, as shown in Fig. 1, is indicative of the continuously varying dynamics inherent to the system of interest.

In the standard binning method, three dimensional spatially- and temporally-resolved data are reduced to a sequence of two-dimensional patterns, where the detector response to the scattering is grouped into non-overlapping time bins of duration t_{w}. As shown in Fig. 1, the detected scattering events are binned with t_{w} = T/10 (between black or colored lines) to form a total of ten scattering patterns over the experiment period, T. By using the standard binning method, all derived material properties have an equal temporal resolution and precision of t_{w}. The temporal precision of the binned data is determined by the bin width, t_{w}, whereas the temporal resolution is determined by the time step between consecutive bins, which in the standard binning method is also t_{w}. Any properties that are changing on time scales faster than t_{w} cannot be detected. The scattering patterns shown result from the data binned between each set of colored lines and contain information about the material properties during that specific portion of the oscillation. In the previous example of a 30 minute scattering experiment with a bin width of one-minute, the standard binning method would result in 30 total bins each of duration one minute. Any structure changes occurring faster than one minute would not be detected using this method.

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c(t) = m(t)⊗b(t,t_{w}) | (4) |

{c(t)} = C(ω) = {m(t)⊗b(t,t_{w})} = M(ω)·B(ω,t_{w}). | (5) |

(6) |

(7) |

(8) |

The results in Fig. 2(a) represent the material response that is obtained when different bin sizes are chosen to process the data. For convenience, the bin duration, t_{w}, is expressed as an even integer fraction of the period of oscillation, T, such that t_{w} = T/N. Here, the temporal resolution is approximately continuous through t/T, whereas the temporal precision is determined by the bin width, T/N. Larger integers thus correspond to shorter bins, better approximating the underlying material response. Clearly, the results of binning alone are dependent on the bin size, as significantly different responses are seen with changing N. In Fig. 2(b), the data from Fig. 2(a) is deconvolved to obtain the true material response. The result of the deconvolution is completely independent of the bin size, a property required of the material response.

On the basis of the data displayed in Fig. 2, a bin as wide as half of the period, T/2, can be used, in theory, to accurately determine the material response via deconvolution. Such analysis has clear implications in the length of time required to carry out a time-resolved scattering experiment. In many studies where binning is used, bin widths on the order of T/30 are employed to provide a reasonable balance between temporal resolution and experiment duration.^{5,14} If a constant number of scattering events per bin is desired (normally around 100000 –250000 or greater, depending on the information sought by the researcher), then scattering events sufficient to fill bins of width T/2 can be collected 15 times faster than the number of scattering events required to fill bins of width T/30. However, experimental limitations and noise in the data limit the practical bin size that can be applied to reduce scattering time, which will be discussed in the next section.

The signals in Fig. 4 are calculated by deconvolving the time-average of the original signal featured in Fig. 2 with different levels of added noise. These calculations are the same as those performed in Fig. 2; however noise was added to the time-averaged signal before performing the deconvolution, such that the results are no longer analytically defined. Gaussian-distributed zero-mean white noise was generated in Matlab, where one standard deviation of the noise amplitude was equal to 0.1%, 0.25% or 0.5% of the mean amplitude of the time-averaged signal. Fig. 4 clearly shows the effect of noise on the deconvolved signal at different bin widths (T/2, T/10, T/32), and of truncating the full Fourier spectrum at different frequencies, ranging from the full frequency spectrum (no truncation) to 1/32 of the full spectrum (number of frequencies used to construct the original signal). As seen in Fig. 4, the closer the truncation frequency is to the number of frequencies used to generate the signal, the more noise is filtered, leading to a smoother signal. However, the end of the Convolution and deconvolution section illustrated a case where a bin width of T/2 can reduce the time that would currently be taken to acquire data sufficient for bins of width T/30 by a factor of 15. However, as Fig. 4 shows, there are limitations in the bin width that should be chosen when performing an experiment. Bins of width T/32 handle the noise the best, as the smaller bin width leads to sharper features in the time-averaged signal, which results in a high signal-to-noise ratio. Using bins of T/32, the deconvolution result after truncation is nearly unaffected by the increasing noise level within the range examined. However, the effect of the added noise is pronounced at bins of T/2, where even the truncated deconvolution results greatly differ from the expected original signal. Smaller bin widths are optimal in the deconvolution procedure for several reasons. The width of the sinc function in frequency space is inversely proportional to the bin width in real time, meaning that smaller bin widths cover a wider frequency range. The number of roots in the sinc function also increases with wider bin sizes in the same frequency range, leading to more information loss with wider bins. Lastly, the amplitude of the sinc function at high frequencies is larger when the bin size is smaller, leading to a the higher signal-to-noise ratio when the deconvolution is performed. However, by truncating the spectrum at all bin widths, the deconvolution result becomes closer to that of the original signal. The truncation of the frequency spectrum is important to perform during this procedure (Fig. 3 and 4) such that only the frequencies with amplitudes significantly above the level of the noise are used to reconstruct the discrete true signal.

If a function x(t) contains no frequencies higher than B Hz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) s apart.

The sampling theorem sets the upper limit on the angular frequencies that can be determined by the Fourier transformation of discretely-measured data. The upper frequency limit of the band, denoted by B, and the size of the time step, t_{s}, are therefore related by:

(9) |

(10) |

(11) |

(12) |

Fig. 5 shows the results for the standard binning method, the sliding binning method, and the full deconvolution (translated for visual aid). First, the alignment factor was calculated for each of the thirty, non-overlapping bins (red) using the standard binning method (t_{w} = t_{s} = T/30). The sliding binning method (gray) was performed on the same data set using overlapping bins with a step size five-fold smaller than the bin width (t_{w} = T/30, t_{s} = T/150 = t_{w}/5), yielding a five-fold improvement in the temporal resolution and 150 total bins. While a value of n = 5 (as seen in Fig. 3) provides sufficient temporal resolution to perform the deconvolution for bins of width T/30 in this experiment, larger values of n may be required when the chosen bin size is larger than T/30. The experimental data processed using the sliding bin method (gray) was then deconvolved to elicit the true (discretized) material response (black). The discrete nature of the standard binning method, which leads to poor temporal resolution, is apparent in Fig. 5, whereas the sliding binning method presents improved temporal resolution. The sliding bin curve (gray) exhibits small oscillations in the alignment factor signal, suggestive of a higher frequency material response, that are not seen with the standard binning method. While portions of the sliding bin curve could be interpolated from the standard bin points, the existence of the higher frequency oscillations in the material structure and the position and value of the maximum alignment would not be resolvable via interpolation. The deconvolved signal (black) displays sharper oscillations in A_{f} that reflect the true material alignment. The residuals in Fig. 5 highlight the improvements to feature sharpness and quality obtained by the deconvolution procedure.

In Fig. 6, we examine the higher frequency oscillations in the alignment factor and compare the deconvolved, true alignment (black) to the measured shear stress (blue). The alignment factor oscillations clearly correspond to similar oscillations in the measured shear stress. These features cannot be resolved using the standard binning method due to poor temporal resolution and limited number of points, making it more difficult to derive information from the stress-SANS law,^{17} or other empirical relationships. Simply reducing the bin size will not improve the situation, as the statistical accuracy of the data decreases correspondingly. The improved temporal resolution and feature clarity gained from the deconvolution provide a more direct and accurate pathway to quantitatively link the structural alignment to the measured bulk rheological properties. The deconvolution process also provides a method for reducing total experiment time required to achieve a given accuracy in scattering experiments. However, the noise level ultimately limits the resolution that can be obtained. Similar to the results shown in Fig. 4, the experimental deconvolution results for this data set are insensitive to bins of width T/20 compared with T/30 (data not shown), leading to a reduction in scattering (data acquisition) time of one-third.

In additional oscillatory shear conditions, we found that the deconvolution results were also unaffected by the reduction in bin width from T/30 to T/20 (data not shown). Depending on the features present in the signal, further time reduction can be obtained by implementing larger bin sizes. The time-averaged signal in Fig. 6 has two small oscillations on the order of T/10 in the alignment factor and stress signal: one from 0.45 < t/T < 0.55 and the other from 0.55 < t/T < 0.65. Therefore, when a bin size on the order of T/10 is used, the magnitude of the Fourier transform at these frequencies is roughly equivalent to the noise, making an accurate deconvolution difficult to perform. As the material stress response contains complementary features to the alignment factor response, the appropriate bin width can be estimated by examining the width of the stress response features and then choosing a smaller bin size. By decreasing the bin size from T/10 to T/20, the deconvolution was successfully performed for this data set.

In summary, we have shown a procedure to determine the true, underlying material response from a time-periodic signal given discrete data points, such as the alignment factor calculated from neutron events. This is accomplished by rigorously deconvoluting the scattering data, which is unavoidably operated on by a sliding boxcar function in time during data processing. When applied to a particular time-resolved data set, the proposed sliding bin method gives the same temporal precision as the standard bin method, while greatly improving the temporal resolution. The full deconvolution procedure yields the true material response, which is unobtainable using binning alone. This procedure is general and can be applied to many forms of time-periodic scattering data.

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## Footnote |

† Present address: University of Illinois at Urbana-Champaign, Department of Chemical and Biomolecular Engineering, 600 S. Matthews Ave., Urbana, IL, USA. Tel: +1 217 333 0020; E-mail: E-mail: sarogers@illinois.edu |

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