computer programs
MATSAS: a smallangle scattering computing tool for porous systems
^{a}Lyell Centre, Institute of GeoEnergy Engineering, HeriotWatt University, Research Avenue South, Edinburgh EH14 4AS, United Kingdom, and ^{b}Heinz MaierLeibnitz Zentrum (MLZ), Forschungszentrum Jülich GmbH, Jülich Centre for Neutron Science (JCNS), Lichtenbergstrasse 1, Garching 85748, Germany
^{*}Correspondence email: ar104@hw.ac.uk, amirsaman.rezaeyan@gmail.com
MATSAS is a scriptbased MATLAB program for analysis of Xray and neutron smallangle scattering (SAS) data obtained from various facilities. The program has primarily been developed for sedimentary rock samples but is equally applicable to other porous media. MATSAS imports raw SAS data from .xls(x) or .csv files, combines smallangle and very small angle scattering data, subtracts the sample background, and displays the processed scattering curves in log–log plots. MATSAS uses the polydisperse spherical (PDSP) model to obtain structural information on the scatterers (scattering objects); for a porous system, the results include (SSA), porosity (Φ), and differential and logarithmic differential pore area/volume distributions. In addition, pore and surface fractal dimensions (D_{p} and D_{s}, respectively) are obtained from the scattering profiles. The program package allows simultaneous and rapid analysis of a batch of samples, and the results are then exported to .xlsx and .csv files with separate spreadsheets for individual samples. MATSAS is the first SAS program that delivers a full suite of pore characterizations for sedimentary rocks. MATSAS is an opensource package and is freely available at GitHub (https://github.com/matsassoftware/MATSAS).
Keywords: MATSAS; smallangle scattering; polydisperse spherical model; porous media; computer programs.
1. Introduction
Smallangle scattering (SAS) of neutrons and Xrays (SANS and SAXS, respectively) is widely used for the nondestructive study of the lowresolution structure of natural and engineered systems, including sedimentary rocks, biological macromolecules, composite nanomaterials and polymers on length scales between ångströms and micrometres in a single or combined experiment (Feigin & Svergun, 1987; Binder et al., 2000; Zemb & Lindner, 2002; Radlinski, 2006; Borsali & Pecora, 2008; Anovitz & Cole, 2015; Melnichenko, 2015; Fritzsche et al., 2016). Advances in SAS instrumentation, such as neutron and highflux Xray synchrotron beamlines, have significantly increased the use of SANS and SAXS experiments (Melnichenko, 2015; Zemb & Lindner, 2002; Heenan et al., 1997). With the availability of these technologies, modern instruments can provide highquality data in time or spaceresolved experiments or measurements under various physical and chemical conditions, such as temperature, pressure, humidity etc. (Konarev et al., 2006; Schrank et al., 2020). The theoretical and methodological developments obtained over the past few decades have allowed the retrieval of structural information from SAS patterns to address questions revolving around the size, shape, distribution and orientation of scatterers (scattering objects) (Konarev et al., 2006; Petoukhov et al., 2012).
Neutron and Xray scattering techniques complement each other, but neutrons and Xrays are different in their charge, energy and interaction with matter, which makes each technique subject to its own experimentation type and/or sample type (Binder et al., 2000; Zemb & Lindner, 2002; Melnichenko, 2015). Fig. 1 illustrates a pinhole SAS experiment. Neutrons or Xrays are collimated and monochromated towards the sample, inside which a neutron or photon is elastically scattered from its wavevector k_{0} into a state with wavevector k under a scattering angle 2θ. The magnitude of a wavevector relates to its wavenumber, which is k = k_{0} = k = 2π/λ for where λ is the neutron or Xray wavelength. The intensity of the scattered radiation dI is therefore measured in the direction k as a function of the momentum transfer (the convention s = k − k_{0}) or the scattering vector Q. The magnitude of the scattering vector is given by Q = 4πsinθ/λ, from which it follows that Q = 2πs, where s = 2sinθ/λ (Radlinski, 2006; Melnichenko, 2015).
The incident Φ_{0}, i.e. Φ_{0} = I_{0}/A, where I_{0} is the incident intensity (neutrons or Xrays per second) and A is the beam crosssectional area at the sample position (Radlinski, 2006). The scattered intensity monitored in the solidangle element dΩ targeted by Q can be expressed as
of the scattering objects is denoted bywhere dΣ is the elemental scattering The quantity dΣ/dΩ is called the differential of scattering (Radlinski, 2006). The aim of SAS experiments is to determine volumeaveraged information on the spatial distribution of the scattering length density (neutrons) or electron density (Xrays) in the sample from the measured dΣ/dΩ as a function of the scattering vector magnitude Q, thus or I(Q) (Melnichenko, 2015).
For a wide range of substances, SAS data for hard and soft matter can generally be interpreted accurately using a twophase approximation (Melnichenko, 2015). In this approximation, the scattering volume is viewed as being composed of abovemolecularsize phases, each characterized by one of two possible values of the physical property that provides the scattering contrast (Δρ*). For instance, for porous media, these two phases are the solid matrix (phase 1) and the pore space (phase 2) (Radlinski, 2006). The twophase approximation is a simplification inherent in the SAS method and has been implicitly or explicitly employed for many years. As such, the general expression of the scattering can be expressed as
where N is the of scatterers N_{p} per unit volume, V_{p} is the volume of the scatterers, and and are the scattering length/electron density of phase 1 and phase 2, respectively. B is the sample background, accounting for scattering in the highQ limit. The highQ background originates from (i) Qindependent caused by hydrogen atoms in organic matter and/or water, and (ii) Qdependent coherent scattering resulting from microscopic inhomogeneities (e.g. small pores in the rock matrix; Bahadur et al., 2015; Blach et al., 2020). P(Q) is the form factor which describes the size and shape of the scatterer. There are analytical expressions for the form factor for simple geometrical objects like spheres, cylinders, discs or parallelepipeds (Melnichenko, 2015). S(Q) is the and contains information about the spatial distribution of the scatterers. The represents the modification of the intensity due to the spatial correlation of the scatterers (Fritzsche et al., 2016), where the positions of the scatterers are frozen in time and space in solid porous materials (Melnichenko, 2015). In softmatter systems, the interaction potential between scatterers is also taken into consideration (Melnichenko, 2015). Form and structure factors need to be specified to determine the structural information in the scattering curves.
Several SAS programs have been developed in different laboratories that consider various data processing and manipulation methods, fitting models, and form and structure factors to characterize the structure of the scatterers (Table 1). Recognizing the increasing application of SAS data to analyse the pore structure of sedimentary rocks, especially lowpermeability rocks such as coal and mudrocks or gas shales (Radlinski, Ioannidis et al., 2004; Radlinski, Mastalerz et al., 2004; Radliński et al., 2009; Mares et al., 2012; Clarkson et al., 2012; Mastalerz et al., 2012; Melnichenko et al., 2012; Bahadur et al., 2014, 2015; Anovitz et al., 2015; Leu et al., 2016; Busch et al., 2017, 2018; Anovitz & Cole, 2018; Sakurovs et al., 2018; Vishal et al., 2019; Blach et al., 2020), we have developed the program package MATSAS. It allows the analysis of data obtained from smallangle and very small angle scattering of neutrons and Xrays [very small angle neutron scattering (VSANS), smallangle neutron scattering (SANS), wideangle Xray scattering (WAXS), ultrasmallangle Xray scattering (USAXS) and smallangle Xray scattering (SAXS)].

MATSAS analyses data from pinholegeometry, timeofflight (TOF) and Bonse–Hart machines and was tested using data acquired at FRMII (Research Reactor Munich II, Garching, Germany) and ORNL (Oak Ridge National Laboratory, Tennessee, USA) (Rezaeyan, Pipich et al., 2019a,b; Rezaeyan, Seemann et al., 2019; Seemann et al., 2019). MATSAS does postprocessing of data obtained from research facilities. It is assumed that initial corrections for sample thickness, transmission, detector sensitivity, instrument background, multiple scattering and noise have been made using the instrumentspecific settings at the facility itself, providing data in absolute units (Hinde, 2004; Melnichenko, 2015). MATSAS is primarily oriented towards the structural analysis of sedimentary rocks using a polydisperse spherical (PDSP) model. The MATSAS software is constantly refined to broaden its functionality, making it applicable to isotropic and partially ordered objects such as biological nanoparticle systems, colloidal solutions, and polymers in solution and bulk. It is an opensource computer tool for academic users and is freely available on GitHub (https://github.com/matsassoftware/MATSAS). Opensource access reflects transparency in the fundamental assumptions and solving approaches employed in the program and allows third parties to interface their inhouse programs with the data analysis framework of the program (Liu et al., 2012) and to help in accelerating its development. In this paper, we summarize the main components of MATSAS and its development framework.
2. Program overview
MATSAS features a scriptbased package in MATLAB (The MathWorks Inc., Natick, MA, USA), which integrates computation and visualization in an easytouse environment. The MATSAS program is a versatile computer tool allowing both users and developers to add additional tools and develop specific novel applications. The flexible userfriendly framework of MATSAS for basic routines, such as intensity calculation or model alignment, allows anyone with basic programming skills to improve or adapt MATSAS to better reflect userspecific needs. Furthermore, the current version of the package includes the PDSP model to analyse SAS data in terms of theoretical intensity computation, the f(r) probability function of pore size distribution and model The PDSP model is the method commonly used for SAS analysis of a polydisperse system of randomly oriented independently scattering particles and is ubiquitous for fractal microstructures (e.g. sedimentary rocks) as well as other porous systems (Radlinski, Ioannidis et al., 2004), provided that the particleshape distribution is independent of the distribution of particle dimensions in the polydisperse system (Schmidt, 1982). The scriptbased MATSAS code allows parameters to be tuned for more features of each routine.
Use of the MATSAS program is divided into three steps: (i) preprocessing of raw or facility postcorrected SAS and very small angle scattering (VSAS) data as well as physical information, (ii) processing of the imported information to produce I(Q) versus Q curves, combine the SAS and VSAS curves, and fit the PDSP model, and (iii) postprocessing to display and export structural information obtained from the samples being analysed. Fig. 2 illustrates the main components of the present version of MATSAS.
Detailed instructions on how to use the package are available on GitHub. Supporting information and command descriptions are embedded in each module. Errors and bugs can be invoked when no parameters or incorrect data are given to the command. We developed the package in Windows and recommend running it in Windows, Mac or Linux, with any Intel or AMD x8664 processor with four logical cores and AVX2 instruction set support, as a minimum. Although the program runs satisfactorily without a specific graphics card, a hardwareaccelerated graphics card supporting OpenGL 3.3 with 1 GB graphical processing unit (GPU) memory is recommended, as displaying figures and generating Microsoft Excel worksheets require more background processing.
3. Data preprocessing
The data preprocessing module is composed of two components: (i) data are prepared in a Microsoft Excel spreadsheet [*.xls(x)] or *.csv file, including (V)SAS data, neutron scattering length densities or Xray electron densities of phases 1 and 2 (e.g. rock matrix and pore), grain density of the sample, data reduction limits (optional), and sample name, and (ii) the MATLAB data_input.m file reads and stores the imported data for the next step. MATSAS allows users to run a batch of samples. The units in the input files can be converted between different unit systems (between nm^{−1} and Å^{−1} for Q, for instance) by changing appropriate codes. The range(s) of data points can be adjusted for each data set individually or simultaneously for selected groups of files.
4. Data processing
The data processing module is used to manipulate and analyse the information imported. The primary data processing script is developed to manipulate scattering curves. The data_manipulation.m program carries out multiple tasks, including I(Q) data sorting, curve fitting, background subtraction, curve merging, curve smoothing and raw data reduction. The secondary data processing script file in data_analysis.m is designed to analyse I(Q)–Q curves and produce structural information. An arbitrary size distribution is created first and the PDSP model is then fitted to the processed scattering curve. Pore characteristics are predicted and fractal dimensions (including the pore fractal dimension, D_{p}, surface fractal dimension, D_{s}, and general fractal dimension, D_{f}) are evaluated from the fit in this module.
4.1. Data manipulation
The program data_manipulation.m is a data processing module encompassing major SAS data processing steps for isotropic systems, from merging of scattering curves to background reduction. This program performs manipulations with onedimensional data sets and calls other analysis and fitting programs via userdefined or builtin function files. The SAS data might have been collected at different sampletodetector distances. Once data from several experimental curves have been combined for one specific instrument (e.g. SANS), they may not be sorted, which leads to numerical problems in further analysis. Data sorting is therefore carried out in the data manipulation package using a builtin function. If the SAS data consist of two scattering profiles obtained from two different instruments (e.g. VSANS and SANS), MATSAS allows users to merge the two curves using a leastsquares fit in the overlapping range, as illustrated for example in Fig. 3. The SAS curve is the basis onto which the VSAS curve is rebinned. The highQ background is subtracted using equation (3),
where the scattering varies with Q^{−a} in the highQ limit before plateauing (Melnichenko, 2015). The value of the background is determined from a linear plot of equation (4),
where is the slope and A is the intercept (Melnichenko, 2015). Fig. 3 shows the background subtraction in the highQ limit for a range that users can change manually in the program.
A noiseremoval operation is embedded to remove the sparse data around the beam stop or detector edge. Raw data reduction, whose cutoff limits are determined in the data input files, is carried out as well. Two data smoothing operations are included in the package, which can be employed to obtain a smooth scattering profile for further structural analysis. Fractal dimensions and slope are determined here. For all operations, the propagation of uncertainty is performed using standard equations (Bevington & Robinson, 2003). A MATLAB plotting operation displays the currently active scattering profiles on a log–log scale. An advanced plotting option included in the plot permits users to change the plotting range, enlargement factor etc. The data manipulation file contains an output section, where the result of each operation can be used in subsequent data analysis. Information about the operation (type of operation, section names, functions, weights, ranges of points used etc.) is written in the package in green and allows modification or change of lines if needed.
4.2. Data analysis
The data analysis program calculates the intensity of SAS from a polydisperse system of scatterers (Porod, 1951, 1952; Guinier & Fournet, 1955). The intensity is expressed in terms of a fractal distribution of scatterers, also called the probability density of the pore size distribution f(r), for greater numerical stability (Ilavsky & Jemian, 2009). SAS curves from sedimentary rocks are usually linear on a log–log scale, particularly in the highQ region, which reflects fractal behaviour (Melnichenko, 2015). Scattering from a fractal surface is equivalent to scattering from a system of polydisperse spherical scatterers (Schmidt, 1982), with a number–size distribution (the number of spheres with radii between R and R + dR) given by
where D_{f} is the fractal dimension determined from the slope of the powerlaw scattering (Melnichenko, 2015). In practice, the distribution described in equation (5) and in the range R_{min} ≤ R ≤ R_{max} shows fractal behaviour between the upper and lower cutoff parameters. f(r) is expressed as
This is valid for R_{max} > R_{min} > 0 and D_{f} ∈ (−1, ∞), where D_{f} = 6 + slope. Scattering from a PDSP featured sample has a linear region with a similar slope −(1 + D_{f}) and is described by (Radlinski, Ioannidis et al., 2004)
where is the volume of a sphere of radius r (volume of a scatterer). In addition, P(Q, r) is the form factor of a sphere of radius r (Guinier & Fournet, 1955):
N is the total number of scatterers, which is related to the number size distribution as N(r) = Nf(r). N(r) is expressed as
where
is the scattering intensity at Q = 0 and is the average volume of the scatterers (Radlinski et al., 2002). Similarly to the approach of Ilavsky & Jemian (2009), MATSAS calculates equation (7) throughout the integration over a continuous size distribution with a summation over a discrete size histogram:
where the subscript i represents different scattering sizes and the subscript j describes bins in the size distribution. Δr_{i, j} is the width of bin j and each scattering size has its own binning index i, j. r is the dimension of the scatterer (radius for spheres) and has limits r_{max i} and r_{min i}. The radius r is calculated using R = 2π/Q, which is R = 2.5/Q in the fractal distribution (Radliński et al., 2000).
MATSAS uses an arbitrary size distribution to model the scattering volume distribution V^{2}(r)P(Q, r) and determine f(r). The user can change the theoretical ranges of the various size distributions in the data analysis program. Numerical calculations call limits on the range of dimensions (r_{min} and r_{max}), the cutoff limits (R_{min} and R_{max}) and the number of bins (N_{bin}). This method results in a natural logarithmic step in dimension and uses three parameters, R_{min}, R_{max} and N_{bin}. The centres of the first (r_{i, 1}) and last () bins are R_{max} and R_{min}, respectively, and extra fractional volumes are discarded for both bins: the volumes associated with and for the first and last bins, respectively. The widths of the bins are equal by selecting associated dimensions at regular increments of the cumulative distribution (Ilavsky & Jemian, 2009), leading to
However, the numerical operation of the data_analysis.m file requires r_{min i, j}, r_{i, j}, r_{max i, j}, f_{i}(r_{i, j}) and IQ_{0i} to fit the PDSP model in equation (7) to the measured I(Q) curve. The fitting procedure employs f(r) and IQ_{0} as fitting parameters for each iteration to attain a match where the summation of square errors (SSQ) tends to a minimum (Hinde, 2004).
To reduce the computation time taken by numerical integration, we found an analytical solution for the scattering volume distribution
that transforms equation (11) into
MATSAS simplifies the intensity calculation by substituting equation (9) into equation (14), leading to
Once the match is reached, the data analysis program yields the structural characteristics of the scatterers using the fitted f(r) and IQ_{0} values. The (SSA) of the scatterers is obtained following Hinde (2004):
where the subscript k represents bins in the size distribution. The of scatterers per unit volume (Φ) is calculated from equation (9), which results in
and the total volume of scatterers (V_{p}) is obtained by
where the subscript k represents bins in the size distribution. Differential (dV/dr or dA/dr) and logarithmic differential (dV/dlogr or dA/dlogr) scatterer size distributions are calculated cumulatively (Meyer & Klobes, 1999).
The scattering intensity decays as Q^{−m} with different powerlaw exponents m; this indicates that m is related to the dimensionality of the pore as understood in terms of the concept of fractality (Mandelbrot, 1983). For a fractal pore scatterer, therefore, D_{p} = m with values 1 < D_{p} < 3, and for a surface fractal D_{s} = 6 − m with values 2 ≤ D_{s} ≤ 3 (Bale & Schmidt, 1984).
The scattering at different length scales indicates the Guinier, mass/pore fractal, surface fractal and Porod regions, suggesting that each fractal region is limited to a specific range of scattering vectors (Fritzsche et al., 2016). Therefore, for sedimentary rocks D_{p} and D_{s} are geared to the ranges of 0.0003–0.003 cm^{−1} and 0.003–0.03 cm^{−1}, respectively. In addition, D_{f} is included to reflect the fractality of the full pore system over the entire scattering vector range, e.g. 0.0003–0.03 cm^{−1} in sedimentary rocks (Rezaeyan, Pipich et al., 2019a,b; Rezaeyan, Seemann et al., 2019). These ranges can be changed by the user.
For demonstration purposes, we tested the analysis operations on SANS and VSANS data obtained from three rock samples (Opalinus Clay) using batch mode. Opalinus Clay is a Jurassic mudrock that was obtained from the Mont Terri Underground Laboratory in Switzerland and has been described in detail previously (Busch et al., 2017). Fig. 4(a) shows the PDSP modelled I(Q) curves and the measured I(Q) curves after two iterations of the fitting operation. The first iteration starts with initial guesses for f(r) and IQ_{0}, which are obtained from the slope of the scattering curves and the Guinier & Fournet (1955) approximation, respectively. SSQ tends to a minimum after the second iteration; two iterations are recommended for most rock samples (Hinde, 2004). Fig. 4(b) shows f(r) after two iterations on a log–log scale. f(r) levels off at scatterer sizes ≳2 µm because the scattering intensities of large scatterers are smeared, possibly due to instrument artefacts at the edge of the detector. The error sensitivity, expressed as dSSQ/dlog(IQ_{0}), relates SSQ to the number of iterations [Fig. 4(c)]. The value of dSSQ/dlog(IQ_{0}) varies around zero for all scatterer sizes. However, as illustrated in Fig. 4(c), this can deviate where the fit is rather poor for large scatterer sizes () due to different instrument resolutions or noise within overlap areas. SSQ is magnified when the number of iterations exceeds two, resulting in an attenuation of f(r). Nevertheless, we recommend attaining a smooth f(r) if the optimum fit requires a larger number of iterations for a specific sample. χ^{2} tolerance can be used for the fit when the user has no preference for the number of iterations.
We also tested the PDSP model on three polydimethylsiloxane (PDMS) polymers with volume fractions of 0.128, 0.25 and 0.5 in toluene to demonstrate the applicability of the fitting operation for a nonpowerlaw nanostructure in solution [Figs. 4(d)–4(f)]. Fig. 4(f) displays the numerical flexibility of the fitting procedure after 20 iterations.
5. Data postprocessing
The data postprocessing module is made of two components, including data_output.m in MATLAB and the results reported in figures and tabulated files. The data_output.m file calls the results of individual samples, reports the results in figures and tables in the MATLAB command window, and writes the results in output.xlsx. The results include measured, processed and predicted scattering curves, fractal distribution fit (f_{r}), (SSA), porosity (Φ), pore volume (V_{p}), pore size distribution (PSD) by pore volume or pore area, fractal dimensions, slopes of scattering curves, pore characteristics divided into macro, meso and micropores, and background subtraction values. Some results for the Opalinus Clay samples are shown in Fig. 5 and Table 2. The results for individual samples are produced and saved in figure formats (*.tif and *.emf), Excel spreadsheets and .csv files for use in further specific analyses. The results are usable if the raw SAS data are provided in absolute units; otherwise users must report pore characteristics in arbitrary units.

6. Conclusions
MATSAS encompasses a set of modules allowing for a full analysis of (V)SANS and (V)SAXS data from porous systems, e.g. sedimentary rocks. MATSAS is written in MATLAB and combines a desktop environment tuned for data processing and structural analyses with pre and postprocessing modules. The preprocessing module is used to import data from Microsoft Excel spreadsheets or .csv files into MATLAB. The main module performs data manipulation and analysis in which I(Q)–Q curves are processed and the PDSP model is fitted to produce structural information for porous systems. The postprocessing module displays results in the form of tables and figures and exports them in Microsoft Excel spreadsheets or .csv files. MATSAS is the first SAS program that provides a full suite of pore characterizations. The programs included in MATSAS are publicly available on GitHub (https://github.com/matsassoftware/MATSAS) for academic users.
Acknowledgements
SANS and VSANS measurements on the rock samples tested in this study were performed on the KWS1 and KWS3 instruments at the Jülich Centre for Neutron Science (JCNS) at the Heinz MaierLeibnitz Zentrum (MLZ) in Garching, Germany. We are very grateful for the beam time obtained. We also thank Masoud Ghaderi Zefreh at the University of Edinburgh for assisting with MATLAB programs, and Gernot Rother at Oak Ridge National Laboratory, Artem Feoktystov at the Forschungszentrum Jülich GmbH and Lester Barnsley at the Australian Synchrotron for testing and reporting on MATSAS.
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