Elsevier

Acta Materialia

Volume 115, 15 August 2016, Pages 155-166
Acta Materialia

Full length article
A new approach to the analysis of short-range order in alloys using total scattering

https://doi.org/10.1016/j.actamat.2016.05.031Get rights and content

Abstract

In spite of its influence on a number of physical properties, short-range order in crystalline alloys has received little recent attention, largely due to the complexity of the experimental methods involved. In this work, a novel approach that could be used for the analysis of ordering transitions and short-range order in crystalline alloys using total scattering and reverse Monte Carlo (RMC) refinements is presented. Calculated pair distribution functions representative of different types of short-range order are used to illustrate the level of information contained within these experimentally accessible functions and the insight into ordering which may be obtained using this new method. Key considerations in the acquisition of data of sufficient quality for successful analysis are also discussed. It is shown that the atomistic models obtained from RMC refinements may be analysed to identify directly the Clapp configurations that are present. It is further shown how these configurations can be enhanced compared with a random structure, and how their degradation pathways and the distribution of Warren-Cowley parameters, can then be used to obtain a detailed, quantitative structural description of the short-range order occurring in crystalline alloys.

Introduction

The effects of short-range order on physical properties have long been noted in a variety of alloys systems. Thermodynamic discontinuities [1] in nickel chromium alloys, along with observed anomalies in electrical resistivity [2], [3] have led to the description of a Komplex (K-) state containing a different order to that expected for a random solid solution. Similarly, changes in magnetic properties, such as an increase in the observed spin wave stiffness of Ni3Mn [4] provide evidence for short-range ordering. In addition, short-range order has been shown to influence dislocation motion [5], [6] and has been observed in a number of systems used for structural applications, including nickel superalloys [7]. In spite of this influence on numerous materials properties, the study of short-range ordering in alloy systems has been largely abandoned over the last 20 years or so. This is due mainly to the lack of experimental evidence available from conventional diffraction experiments and the difficulties associated with the calculation of the ordering parameters from the diffuse scattering observed in single crystal experiments.

In this paper, a methodology is presented by which total scattering techniques may be used for the direct observation of short-range order in crystalline alloy systems. Simulated supercells based on the face-centred cubic (fcc) structure, containing various types and degrees of short-range order, are used to explore the information content of pair distribution functions, which are experimentally accessible through Fourier transformation of total scattering data. The atomistic models that may be obtained from the analysis of such experimental data can be used to provide quantitative information about the types of short-range order present, and approaches by which this can be achieved are discussed.

Standard diffraction experiments, particularly of powder samples, derive structural information from the characteristic Bragg pattern [9]. Inherently, this analysis is based on the long-range average structure of the material and information on the short-range ordering that may be present in the system is lost in the averaging. The study of amorphous materials, which by definition lack the long-range order that leads to Bragg peaks, motivated the development of the total scattering technique, in which the Bragg and diffuse scattering from a sample are measured and analysed simultaneously. The technique has, more recently, been applied with great success to study many crystalline and disordered-crystalline systems [10] providing insight into the local information through analysis of deviations from the average structure. The basic scattering function for neutrons [11] is given as:S(Q)=1Ni,jbibjexp(iQ·[rirj])where i and j are atomic labels, r the instantaneous position of an atom, b the atomic scattering length, N the number of atoms and Q the magnitude of the scattering vector Q i.e. the momentum transfer of the incident radiation. An equivalent expression for X-rays can be derived, replacing the neutron scattering length bwith the Q-dependent atomic form factor. For a periodic crystal this equation simplifies to the Bragg condition. In the case of an amorphous glass this may be written as:S(Q)=1Ni,jb¯ib¯jsin(Q|rirj|)Q|rirj|Separating the self-scattering components:S(Q)=1Nibi¯2+1Nijb¯ib¯jsin(Q|rirj|)Q|rirj|which can be recast in the form:S(Q)=F(Q)+icibi¯2where ci is the concentration of species i and the total scattering function, F(Q), is given by:F(Q)=ρ004πr2G(r)sinQrQrdrin which ρ0 is the average density of the structure and:G(r)=i,jcicjb¯ib¯j(gij(r)1)where the gij(r) terms are the partial pair distribution functions. The function G(r) is often known as the pair distribution function (PDF). The inverse Fourier transform of Eq. (5), used for the calculation of a PDF, is given by:G(r)=1(2π)3ρ004πQ2F(Q)sinQrQrdQ

Alternative normalisations of the PDF are sometimes used for convenience, to emphasise specific features of the function related to the properties and length-scales of interest. A full description of the other formulations of the PDF has been provided by Keen [12]. Unless otherwise stated, the terms G(r) and PDF are used interchangeably here.

The partial PDFs are described mathematically as:gij(r)=nij(r)4πr2ρidrwhere nij(r) is the number of atoms lying within r and r+dr, and ρi = ci ρ0, where ρ0 is the density of the substance and ci the concentration of species i. Critically, this description is as applicable to periodic crystals as to amorphous materials.

The PDF, by definition, is a weighted histogram of the distribution of distances between atoms in the structure. A peak in the PDF will indicate the average distance of one atom relative to another; its width being dependent on the distribution of the inter-atomic distances (thermal vibration or static displacements) and its area being governed by the number and scattering length of the correlating atoms.

As is apparent from Eqs. (5), (7)), the PDF and F(Q) are effectively different representations of the same information (both contain the local structural information lacking in the Bragg peaks alone). Here the PDF is considered, as it is intuitively understandable as a visual representation of the local structure, but an equivalent analysis could be achieved through consideration of the F(Q).

Whilst other local structural probes exist, such as EXAFS and NMR spectroscopy, they are of limited use for the analysis of short-range order in alloys systems, owing to the fact that ordering information for a large number of coordination shells can be extracted from a PDF, whilst EXAFS and NMR provide information for only the first couple of shells.

It is straightforward to achieve a qualitative understanding of the information provided by a PDF, however producing structural models is more involved. There are two popular approaches. The first, often known as ‘small-box’ modelling, utilises tools such as PDFgui [13], and is analogous to Rietveld refinement though the models are constrained by the PDF instead of the Bragg diffraction pattern. This technique is not well suited to the characterisation of short-range order in alloys since it produces a crystallographic description of a structure (i.e. one that comprises cell parameters, thermal parameters, fractional site occupancies), albeit one that is biased towards the local, rather than the average, structure. The second, a ‘large-box’ modelling technique using the reverse Monte Carlo [14] algorithm, is arguably better suited for the analysis of ordering in alloy systems as it is unconstrained by symmetry and produces large (>10,000 atom) models that provide the ability to probe the ordering across appropriate length scales.

Traditionally, analysis of total scattering data involves visual inspection of the scattering functions and the use of peak fitting to obtain information about the arrangement of the first one or two coordination shells. It is also common to use molecular dynamics or hard-sphere Monte-Carlo to calculate structural models, from which theoretical total scattering functions can be calculated and compared with the observed data. Analysis of the system is then possible only if the two are found to be in good agreement. The tuning of initial potentials to create a structural model that more accurately reflects the data is theoretically possible, but the direct effect of the potentials on the diffuse scattering is often difficult to quantify and the process of fine-tuning is laborious. To this end, the empirical potential structure refinement (EPSR) technique [15] combines direct Monte-Carlo simulations with the refinement of an empirical potential in order to obtain calculated scattering functions that are in good agreement with the observed data. In contrast, the reverse Monte Carlo (RMC) technique [14] is an iterative process in which goodness-of-fit parameters, which describe the statistical quality of the agreement between the model and the experiment, are minimised.

In the RMC method, an initial arrangement of atoms is created that reflects the average structure of the system, as determined by analysis of the Bragg diffraction data. The PDF calculated from this will consist of a series of delta functions, as every atom will be on its ideal position, with no allowance made for positional variations associated with thermal vibrations. However, a more realistic initial model can be obtained by applying small off-site displacements to each atom according to a pseudo-random Gaussian distribution generated using the Box-Muller method [16]. From this, a number of scattering functions may then be calculated and compared with the experimental data.

In general, the quality of the fit is defined by an agreement factor, χ2, which is calculated as follows:χ2=j(yjexpyjcalc)/σj2where j indicates a data point, yjexp the experimental value at that point, yjcalc the value calculated from the refined box and σj the weighting factor for an individual data set. The χ2 parameters for different data sets are then summed as follows:χ2RMC=χ2 F( Q)+χ2 G( r)+χ2Bragg+χ2potentials+

Any number of data sets can be summed, allowing the independent fitting of multiple different data sets simultaneously. The implementation of physical constraints (e.g. χ2potentials) helps to ensure physically realistic results are obtained from the refinement.

The refinement makes random atomic translations and swaps and calculates the change in the scattering functions that result. Changes that lower χ2RMC will be automatically accepted, whilst those that increase χ2RMC by Δχ2RMC will be accepted with a probability of:P=exp(ΔχRMC22)

By accepting a certain number of ‘bad’ moves, the system may be prevented from becoming stuck in false minima. The result of the RMC refinement is an atomistic model that adequately fits all datasets and obeys all applied constraints.

The RMC algorithm has been implemented in a number of software packages (for example [8], [15], [17], [18]) for the analysis of local structure and disorder. However, for accurate analysis of the structure of crystalline alloy systems, it is necessary to explicitly fit the Bragg pattern as well as the total scattering data. The RMCProfile [8] code, developed and maintained by some of the authors of this work, which incorporates this functionality, is therefore optimised for studies of this nature.

Section snippets

Super-lattice structures in binary alloys

The phase diagrams of many alloy systems show regions where long-range ordered structures form, that differ from the disordered parent phases only in that the atoms are not randomly distributed but occupy specific lattice sites, either within a single cell or a super-cell of the parent phase. This results in the breaking of the symmetry of the parent structure and the appearance of super-lattice peaks in the diffraction pattern. To demonstrate the efficacy of analysing short-range order though

The measurement of high-quality PDFs

Owing to the low intensity of the diffuse scattering signal, compared with the much stronger Bragg peaks, high quality data are required in order to obtain a statistically significant measurement of the diffuse scattering in a total scattering experiment. Care must be taken in both the acquisition of the data and the subsequent correction, processing and analysis.

It can be seen from Eq. (7) above, that the integral should be calculated over an infinite Q-range, and while this is obviously not

Discussion and conclusions

It is clear from the preceding sections that the use of total scattering for the analysis of short-range order in crystalline alloys could provide valuable insight into the structure and properties of many complex alloy systems. To increase confidence in the results obtained in such a manner, it is essential to establish a systematic methodology for the collection and analysis of data.

Acknowledgements

This work was supported by the STFC ISIS Facility and the Rolls-Royce plc/EPSRC Strategic Partnership under EP/H022309/1 and EP/M005607/1. The authors gratefully acknowledge STFC for the provision of beamtime at Diamond Light Source Ltd (EE10354, EE11665) and the ISIS Facility (RB1510579, RB1520332), and thank Dr Stephen Hull for useful discussions.

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