polymers

Article The Use of Scattering Data in the Study of the Molecular Organisation of Polymers in the Non-Crystalline State

Thomas Gkourmpis 1,* and Geoffrey R. Mitchell 2 1 Innovation & Technology, Borealis AB, SE-444 86 Stenungsund, Sweden 2 Centre of Rapid and Sustainable Product Development, Polytechnic of Leiria, 2430-028 Marinha Grande, Portugal; geoff[email protected] * Correspondence: [email protected]; Tel.: +46-303-205-576



Received: 10 November 2020; Accepted: 3 December 2020; Published: 5 December 2020


Abstract: Scattering data for polymers in the non-crystalline state, i.e., the glassy state or the molten state, may appear to contain little information. In this work, we review recent developments in the use of scattering data to evaluate in a quantitative manner the molecular organization of such polymer systems. The focus is on the local structure of chain segments, on the details of the chain conformation and on the imprint the inherent chemical connectivity has on this structure. We show the value of tightly coupling the scattering data to atomistic-level computer models. We show how quantitative information about the details of the chain conformation can be obtained directly using a model built from definitions of relatively few parameters. We show how scattering data may be supplemented with data from specific deuteration sites and used to obtain information hidden in the data. Finally, we show how we can exploit the reverse Monte Carlo approach to use the data to drive the convergence of the scattering calculated from a 3d atomistic-level model with the experimental data. We highlight the importance of the quality of the scattering data and the value in using broad Q scattering data obtained using neutrons. We illustrate these various methods with results drawn from a diverse range of polymers.

Keywords: polymer melt; polymer glass; conformational analysis; chain segment packing; neutron scattering

1. Introduction Amorphous polymers are characterised by the absence of long-range order, leading to a level of order that is essentially associated with the inherent atomic connectivity and is concentrated in the vicinity of the polymer chain. Consequently, the short-range order of the chain will be dominated by the chemical structure through the spatial organisation of chain segments whose continuous assembly represents the polymer trajectory. The freedom of rotation enjoyed by the chemical bonds leads to a vast array of macromolecular structures, where these continuous assemblies of bonds are able to explore different possible spatial configurations. These chain trajectories are traditionally treated in terms of rotational states that can be discrete or continuous [1]. In principle, large numbers of these states can be considered and the chain segment can be defined as a part of the macromolecule where the chain conformation is assumed to be broadly constant [2]. Neutron and X-ray scattering methods are commonly employed in the study of polymeric short-range order due to their sensitivity to the local structural fluctuations. Contrary to NMR methods that provide information only on a very localised level, scattering methods are able to probe a large range of length scales. In the simple case of a disordered system comprised of N identical nuclei,

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scattered in a solid angle element Ω to Ω + dΩ relative to the number of incident neutrons [3,4]. The the experimentally measured quantity is the differential cross-section σ, namely the ratio of neutrons differential cross-section contains coherent and incoherent components, and most scattering scattered in a solid angle element Ω to Ω + dΩ relative to the number of incident neutrons [3,4]. experiments will record the total cross-section. In the case of structural studies as we are discussing The differential cross-section contains coherent and incoherent components, and most scattering here, the component of interest is of coherent nature and can be expressed as experiments will record the total cross-section. In the case of structural studies as we are discussing here, the component of interest is of coherent nature and can be expressed as 1 = Qr (1) Ω! * N + ∂σ 1 X,   = N b b exp iQr (1) ∂Ω N i j ij coh i,j=1 where bi, bj are the coherent scattering lengths of atoms i and j, respectively, N is the total number of atoms and rij is the position vector between atoms i and j. The angular brackets indicate the thermal where bi, bj are the coherent scattering lengths of atoms i and j, respectively, N is the total number average and Q is the scattering vector commonly expressed as =4sin⁄ , where 2 is the of atoms and rij is the position vector between atoms i and j. The angular brackets indicate the scattering angle and is the incident wavelength [3,4]. In a neutron scattering experiment over a thermal average and Q is the scattering vector commonly expressed as Q = 4π sin θ/λ, where 2θ is broad Q range, we can obtain information about bond lengths and segmental interactions at distances the scattering angle and λ is the incident wavelength [3,4]. In a neutron scattering experiment over a of approximately 0.1–100 Å. The experimentally observed structure factor S(Q) is defined as the broad Q range, we can obtain information about bond lengths and segmental interactions at distances spatial Fourier transform of the atomic pairwise correlation functions of approximately 0.1–100 Å. The experimentally observed structure factor S(Q) is defined as the spatial Fourier transform of the atomic pairwise correlation functions Q =1+ −1Qrr (2) Z 0 with ρo the average density ofS( Qthe) = system1 + ρ and (g(g(rr)) the1 )pairexp( distributioniQr)dr function that quantifies (2) the o − probability density of an atom existing at a distance∞ r from the origin. In the case of multiple chemical withspecies,ρo the both average S(Q) density and g(r) of can the systembe split and intog(r) partialthe pairterms. distribution The experimentally function that observed quantifies structure the probabilityfactor arises density from of a an wide atom range existing of structural at a distance correlationsr from the in origin. the bu Inlk. the From case this of multiple expression, chemical we can species,see that both structuralS(Q) and correlationsg(r) can be can split be separated into partial into terms. those The arising experimentally from correlations observed within structure the chain factor(intrachain) arises from and a those wide rangearising of by structural different correlations chains (interchain) in the bulk. [2,5,6]. From The this interchain expression, correlations we can seeprovide that structural information correlations on the different can be separated ways polymeric into those segments arising fromarrange correlations themselves within in the the bulk. chain The (intrachain)intrachain and contribution those arising to the by di scatteringfferent chains can be (interchain) extracted [from2,5,6]. the The observed interchain structure correlations factor provide at values informationof the momentum on the di fftransfererent ways ≥2−2.5 polymeric Å− segments1, using atomistic arrange themselvesmodelling techniques in the bulk. [6]. The In intrachain Figure 1, a contributionschematic torepresentation the scattering of can the be different extracted structural from the observedregimes accessible structure factorfor neutron at values and of theX-ray momentumscattering transfer can be seen.Q 2 2.5 Å 1, using atomistic modelling techniques [6]. In Figure1, a schematic ≥ − − representation of the different structural regimes accessible for neutron and X-ray scattering can be seen.

Figure 1. Schematic representation of the different correlations available by the use of diffraction methodsFigure in 1. the Schematic study of representation amorphous polymers. of the different correlations available by the use of diffraction methods in the study of amorphous polymers.

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Traditionally, most experimental studies on glassy and liquid polymers focus on large-scale properties probed by small-angle scattering [7] and dynamics of the glass transition [8]. Studies on amorphous polymers probing the local conformation over extended Q ranges have been made mostly by X-rays [9], and the advantage of neutrons to extend the available reciprocal space has not been used to the same extent. Conventionally, computer simulations are used to extract information regarding the different correlations and conformations via comparison with diffraction data. Due to the complex nature of the systems under investigation, a computational model tends to simplify the details of the chemical characteristics, in favour of the long-range interactions. In techniques such as molecular mechanics and molecular dynamics, the overall target is a realistic representation of the thermodynamic and macroscopic properties of the material under investigation [10]. Therefore, the chain conformation is generally represented in terms of multiple-body intramolecular potentials that relate to bond stretching, valence angle bending, and dihedral rotations. Nonbonded interactions are treated via van der Waals and electrostatic interactions. The main shortcoming of these techniques is that despite the fact that the force field is obtained via highly accurate ab initio calculations or experiments on small molecules, it does not always provide the necessary detailed description of a complex polymeric system [11]. The reverse Monte Carlo (RMC) method is a deceptively simple approach that allows for the extraction of detailed atomistic structural parameters of liquids and solids from diffraction data [12,13]. A number of variations of the base method has been proposed since its introduction, dealing with specific types of disordered materials, and more has recently been extended to incorporate the existence of crystals within the structure [13,14]. In this work, we will discuss the advantages of neutron scattering at small angles and deuterium labelling, and how this offers an approach to unlocking the secrets of the chain conformation and revealing the polymer structure on a multitude of length scales. At the same time, we will discuss the role of the intimate coupling of the experimental data with computer-generated models that are centred on a range of different approaches originating from the RMC method that allows for the creation of realistic three-dimensional models of amorphous and semi-crystalline polymers.

2. Diffraction and Experimental Requirements Most neutron diffraction experiments over extended Q ranges are total scattering experiments, meaning that all neutrons that reach the different detector banks are considered. The existence of hydrogen due to its incoherent cross-section (or scattering length) can cause significant problems due to the substantial ‘background’ signal which contains no structural information. One of the most direct solutions to this problem is the substitution of hydrogen atoms with deuterium that allows contrast variation and minimises the incoherent scattering from the sample [3,4,15]. Diffractometers with monochromatic incident beams are able to extend the Q range by use of short wavelengths and high scattering angles. In a pulsed neutron experiment, a relatively large number of epithermal neutrons of short wavelengths are available, something that is usually not possible for traditional reactor setups. These short-wavelength neutrons allow a wide range of Q values to be explored at relatively small angles, leading to more straightforward corrections of inelasticity effects that are further reduced with scattering angle [16]. A prime example of instruments that take advantage of these possibilities by scanning all scattering angles with a continuous range of detectors is SANDALS [17,18] and NIMROD [19,20] diffractometers at the ISIS Pulsed Neutron Source. The main disadvantage of using detectors at small angles is the reduced resolution ∆Q/Q, but for most instruments, a resolution of the order of ~2.5–4% is adequate for the majority of studies of disordered materials. Here, we must note that X-ray diffractometers tend to be limited due to the incident wavelength used. Moreover, the extent of Compton scattering increases greatly at higher Q and so tends to generate problems unless removed experimentally (e.g., as was the case of the 9.1 station at the Daresbury SRS) [21,22] or using an energy-dispersive detector [23]. The latter is Polymers 2020, 12, 2917 4 of 21 limited in usage to MoKα in a laboratory environment. For these reasons, we chose to focus on neutron Polymers 2020, 12, x FOR PEER REVIEW 4 of 21 scattering, although the same principles and methodology could be applicable to X-rays.

3. Use Use of Partial Structure Factors In thethe casecase of of multicomponent multicomponent disordered disordered materials, materials, the experimentallythe experimentally observed observed structure structure factor Dfactor(Q) can D(Q be) can expressed be expressed as as X   D(Q) = 2 δaβ cacβbabβSaβ(Q) (3) Q =2−− Q (3) α,β α ,≥ with S the partial structure factors with weights determined by the products of the atomic fractions with Sαβ the partial structure factors with weights determined by the products of the atomic fractions c , c and the scattering length. The Kronecker delta δ is introduced to avoid multiple counting of cα, cββ and the scattering length. The Kronecker delta δαβ is introduced to avoid multiple counting of similar items. According According to to this, this, the partial structure factor can be related to the appropriate partial distribution function g (r) by distribution function gαβ(r) by Z∞ Q = r−1Qrr Sαβ(Q) = ρ gαβ(r 1)exp(iQr)dr(4) (4) − 0 If the set of materials used is isostructural, then a set of total diffraction patterns D(Q) can be If the set of materials used is isostructural, then a set of total diffraction patterns (Q) can be used used to produce a complete set of partial structure factors accompanied by the associatedD radial to produce a complete set of partial structure factors accompanied by the associated radial distribution distribution functions [5]. functions [5]. These partial structure factors can be obtained from computational models [24,25] or directly fromThese the experiments partial structure themselves factors [5]. can For be a system obtained of fromatactic computational polystyrene (a-PS), models the [ 24scattering,25] or directly can be distinguishedfrom the experiments between themselves that which [ 5originates]. For a system from ofthe atactic side groups polystyrene and that (a-PS), originating the scattering from canthe backbonebe distinguished atoms. betweenConsequently, that which these originates structural from correlations the side groups can andbe thatrepresented originating via from partial the correlationbackbone atoms. functions Consequently, related to theseinteractions structural between correlations specific can sites be represented r, viar and partial correlationr with b ( ) ( ) ( ) andfunctions s indicating related the to interactionsbackbone and between side group, specific respec sitestively.gbb r These, gbs r partialand g corress r withlationb functionsand s indicating can be probedthe backbone via selective and side substitution group, respectively. of hydrogen These with partialdeuterium, correlation and high-quality functions canneutron be probed scattering via selective substitution of hydrogen with deuterium, and high-quality neutron scattering data were data were collected for four glassy polystyrene samples D0, D3, D5 and D8, with the number indicating collected for four glassy polystyrene samples D0,D3,D5 and D8, with the number indicating the level the level of deuteration with respect to the monomer unit [2]. D3 involves the deuteration of the of deuteration with respect to the monomer unit [2]. D3 involves the deuteration of the skeletal chain, skeletal chain, whereas D5 refers to the deuteration of the side group. The existence of different whereasdeuteration D5 levelsrefers leads to the to deuteration significant ofchanges the side in group.the observed The existence structure of factor, different something deuteration that levelsis not visibleleads to with significant X-rays. changes Further in manipulation the observed structureof these factor,functions something leads to that a set is not of visiblepartial withcorrelation X-rays. functionsFurther manipulation (see Figure 2). of these functions leads to a set of partial correlation functions (see Figure2).

Figure 2.2. StructureStructure factors factors (left) (left) and and radial radial distribution distribution functions functions (right) (right) for selectively for selectively deuterated deuterated glassy polystyrene.glassy polystyrene. The level The of level deuteration of deuteration can be assessed can be byassessed the number by the next number to the next label to B the (see label text forB (see full

textdetails). for full (a) indicates details). the(a) totalindicates correlation the total function correlation for system function D8 and for (b) system the partial D8 correlationand (b) the function partial correlationgbb describing function the correlations gbb describing involving the correlations only the backbone involving atoms. onlyBoth the backbone graphs were atoms. reproduced Both graphs from werereference reproduced [2]. from reference [2].

These partial correlation functions indicate the existence of a broad feature at r ~ 10–10 Å that is not visible in the total correlation function obtained from D8. This effect has been attributed to the correlation between polymer backbones, and the peak is fairly broad and weak, indicating very

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These partial correlation functions indicate the existence of a broad feature at r ~ 10–10 Å that is not visible in the total correlation function obtained from D8. This effect has been attributed to the correlation between polymer backbones, and the peak is fairly broad and weak, indicating very limited backbone correlations. Such a feature is much weaker than the corresponding features in simple polymers like polyethylene [26] or poly(tetrafluoroethylene) [27], thus concluding that no substantial chain correlations are present in glassy polystyrene [2]. These limited backbone correlations further indicate that correlations between the bulky side groups dominate the structure, a model previously proposed on the basis of X-ray scattering analysis [28].

4. Local Mixing in Polymer Blends Simple theory assumes that mixing between polymers is random at the segmental level, although experimental evidence suggests otherwise. It has been shown that the miscibility of polyolefins is essentially controlled by the values of their solubility parameters that can be determined by small-angle neutron scattering (SANS) measurements [29]. Solubility parameters can also be derived directly from knowledge of chain dimensions usually expressed via the packing length. In other words, the local chemical topology (branches, unsaturated bonds, specific interactions, etc.) is expected to play a very strong role in the local mixing process, something seen in the case of mixtures of chemically similar but topologically dissimilar polymers like 1,4 and 1,2 polybutadiene [30]. Let us consider blends of deuterated and protonated 1,2 and 1,4 polybutadiene, namely d-14pbd/d-12pbd, d-14pbd/h-12pbd, h-14pbd/d-12pbd and h-14pbd/h-12pbd where the d or h indicate a deuterated or protonated system, respectively. For a binary mixture, we can express the interchain component in terms of partial structure factors PAA(Q), PAB(Q) and PBB(Q) where A and B are segments of the two chemically different polymers. According to this, the total structure factor can be expressed as

S (Q) = c c F2 P (Q) + 2c c F F P (Q) + c c F2 P (Q) (5) i A A AH,D AA A B AH,D BH,D AB B B BH,D BB with cA, cB the concentrations of segments A and B, respectively, and FA(Q), FB(Q) the molecular form factors for the given segments. Here, we must note that these expressions are valid for samples that are isotropic in which the local mixing is also isotropic. Extracting the partial structure factors from such an expression requires a series of isostructural blends where either composition of the deuteration level (that alters the molecular form factors) is systematically varied. This approach can only be carried out with neutron scattering methods, as X-ray diffraction is insensitive to the isotope content, thus confirming that the blends are largely at least isostructural. In Figure3, the broad Q neutron scattering results for two of the individual components (protonated and deuterated 1,4 and 1,2 polybutadiene) and the resulting blends can be seen. Information regarding the sample preparation and experimental procedure for the individual components can be found elsewhere [6,31] and for the blends in the Supportive Information. Clearly, qualitative differences between the different structure factors exist (as expected) but the data quality and extent of Q range are visible. In Figure4, we can see the inter and intra chain parts of one of the blends (d12 /d14) as an illustration of the overall behaviour of all the different components. The main peak in the structure factor arises from correlations between chain segments, and is sensitive to the level of mixing. Therefore, if the system is to be totally immiscible, then the main peak will be the addition of the two peaks of the individual components weighted by their appropriate contributions. From Figure4, we can see that this is clearly not the case, as the artificial structure factor (d12/d14 addition) that has been created by simple addition of the individual components is not the same as the observed structure factor for the blend. In other words, a certain level of mixing is present in the system and the decomposition with respect to the partial structure factors is, to a large extent, valid. Another important element we see from Figure4 is the intrachain scattering that is very similar to the simple addition of Polymers 2020, 12, 2917 6 of 21

the intrachain scattering of the individual components, indicating that the local conformation remains Polymers 2020, 12, x FOR PEER REVIEW 6 of 21 largelyPolymers 2020 unchanged, 12, x FOR during PEER REVIEW mixing. 6 of 21

Figure 3. Experimentally observed structure factors of protonated and deuterated 1,2 and 1,4 Figure 3.3. ExperimentallyExperimentally observed observed structure structure factors factors of of protonated protonated and and deuterated 1,2 and 1,4 polybutadiene (left) and blends consisting of these components (right). In both graphs, the term d or polybutadiene (left) and and blends blends cons consistingisting of of these these components components (right). (right). InIn both both graphs, graphs, the the term term d or d h indicate the deuterated or protonated system, respectively, and the term 1,2 or 1,4 indicate the orh indicate h indicate the the deuterated deuterated or orprotonated protonated system, system, re respectively,spectively, and and the the term term 1,2 1,2 or or 1,4 indicate the stereoregularity of the polybutadiene chain, respectively. stereoregularity ofof thethe polybutadienepolybutadiene chain,chain, respectively.respectively.

FigureFigure 4. Comparison ofof thethe experimentally experimentally observed observed structure structure factor factor for for a blend a blend of deuterated of deuterated 1,2 (d12) 1,2 Figure 4. Comparison of the experimentally observed structure factor for a blend of deuterated 1,2 (d12)and 1,4 and (d14) 1,4 polybutadiene(d14) polybutadiene (d12/d14) (d12/d14) with an artificialwith an structureartificial factorstructure (d12 factor/d14 addition) (d12/d14 comprising addition) (d12) and 1,4 (d14) polybutadiene (d12/d14) with an artificial structure factor (d12/d14 addition) comprisingthe sum of thethe experimentally sum of the observedexperimentally structure observed factors ofstructure the two individualfactors of componentsthe two individual weighted comprising the sum of the experimentally observed structure factors of the two individual componentsby their relevant weighted concentration. by their relevant See text concentration. for more detailed See discussion.text for more detailed discussion components weighted by their relevant concentration. See text for more detailed discussion InIn this this work, work, the the interchain interchain peak peak is is of of interest interest and, and, in in our our approach, approach, we we decompose decompose the the observed observed In this work, the interchain peak is of interest and, in our approach, we decompose the observed structurestructure factorfactor intointo inter inter and and intrachain intrachain components. components. We We have have developed developed atomistic atomistic models models that match that structure factor into inter and intrachain components. We have developed atomistic models that matchthe intrachain the intrachain scattering scattering for 1,4 [for6] and 1,4 1,2[6] and polybutadiene 1,2 polybutadiene [31], and [31], this and is subtracted this is subtracted from the observedfrom the match the intrachain scattering for 1,4 [6] and 1,2 polybutadiene [31], and this is subtracted from the observedstructure factorstructure yielding factor the yielding scattering the scattering that arises that from arises the intersegmental from the intersegmental scatteringalone. scattering The partialalone. observed structure factor yielding the scattering that arises from the intersegmental scattering alone. Thestructure partial factors structure can thenfactors be can obtained then be by obtained inversion by of theinversion previous of expression,the previous and expression, the results and can the be The partial structure factors can then be obtained by inversion of the previous expression, and the resultsseen in can Figure be 5seen. From in Figure this analysis, 5. From we this can analysis, clearly see we that can thereclearly is verysee that limited there mixing is very at limited the local mixing level, results can be seen in Figure 5. From this analysis, we can clearly see that there is very limited mixing atas the indicated local level, by the as cross-termindicated by (1,2–1,4). the cross-term From the (1,2–1,4). widthof From the interchainthe width peak,of the weinterchain can estimate peak, thatwe at the local level, as indicated by the cross-term (1,2–1,4). From the width of the interchain peak, we canthe extentestimate of that spatial the correlation extent of spatial between correlation chemically between similar chemically segments is similar limited segments to ~2 chain is limited segments. to can estimate that the extent of spatial correlation between chemically similar segments is limited to ~2 chainBroad segments. Q neutron scattering on isotopically labelled blends provides a route to extracting partial ~2 chain segments. structureBroad factors Q neutron that arise scattering from correlations on isotopically between labelled chain blends segments provides belonging a route to theto extracting same or di partialfferent Broad Q neutron scattering on isotopically labelled blends provides a route to extracting partial structurepolymers, factors thus allowing that arise a levelfrom ofcorrelations quantification between in the chain extent segments of local mixingbelonging in ato blend. the same This or is structure factors that arise from correlations between chain segments belonging to the same or differentcritical as polymers, it allows thethus quantification, allowing a level mapping, of quantification and insight in intothe extent a number of local of di mixingfferent factorsin a blend. that different polymers, thus allowing a level of quantification in the extent of local mixing in a blend. Thisare condensed is critical as into it allows a single the interaction quantification, parameter mapping, in the and conventional insight into Flory–Huggins a number of different theory factors and its This is critical as it allows the quantification, mapping, and insight into a number of different factors thatvariations are condensed [32]. In the into case a single of polybutadiene-based interaction parameter blends in the considered conventional in this Flory–Huggins work, it has beentheory found and that are condensed into a single interaction parameter in the conventional Flory–Huggins theory and itsthat variations the very local[32]. stateIn the is case largely of polybutadiene-based unmixed in contrast withblends larger considered length scales in this at work, which it the has system been its variations [32]. In the case of polybutadiene-based blends considered in this work, it has been foundappears that homogeneous the very local and state optically is largely clear. unmixed in contrast with larger length scales at which the found that the very local state is largely unmixed in contrast with larger length scales at which the system appears homogeneous and optically clear. system appears homogeneous and optically clear.

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FigureFigure 5. Partial structurestructure factors factors for for the the diff differenterent correlations correlations derived derived from from experimental experimental data on data blends on blendsof 1,2 and of 1,2 1,4 polybutadiene.and 1,4 polybutadiene. 1,2-1,2 (red), 1,2-1,2 1,2-1,4 (red), (blue) 1,2-1,4 and (blue) 1,4-1,4 and (black) 1,4-1,4 indicate (black) the indicate correlations the correlationsbetween segments between of segments 1,2 and 1,4 of polybutadiene 1,2 and 1,4 polybu respectively.tadiene Seerespectively. text for more See detailedtext for more discussion. detailed discussion. 5. Coupling Diffraction with Modelling 5. CouplingFrom the Diffraction discussion with so far, Modelling it is clear that the structure factor contains a considerable amount of structuralFrom information,the discussion making so far, the it is use clear of molecular that the structure modelling factor an attractive contains option a considerable for probing amount all these of structurallength scales. information, Before jumping making to the the use modelling of molecular techniques modelling though, an attractive some important option for considerations probing all theseneed tolength be made. scales. By comparingBefore jumping the experimental to the modelling data to the model,techniques how dothough, we quantify some theimportant possible considerationsdifferences? What need do theseto be saymade. about By the comparing model used? the Doesexperimental this mean data that theto modelthe model, is totally how wrong do we or quantifyother ways the of possible introducing differences? and/or analysing What do segmentalthese say about orientations the model are available?used? Does Without this mean information that the modelof this sort,is totally small wrong deviations or other between ways observed of introducing and predicted and/or structure analysing factors segmental may be vitallyorientations important are available?or just trivial Without computational information limitations of this sort, (e.g., small size of deviations the model). between Without observed a concrete and and predicted robust structureunderstanding factors of may the levelbe vitally of information important available or just trivial from thecomputational experiment limitations and how this (e.g., information size of the is model).coupled Without with the a model, concrete any and further robust validation understanding is based of on the uncertain level of foundations.information Inavailable Figure6 from, we canthe experimentsee an example and how of such this a information case where twois coupled structure with factors the model, of amorphous any further deuterated validation polyethylene is based on uncertain(d-PE) prepared foundations. with di Infferent Figure methods 6, we can are see shown. an example The experimental of such a case procedure where wastwo similarstructure to thefactors one ofused amorphous for the polybutadiene deuterated polyethylene samples discussed (d-PE) earlier prepared and can with be founddifferent in themethods Supportive are shown. Information. The experimentalThe first sample procedure is molten was polyethylene similar to irradiatedthe one used at 160 for◦ theC with polybutadiene high-energy samples electrons discussed (1 MeV, 50 earlier MGy) andthat can induce be found both cross-linkingin the Supportive and Information. chain scission The [33 first]. The sample observed is molten structure polyethylene factor is irradiated similar to atthe 160 one °C observed with high-energy for molten electrons polyethylene (1 MeV, [26]. 50 The MGy) second that sample induce has both been cross-linking prepared by and extensive chain scissionirradiation [33]. in The the observed absence of structure oxygen byfactor high-energy is similar electronsto the one (1 observed MeV, 50 MGy)for molten of semi-crystalline polyethylene [26].polyethylene The second at 20sample◦C. The has irradiation been prepared of crystalline by extens polyethyleneive irradiation with in the doses absence greater of oxygen than 30 by MGy high- is energyknown toelectrons lead to the(1 MeV, destruction 50 MGy) of crystals of semi-crystal [34]. Theline two polyethylene structure factors at 20 seen °C. in The Figure irradiation6 exhibit noof crystallineBragg peaks polyethylene and are similar with to dose thes ones greater expected than from30 MGy amorphous is known materials. to lead to Still, the theydestruction do exhibit of crystals [34]. The two structure factors seen in Figure 6 exhibit no Bragg peaks and are similar to the

Polymers 2020, 12, 2917 8 of 21 Polymers 2020, 12, x FOR PEER REVIEW 8 of 21 someones expected clear diff erences,from amorphous indicating materi thatals. these Still, two they amorphous do exhibit structures some clear are differences, different. Theindicating feature that at these1 two amorphous structures are different. The feature at ~5 Å−1 for the 20 °C sample is indicative ~5 Å− for the 20 ◦C sample is indicative of extensive trans sequences along the backbone [35] with of extensive trans sequences along the backbone [35] with1 the interchain peak maximum occurring1 the interchain peak maximum occurring at Q = 1.5 Å− in comparison with Q = 1.35 Å− for the −1 −1 sampleat Q = 1.5 irradiated Å in comparison in the melt, with suggesting Q = 1.35 Å a closer for the packing sample ofirradiated chains. Fromin the thismelt, superficial suggesting analysis a closer packing of chains. From this superficial analysis of the structure factor, we can deduce that the sample of the structure factor, we can deduce that the sample irradiated at 20 ◦C probably contains some smallirradiated clusters at of20 parallel °C probably segments contains in the trans some configuration small clusters while of the parallel sample irradiatedsegments inin thethe melt trans is representativeconfiguration ofwhile a more the sample random irradiated coil. In other in the words, melt theseis representative two structure of a factors more representrandom coil. two In totally other diwords,fferent these models two of structure orientation factors correlations represent despite two totally the experimentally different mode observedls of orientation broad similarities.correlations Thisdespite example the experimentally is quite informative observed and underlinesbroad simila therities. need forThis detailed, example robust, is quite and accurateinformative models and andunderlines the level the of sensitivityneed for detailed, of the di ffrobust,erent structural and accurate parameters models on and the the overall level structure of sensitivity factor. of the different structural parameters on the overall structure factor.

FigureFigure 6.6.Experimentally Experimentally observed observed structure structure factors factors from samplesfrom samples of deuterated of deuterated polyethylene polyethylene prepared withprepared different with methods. different Themethods. solid The line solid indicates line theindi samplecates the irradiated sample irradiated in the melt in (160the melt◦C), (160 and the°C), dashedand the linedashed indicates line indicates the sample the irradiatedsample irradiated at room at temperature room temperat (20ure◦C). (20 On °C). the On right-hand the right-hand side, theside, di fftheraction diffraction of the of semi-crystalline the semi-crystallin deuteratede deuterated polyethylene polyethylene can be can seen. be seen. 6. Real vs. Reciprocal Space Functions—The Reverse Monte Carlo Method 6. Real vs. Reciprocal Space Functions—The Reverse Monte Carlo Method The patterns shown in Figure2 contain the same information, but it is obvious that the level of The patterns shown in Figure 2 contain the same information, but it is obvious that the level of information that is pronounced is different between the data representation in real (Figure2 right) information that is pronounced is different between the data representation in real (Figure 2 right) and reciprocal space (Figure2 left). The real-space representation relates with the high level of and reciprocal space (Figure 2 left). The real-space representation relates with the high level of accuracy to distances that are determined via covalent bonding. Larger distances associated with accuracy to distances that are determined via covalent bonding. Larger distances associated with spatial arrangements and rotations such as valence and torsional angles are far more difficult to spatial arrangements and rotations such as valence and torsional angles are far more difficult to extract, making the identification of the local conformational characteristics a very challenging affair. extract, making the identification of the local conformational characteristics a very challenging affair. An alternative to such problems is the use of the reciprocal space function as obtained by the experiment. An alternative to such problems is the use of the reciprocal space function as obtained by the This can be done by ‘reconstructing’ the radial distribution function using a computer model and experiment. This can be done by ‘reconstructing’ the radial distribution function using a computer comparing the calculated with the observed scattering pattern SE(Q). The calculation of the scattering model and comparing the calculated with the observed scattering pattern SE(Q). The calculation of from a model SC(Q) can be obtained using Debye’s relationship [6]. the scattering from a model SC(Q) can be obtained using Debye’s relationship [6].   11 X X sinsinQrQrij SC(QQ)== bibj (6)(6) N QrQr i i,j ij

where N is the number of atoms, bi, bj are the neutron scattering lengths [3,4] of atoms i and j, where N is the number of atoms, bi, bj are the neutron scattering lengths [3,4] of atoms i and j, respectively, and rij is the equivalent interatomic distance. We have chosen to make the comparison respectively, and rij is the equivalent interatomic distance. We have chosen to make the comparison usingusing thethe reciprocal-reciprocal- insteadinstead ofof thethe real-spacereal-space functionfunction duedue toto thethe simplicitysimplicity ofof separatingseparating interinter andand intrachainintrachain contributions.contributions. TheThe computer-generatedcomputer-generated structurestructure factorfactor isis easilyeasily obtainedobtained asas allall thethe atomicatomic coordinatescoordinates areare known;known; therefore,therefore, by by suchsuch reconstructionreconstruction of of thethe scatteringscattering andand subsequentsubsequent comparisoncomparison withwith thethe actualactual experimental data, it allows a vast volume of structural parameters to be probed and their influence in the overall scattering to be established.

Polymers 2020, 12, 2917 9 of 21 experimental data, it allows a vast volume of structural parameters to be probed and their influence in the overall scattering to be established.

The RMC Method The RMC technique is based on the traditional Metropolis Monte Carlo (MC) method [36] and aims to generate an atomistic configuration (model) that is consistent with an experimentally observed data set within its errors and is subject to a set of pre-defined constraints. All errors are assumed to be statistical and have a normal distribution [13]. In the general case, we assume N nuclei placed randomly in a cell under periodic boundary conditions. As all the positions of the atoms are known, we can calculate the partial radial distribution function from the given configuration

Co ( ) nαβ r Co ( ) = (7) gαβ r 2 4πr drρcα

Co ( ) where ρ is the atomic density, cα is the concentration of nuclei of atom α and nαβ r is the number of nuclei of type β located at a distance between r and r + dr from the nuclei of type α, averaged over Co ( ) all nuclei of type α. By performing a Fourier transform on nαβ r , we can obtain the partial structure Co ( ) Co ( ) factor Fαβ Q and the total structure factor S Q :

Z ∞  sin Qr FCo (Q) = ρ 4πr2 gCo (r) 1 dr (8) αβ αβ − Qr 0 X   Co Co S (Q) = cαc bαb F (Q) 1 (9) β β αβ − where Q is the momentum transfer and bαbβ are the coherent neutron scattering lengths for nuclei α and β, respectively. At this moment, we have two structure factors, that obtained by the experiment SE(Q) and that obtained by the atomistic model SCo (Q). In order to test the accuracy of our model, we need an impartial way of comparing the two structure factors. This can be done via a statistical χ2 test:

 2  Xm  SCo (Q ) SE(Q )  2  i i  χ =  −  (10) o  σ2(Q )  i=1 i 

With the sum taken over m experimental points, σ corresponds to the experimental error. Furthermore, we note that the minimum Qi value needs to be larger than 2π/L with L corresponding to the minimum dimension of the configuration under investigation. As soon as the initial (old) configuration is compared with the experimental one, we can move one atom at random (limitations on the maximum distance that these move are allowed to exist, but this is beyond the scope of this work [12,13]). Due to this move, a new configuration is available and, thus, Cn ( ) Cn ( ) Cn ( ) a new pair distribution function gαβ r , followed by a new partial Fαβ Q and total S Q structure factor, respectively. As with the old configuration, we can again calculate the accuracy of the new configuration against our model via the new χ2 test:

 2  Xm  SCn (Q ) SE(Q )  2  i i  χ =  −  (11) n  σ2(Q )  i=1 i  Polymers 2020, 12, 2917 10 of 21

If χ2 < χ2, the move is accepted and the new configuration becomes the old one. If χ2 > χ2, n o     n o the move is accepted with a probability exp χ2 χ2 /2 ; otherwise, it is rejected. The procedure − n − o continues until no further changes below a pre-defined limit can be observed in the value of the statistical test. In practice, as the calculated configuration approaches the experimentally observed one, the value of the statistical test reaches an equilibrium and oscillates around a mean value. The configuration that corresponds to the minimal statistical test is a three-dimensional structure that is consistent with the experimental data within the relevant experimental error. In practice, both the radial distribution function and the structure factor can be used for fitting, although the use of the reciprocal space functions has been seen to be more sensitive to small alterations of the chain conformation [6]. In an RMC process, the quantity that is sampled is equivalent to the sampling quantity of the   Metropolis Monte Carlo U2 U2 /kT with U the potential energy of the configuration for a given n − o interatomic potential, T the temperature and k the Boltzmann constant. In other words, the structure factor can be considered the driving force of the model (i.e., energy), and σ corresponds to the temperature. In principle, any type of statistical test for comparison of that observed with the simulated structure factor can be used, but the χ2 is typically used due to simplicity [6,13,31]. It can be shown that for a structural system with only pairwise forces, the RMC procedure leads to an exact solution.

RMC Variations The RMC protocol discussed previously has been used with various degrees of success in the study of the atomic structure of disordered materials like simple liquids [37], molten salts [38], glasses [39] and polymers [26,27]. The basic idea of the method, namely the comparison of a computer-generated model with experimental data following a stochastic set of pre-defined rules, has been extended to approaches that treat the intrachain contribution [6,31], the establishment of actual experimentally based force fields [40] and the role of crystallinity in the overall structure [14,41]. Extending the simulation towards coarser length scales and introducing crystals by arranging the segments in specified ways while keeping them consistent with the chain conformation have been seen to offer possibilities for an in-situ study of time-resolved crystallization [42,43], thus indicating the potential of such intimate coupling of an RMC-based procedure with experimental data [44,45].

7. The Role of the Scattering Data Based on our previous discussion, we can clearly see that the extent, resolution and quality of the experimentally observed structure factor will have an influence on the outcome of the RMC modelling procedure. In this section, we will discuss the role of the diffraction pattern on the computational model and provide some examples on a number of different polymeric systems.

7.1. Intrachain Correlations In this section, we will discuss a variation in the traditional RMC procedure that focuses on the extraction of structural information regarding the intrachain correlations and the chain conformation. In this procedure, we utilise only part of the experimental data that corresponds to the intrachain contribution to the diffraction pattern. The model can be generated by using internal coordinates for the definition of the polymer chain in terms of bond lengths, valence, and torsion angles with values assigned in each parameter through a stochastic Monte Carlo procedure where the relevant parameters are drawn from probability distributions representing the possible value ranges [6]. In most cases, the distributions used are Gaussian, but this is not exclusive and, if necessary, other more complex distributions can be used. Using this simple procedure, we can scan through a large number of parameters and, in order to acquire good statistics, we can average over a number of different models (usually 100). Each parameter representing a probability distribution is varied in a systematic way using a grid method to find the values that minimise the statistical test. In Figure7A,B, we can see the intrachain part of the experimentally observed structure factor for a system of deuterated 1,2 polybutadiene as compared with the initial (A) and final (B) model Polymers 2020, 12, 2917 11 of 21 after the refinement has been completed [31]. Using these types of iterative methods allows us to identify the impact different types of structural parameters and their resulting correlations have on the structure factor (Figure7B). In Figure7C, we can see an example of the parameter search through the multidimensional space with a plot of the statistical test against the values of the C-C and C=C positions (lengths) for the case of 1,4 polybutadiene [6]. This method has been tested for a series of high-quality data for 1,2 and 1,4 polybutadiene for a wide range of temperatures covering almost all the experimentally accessible range, allowing us to obtain information both in the liquid and glass Polymersstate [6 2020,31]., 12 In, x otherFOR PEER words, REVIEW each temperature of each material (9 for 1,2 polybutadiene and11 11 of for 21 1,4 polybutadiene) can be considered an independent measurement that can be used to validate the model.model. In In Figure Figure 77D,D, we can see thethe statisticalstatistical testtest resultresult forfor thethe optimaloptimal modelmodel structurestructure forfor eacheach ofof thethe temperatures temperatures for for the the sample sample of of 1,2 1,2 polybutadien polybutadiene.e. Although Although some some variation variation exists, exists, we we can can clearly clearly seesee that that all the χ2 resultsresults are are clustered around the same me meanan value, indicating the robustness and limitationslimitations of of the the method. method.

FigureFigure 7. ComparisonComparison between between the the experimentally experimentally observ observeded structure structure factor factor (solid (solid line) line) and and that calculatedcalculated from from a a model model (dashed (dashed line) line) for for 1,2 1,2 polybutadiene polybutadiene at 250 at 250 K ( KA, (BA)., BIn). ( InB), ( Bthe), theareas areas where where the differentthe different structural structural parameters parameters have have the themost most impact impact on onthe thestructure structure factor factor is indicated. is indicated. In In(C), (C a), surfacea surface plot plot of ofthe the statistical statistical test test as as a afunction function of of the the C-C C-C and and C=C C=C bond bond lengths lengths can can be be seen seen for for a modelmodel of 1,4 1,4 polybutadiene. polybutadiene. In In 7 (D,), the the final final result result of of the the best-fit best-fit statistical statistical test test for for the the different different models models ofof 1,2 1,2 polybutadiene polybutadiene can be seen.

7.2.7.2. The The Case Case of of Selenium Selenium InIn orderorder toto highlight highlight the the role role of theof the experimental experimental data data and theirand their influence influence on the on modelling the modelling output, output,we will usewe will a set use of neutron a set of scatteringneutron scattering data for vitreous data for selenium. vitreous selenium. Selenium wasSelenium considered was considered by Flory as byan exampleFlory as an of aexample simple chainof a simple in having chain a one-bondin having repeat a one-bond [1]. In repeat Figure 8[1], the (. In total Figure structure 8, the factor total structurefor a sample factor of for vitreous a sample selenium of vitreous can be selenium seen. Sample can be preparation seen. Sample and preparation experimental and detailsexperimental can be detailsfound elsewherecan be found [46 ,elsewhere47]. A series [46,47]. of broad A series and di offf usebroad peaks and indicatingdiffuse peaks the absenceindicating of anythe absence crystalline of anyfeatures crystalline are present features in the are scattering present in pattern. the scatteri Observationng pattern. of experimental Observation structure of experimental factors [48 structure,49] and factorscomputational [48,49] and predictions computational [50] show predictions that the position[50] show and that width the position of the first and intense width peakof the changes first intense with peak changes with temperature. The rest of the scattering curve remains though practically unaffected. This difference can be attributed to the different nature of correlations probed by the different parts of the scattering curve. The first intense peak arises from the packing of selenium in the bulk, and it is of intermolecular origin. The rest of the curve is thus probing interactions of intramolecular origin. As discussed previously, structural information carried by the diffraction pattern can be extracted via the use of real-space functions through the reduced correlation function and the radial distribution function [6]. Structural investigation in real space has the advantage of locating with high accuracy near-neighbour distances as they appear in the form of sharp and well-defined peaks. Consequently, the level of information probed from a broad Q scattering experiment is on the atomic connectivity and co-ordination. This can be achieved by deconvolution of the real-space function to

Polymers 2020, 12, 2917 12 of 21 Polymers 2020, 12, x FOR PEER REVIEW 12 of 21 itstemperature. constituent Thecomponents rest of the [46]. scattering From the curve radial remains distribution though function practically (inset una inff ected.Figure This8), we di ffcanerence see acan very be well-defined attributed to first the disharpfferent peak nature that ofindicates correlations the near-neighbour probed by the didistance.fferent parts Correlations of the scattering seem to becurve. weak The at distances first intense above peak 10 arises Å in fromagreement the packing with previously of selenium reported in the bulk, experimental and it is of results intermolecular [48] and computationalorigin. The rest predictions of the curve [51]. is thus probing interactions of intramolecular origin.

FigureFigure 8. Total structurestructure factor factor of of vitreous vitreous selenium selenium obtained obtained from from neutron neutron scattering scattering data. data. In the In inset, the inset,the radial the radial distribution distribution function function of the of same the scatteringsame scattering curve curve can be can seen. be seen.

TheAs discussed mean value previously, for the near-neighbour structural information distance carried has been by the found diffraction to be 2.346 pattern ± 0.013 can beÅ, extracted in good agreementvia the use with of real-space previously functions reported through values the [52–55]. reduced The correlation coordination function number and thewas radial found distribution [47] to be 1.974function ± 0.205, [6]. Structuralin good agreement investigation with in previously real space reported has the advantage values [55–57]. of locating The mean with value high accuracyof 1.974 fornear-neighbour the coordination distances number as theyis very appear close into thethe formideal of value sharp of and2. Having well-defined a coordination peaks. Consequently, number of 2 (orthe close level to of 2) information fits both chains probed and from rings a in broad the amorphous Q scattering selenium experiment structure. is on the As atomicthe value connectivity extracted isand less co-ordination. than 2, it indicates This can that be undercoordinated achieved by deconvolution atoms (chain of theends) real-space exist in the function structure. to its The constituent second peakcomponents has a maximum [46]. From theat ~3.75 radial Å distribution [47], in good function agreement (inset inwith Figure previously8), we can reported see a very experimental well-defined valuesfirst sharp [52–55]. peak This that indicatespeak is believed the near-neighbour to arise from distance. the superposition Correlations of seem the second to be weak near-neighbour at distances interactionsabove 10 Å and inagreement intermolecular with correlations previously [54]. reported The coordination experimental number results was [48 ]found and computational[47] to be ~6.7, indicatingpredictions the [51 existence]. of two second neighbours (covalently bonded to the first neighbours) and The mean value for the near-neighbour distance has been found to be 2.346 0.013 Å, in good elements of intermolecular (in terms of rings or chains) interactions caused by van± der Waals forces. Theagreement valence with angle previously was calculated reported to be values located [52 at– 5510].4° The with coordination a RMS deviation number of 5° was in good found agreement [47] to be 1.974 0.205, in good agreement with previously reported values [55–57]. The mean value of 1.974 with previously± reported values [58]. for theDue coordination to the weak number diffuse ispeaks very closefollowing to the the ideal near-n valueeighbour of 2. Having one, it a is coordination very difficult number to extract of 2 information(or close to 2) concerning fits both chains the torsional and rings rotation in the amorphousfrom the radial selenium distribution structure. function. As the The value superposition extracted is ofless the than sine 2, waves it indicates of the thatFourier undercoordinated transform leads atoms to a series (chain of ends)broad exist and indiffuse the structure. peaks that, The although second theypeak carry has a information maximum atconcerning ~3.75 Å[ 47the], valence in good angles agreement and the with torsion previously rotation, reported it is challenging experimental to extract.values [52The–55 identification]. This peak isof believedthe torsion to arisedistributi fromon the is superposition of great importance of the second as it defines near-neighbour the local conformation.interactions and As intermolecular the inter and correlationsintramolecular [54 ].terms The coordinationof the scattering number are merged was found in the [47] real-space to be ~6.7, function,indicating a theway existence to deconvolute of two them second is required. neighbours (covalently bonded to the first neighbours) and elementsThis deconvolution of intermolecular can (in be termsachieved of rings using or the chains) techniques interactions discussed caused in the by previous van der Waalssections. forces. For thisThe work, valence we angle have was utilised calculated a method to be of located intramolecular at 104◦ with structural a RMS deviationanalysis that of 5 has◦ in been good developed agreement andwith used previously for polymers reported with values very [ 58successful]. results [6,31]. Initially, the model was built by assigning values taken from the literature for bond lengths (2.373 Å), valence angles (103.1°) and torsion angles (±150°) (see schematic in Figure 9). The comparison of the initial model with the experimentally

Polymers 2020, 12, 2917 13 of 21

Due to the weak diffuse peaks following the near-neighbour one, it is very difficult to extract information concerning the torsional rotation from the radial distribution function. The superposition of the sine waves of the Fourier transform leads to a series of broad and diffuse peaks that, although they carry information concerning the valence angles and the torsion rotation, it is challenging to extract. The identification of the torsion distribution is of great importance as it defines the local conformation. As the inter and intramolecular terms of the scattering are merged in the real-space function, a way to deconvolute them is required. This deconvolution can be achieved using the techniques discussed in the previous sections. For this work, we have utilised a method of intramolecular structural analysis that has been developed andPolymers used 2020 for, 12 polymers, x FOR PEER with REVIEW very successful results [6,31]. Initially, the model was built by assigning13 of 21 values taken from the literature for bond lengths (2.373 Å), valence angles (103.1◦) and torsion angles observed structure factor showed us that the general trend and the basic form and shape of the ( 150◦) (see schematic in Figure9). The comparison of the initial model with the experimentally experimental± structure factor is being reproduced by the initial model, but refinement of the observed structure factor showed us that the general trend and the basic form and shape of the structural parameters was required in a manner similar to the work in polybutadiene (see Figure 7A) experimental structure factor is being reproduced by the initial model, but refinement of the structural [31]. parameters was required in a manner similar to the work in polybutadiene (see Figure7A) [31].

Figure 9. Schematic representation of vitreous selenium indicating the various structural parameters usedFigure in 9. the Schematic modelling representation procedure. of vitreous selenium indicating the various structural parameters used in the modelling procedure. Having built the initial configuration, all parameters are kept constant and the refinement is performedHaving for built the bondthe initial length. configuration, The Se–Se distance all parameters is assigned are by kept a Gaussian constant probability and the refinement distribution is atperformed a particular for meanthe bond and length. standard The deviation. Se–Se distance The position is assigned of the by distributiona Gaussian probability is searched distribution initially as itat is a knownparticular that mean the impact and standard on the scatteringdeviation. curveThe position will be moreof the visible.distribution The modelis searched is built initially and the as bondit is known length that is assigned the impact via aon normal the scattering distribution curve of will diff beerent more lengths visible. at eachThe model iteration. is built Every and time the thebond length length changes, is assigned the scattering via a normal pattern distribution is calculated of different and compared lengths at via each the iteration. statistical Every test with time thethe experimentallength changes, one. the The scattering result of pattern the search is calculat in theed case and of thecompared bond length via the can statistical be seen intest Figure with the10, andexperimental a clear minimum one. The exists result at of a the distance search of in 2.38 the Å.case Furthermore, of the bond thelength scattering can be seen calculated in Figure for a10, bond and lengtha clear of minimum 2.38 Å yields exists a very at a gooddistance comparison of 2.38 Å. between Furthermore, calculated the and scattering observed calculated scattering for in high-Qa bond vectors.length of Consequently, 2.38 Å yields a establishing very good comparison that the oscillating between sine calculated wave in and the observed structure factorscattering arises in fromhigh- seleniumQ vectors. atoms Consequently, joins together establishing by covalent that the bonding. oscillating A similar sine wave search in forthe thestructure position factor of the arises valence from angleselenium distribution atoms joins can together be seen inby Figure covalent 10. bonding. The value A of similar the mean search of thefor distributionthe position ofof thethe valencevalence angleangle wasdistribution found to can be be at seen 105◦ .in The Figure minimum 10. The ofva thelue searchof the mean curve of was the locateddistribution by identifying of the valence the positionangle was where found the firstto be derivative at 105°. The is equal minimum to zero. of A the smooth search spline curve curve was was located fitted toby the identifying search result the inposition order towhere minimise the first the derivative level of noise, is equal and to its ze firstro. derivativeA smooth wasspline calculated. curve was Due fitted to theto the complex search natureresult in of theorder torsion to minimise angle distribution, the level of a noise, series ofand di ffitserent first models derivative were was tested calculated. and the searchDue to was the performedcomplex nature by averaging of the torsion over 100 angle models distribution, at each iteration a series to of ensure different better models statistics. were In tested the case and of thethe torsionsearch angles,was performed the parameters by averaging used in over the search100 models were theat each mean iteration and standard to ensure deviation better ofstatistics. each state In andthe thecase relative of the fractiontorsion ofangles, the individual the parameters states with used respect in the to search each other. were the mean and standard deviation of each state and the relative fraction of the individual states with respect to each other.

Figure 10. χ2 test results for the selenium bond length (left) and valence angle (right).

Polymers 2020, 12, x FOR PEER REVIEW 13 of 21

observed structure factor showed us that the general trend and the basic form and shape of the experimental structure factor is being reproduced by the initial model, but refinement of the structural parameters was required in a manner similar to the work in polybutadiene (see Figure 7A) [31].

Figure 9. Schematic representation of vitreous selenium indicating the various structural parameters used in the modelling procedure.

Having built the initial configuration, all parameters are kept constant and the refinement is performed for the bond length. The Se–Se distance is assigned by a Gaussian probability distribution at a particular mean and standard deviation. The position of the distribution is searched initially as it is known that the impact on the scattering curve will be more visible. The model is built and the bond length is assigned via a normal distribution of different lengths at each iteration. Every time the length changes, the scattering pattern is calculated and compared via the statistical test with the experimental one. The result of the search in the case of the bond length can be seen in Figure 10, and a clear minimum exists at a distance of 2.38 Å. Furthermore, the scattering calculated for a bond length of 2.38 Å yields a very good comparison between calculated and observed scattering in high- Q vectors. Consequently, establishing that the oscillating sine wave in the structure factor arises from selenium atoms joins together by covalent bonding. A similar search for the position of the valence angle distribution can be seen in Figure 10. The value of the mean of the distribution of the valence angle was found to be at 105°. The minimum of the search curve was located by identifying the position where the first derivative is equal to zero. A smooth spline curve was fitted to the search result in order to minimise the level of noise, and its first derivative was calculated. Due to the complex nature of the torsion angle distribution, a series of different models were tested and the search was performed by averaging over 100 models at each iteration to ensure better statistics. In

Polymersthe case2020 of, 12 the, 2917 torsion angles, the parameters used in the search were the mean and standard14 of 21 deviation of each state and the relative fraction of the individual states with respect to each other.

FigureFigure 10. 10.χ χ22 testtest resultsresults forfor thethe seleniumseleniumbond bond length length (left) (left) and and valence valence angle angle (right). (right).

7.2.1. First-Order Probabilities The simplest possible model for the torsion distribution is that where no correlation is assumed between each individual torsion state and its close environment. This yields a picture in which the torsion states are assigned randomly (following specific rules taken from pre-defined distributions) in an unconditional manner having only first-order probabilities and disregarding the previous and following torsion states. That way, two Gaussian distributions were assigned of equal weighting and the position of the torsion distribution was searched in a manner similar to the bond length and valence angle. The distributions were initially assigned at 0◦ or trans defining the symmetry of the rotation. The search was performed in steps of one degree with one distribution taking values from 0◦ to 180 and the other from 0 to 180 . Following this procedure, the torsional angle has been detected − ◦ ◦ ◦ at a position of 145 , indicating the existence of a conformation similar to that found in the trigonal ± ◦ structure [59]. The relative fraction of the two distributions was also searched, and we have found out that the probability of a torsion angle followed by one of the same sign (+ + or ) is almost − − 100%. This indicates that although the helix is highly distorted due to the valence and torsion angle fluctuations, the energy penalty of the torsion angle to change in sides of rotation is very high.

7.2.2. Conditional Probabilities In order to include conditional (second-order) probabilities and make the torsion assignment conditional on its neighbours, we have set-up a matrix formulation similar to that used in the RIS model for polymers [1]. That way, the conditional probabilities of the possible helical configurations will be represented as ! α 1 α − (12) 1 α α − where α is the probability of a torsion angle to be followed by one of a same sign. Using the probability α as a parameter in the refinement technique yields a solution of ~1, indicating that long + + + + or sequences are preferable. This approach of confining the torsional rotations through a − − − − Markov Chain representation has been used in the past for a number of polymers like polyethylene [26], PTFE [27] and polybutadiene [6,31].

7.2.3. Predicted Models There have been reports [56] of flat torsional distributions that allow selenium to adopt a highly flexible chain configuration similar to the freely rotating chain model for polymers in solution. We have created a flat torsion angle distribution, keeping bond lengths and angles at their optimum values. The comparison between the experimental data, the ‘random coil’ and the distorted helix can be seen in Figure 11A,B. Although the differences between the two configurations are difficult to observe with Polymers 2020, 12, 2917 15 of 21 the naked eye, the statistical test results show a completely different picture. The χ2 test for the helix is 0.0044 0.0001 compared with 0.0065 0.0001 of the ‘random coil’ being at a value 50% larger. ± ± Comparison of the structural parameters extracted with the technique presented here with values reportedPolymers 2020 in, the12, xliterature FOR PEER REVIEW can be seen in Table S2 in the Supportive Information. 15 of 21

Figure 11. Comparison between the experimentally obtained structure factor (solid line) for vitreous selenium and the scattering pattern obtained from a model of ( A) a distorted distorted helix, ( B) a a flexible flexible chain, chain, ((C)) six-membersix-member ringsrings andand ((DD)) eight-membereight-member ringsrings (open(open circles).circles).

7.2.4.The Chain existence Length of defects can be handled by inclusion of planar zig-zag trans conformations on the chain. The inclusion of trans states was performed in the same manner as every torsion angle. Initially, The majority of atoms in amorphous selenium have two-fold coordination, as seen from the trans sequences were included in an unconditional manner in a 3-fold torsion distribution of 145 and radial distribution function analysis. In order to make an approximate estimate on the chain± length, trans (0 ), allowing a large probability of + and + sequences. The fit was not satisfactory with a χ2 we used◦ the scattering pattern to try to establish− the− level of sensitivity the number of atoms in the test of the order of 0.008 significantly larger than the value obtained for the helical chain. Inclusion of model has on the statistical test. Obviously, such an approach provides a rough estimation on the conditionality in the probability matrix yields a picture of almost equally probable –trans+ and +trans– level of information the diffraction pattern carries and its sensitivity on the model size. For this, we sequences with a significantly lower statistical test value of 0.0025 0.0001. used the ″best-fit″ model of the distorted helix. We calculate the± scattering of the model and we χ2 7.2.4.performed Chain the Length test in an iterative manner, changing the number of atoms in the model each time. We concluded that based on the scattering data, the diffraction pattern stays almost unaffected by the numberThe majority of atoms of atoms in the in chain amorphous above seleniumN > 100. This have can two-fold be seen coordination, in Figure 12 as where seen from results the of radial the distributioncomparison functionbetween analysis.similar models In order of todifferent make an chain approximate lengths are estimate plotted on against the chain the length,experimentally we used theobserved scattering scattering pattern curve. to try toBased establish on this the levelinformat of sensitivityion, we can the numbersay that of the atoms model in the is modelnot sensitive has on theenough statistical to distinguish test. Obviously, between such models an approach of very provides large chains. a rough It indicates estimation though on the that level the of chains information have theto be di atffraction least 100 pattern atoms carrieslong toand repres itsent sensitivity in an adequate on the modelway the size. scattering. For this, we used the ”best-fit” model of the distorted helix. We calculate the scattering of the model and we performed the χ2 test in an iterative manner, changing the number of atoms in the model each time. We concluded that based

Polymers 2020, 12, 2917 16 of 21 on the scattering data, the diffraction pattern stays almost unaffected by the number of atoms in the chain above N > 100. This can be seen in Figure 12 where results of the comparison between similar models of different chain lengths are plotted against the experimentally observed scattering curve. Based on this information, we can say that the model is not sensitive enough to distinguish between models of very large chains. It indicates though that the chains have to be at least 100 atoms long to represent in an adequate way the scattering. Polymers 2020, 12, x FOR PEER REVIEW 16 of 21

2 FigureFigure 12. ComparisonComparison between between the the different different χχ2 testtest results results for for a a model model of of selenium selenium based based on on the the disordereddisordered helix helix as as a a function function of of chain chain length length (indicated (indicated by by the the nu numbermber of of atoms atoms in in the the model) model) and and the different areas of the experimentally observed structure factor. Qmax = 30 Å−11 indicates the use of the different areas of the experimentally observed structure factor. Qmax = 30 Å− indicates the use of the entire scattering curve, Qmax = 5 Å−1 1indicates the use of the first peak of the scattering curve and the entire scattering curve, Qmax = 5 Å− indicates the use of the first peak of the scattering curve and Qmax = 10 Å−1 indicates1 the use of the first two peaks of the scattering curve. In all cases, we can see Qmax = 10 Å− indicates the use of the first two peaks of the scattering curve. In all cases, we can see thatthat the the statistical statistical test test becomes becomes invariant invariant fo forr a a model of at least 100 selenium atoms.

7.2.5.7.2.5. Rings Rings TheThe presence presence of of six- six- and and eight-member eight-member rings rings has has been been debated debated in in the the past past [60], [60], although although it it has has beenbeen argued argued [61] [61] based based on on optical optical spectroscopy spectroscopy studies studies that that ring ringss do do not not appear appear to to any any great great extent. extent. ToTo investigate the existence of rings, we we construc constructedted a a model model of of 1000 1000 six- six- and and eight-member eight-member rings, rings, takingtaking thethe values values for for their their structural structural characteristics characteri fromstics their from relative their crystalrelative structure. crystal Thestructure. comparison The comparisonbetween the between experimentally the experimentally observed scatteringobserved scat patterntering and pattern the scatteringand the scattering from the from rings the can rings be canseen be in seen Figure in 11FigureC,D. The11C,D. comparison The comparison is clearly is notclearly satisfactory not satisfactory especially especially when compared when compared with the withhelical-based the helical-based model (Figure model 11B). (Figure Adding 11B). up theAdding structure up factorsthe structure according factors to their according weighted to average their weightedleads to a average possible leads fraction to a of possible rings in fraction the sample of rings to be in less the than sample 2%. to be less than 2%. FromFrom this this work, work, we we can can see that the model is fairly sensitive and has the ability to distinguish betweenbetween small small variations variations in in the the local conformation conformationalal characteristics characteristics and and their their expression expression via via the the diffractiondiffraction pattern. pattern. As As discussed discussed previously, previously, simila similarr behaviour behaviour has has been been observed observed for for a a number number of of differentdifferent polymeric systems [6,26,27,31]. [6,26,27,31]. The The main main re reasonason for for this this ability ability is is the chain conformation andand connectivity connectivity that that restricts restricts the the potential potential number number of of configurations configurations that that are are available available to to the the system. system. ThisThis restriction restriction and deviations from it manifest it itselfself with with the the existence existence of of unaccounted unaccounted peaks peaks on on the the calculated structure factor that are not represented by the experimentally observed one. This has, as a consequence, a nonsatisfactory comparison between the model and the data that leads to a significant increase in the value of the statistical test. An example of this effect can be seen in Figure 13, where the experimentally observed structure factor for a system of deuterated 1,4-polybutadiene is compared with an equivalent model that has different values for the C-D bond length. For 1,4- polybutadiene, the C-D bond length has been estimated from the model at 1.13 Å and, as we can see from Figure 13, the comparison with the experimental data especially in the high-Q region is fairly good. For unrealistic values of the bond length (1 and 1.5 Å in this case), we can see that the comparison between the experimental data and the model prediction is fairly poor [6]. This effect is

Polymers 2020, 12, 2917 17 of 21 calculated structure factor that are not represented by the experimentally observed one. This has, as a consequence, a nonsatisfactory comparison between the model and the data that leads to a significant increase in the value of the statistical test. An example of this effect can be seen in Figure 13, where the experimentally observed structure factor for a system of deuterated 1,4-polybutadiene is compared with an equivalent model that has different values for the C-D bond length. For 1,4-polybutadiene, the C-D bond length has been estimated from the model at 1.13 Å and, as we can see from Figure 13, the comparison with the experimental data especially in the high-Q region is fairly good. For unrealistic Polymersvalues 2020 of the, 12, bondx FOR PEER length REVIEW (1 and 1.5 Å in this case), we can see that the comparison between17 of the 21 experimental data and the model prediction is fairly poor [6]. This effect is very pronounced in the case veryof bond pronounced lengths (especially) in the case and of valencebond lengths angles (especially) (to a slightly and lesser valence degree) angles due (to to thea slightly large number lesser degree)of these due configurations to the large in number the system. of these Therefore, configurations we can usein the these system. structural Therefore, parameters we can as use a way these of structuralexcluding unrealisticparameters configurations as a way of whileexcluding keeping unrealistic the total numberconfigurations of conformations while keeping manageable the total for numberthe model of [conformations2]. manageable for the model [2].

Figure 13. Effect of the carbon–deuterium bond on the overall picture of the diffraction pattern. Figure 13. Effect of the carbon–deuterium bond on the overall picture of the diffraction pattern. The The best-fit distance of 1.13 Å (short-dashed line) shows a significant resemblance to the experimental best-fit distance of 1.13 Å (short-dashed line) shows a significant resemblance to the experimental data (solid line) when compared with two artificial C-D values of 1 Å (dotted line) and 1.5 Å (short dashed data (solid line) when compared with two artificial C-D values of 1 Å (dotted line) and 1.5 Å (short line). All the curves have been shifted for clarity. The image has been reproduced with permission dashed line). All the curves have been shifted for clarity. The image has been reproduced with from Reference [6] © American Chemical Society. permission from Reference [6] © American Chemical Society. 8. Time-Resolved Crystallisation 8. Time-Resolved Crystallisation With modern equipment and increases in time resolution available for neutron diffraction, it is possibleWith to modern extend theequipment ideas mentioned and increases previously in time in exploringresolution possibilities available for of usingneutron broad diffraction, Q diffraction it is possiblemethods to in theextend study the of phaseideas transitionsmentioned such previously as crystallisation. in exploring As the possibilities main interest of isusing understanding broad Q diffractionthe molecular-level methods changesin the study which of are phase involved transitions in this such phase as transformation, crystallisation. itAs is the clear main that interest unless we is understandinghave a high-quality the molecular-level model of the melt changes phase, which attempting are involved to identify in this small phase deviations transformation, from it, in it the is clearvery earlythat unless stages, we is mosthave likelya high-quality to lead to model no satisfactory of the melt conclusion. phase, attempting In Figure 14to, weidentify can see small the deviationsstructure factors from it, and in radialthe very distribution early stages, functions is most obtained likely to fromlead ato systemno satisfactory of partially conclusion. deuterated In Figurepoly(ε -caprolactone)14, we can see inthe the structure melt and factors at diff anderent radi processesal distribution of crystallization functions obtained [42,44,45 ].from The a di systemfferent of partially deuterated poly(ε-caprolactone) in the melt and at different processes of crystallization [42,44,45]. The different length scales associated with different parts of the experimentally observed structure factor can be clearly seen and the different features observed in the high-r region of the radial distribution function gives an indication of the power and convenience in utilising the reciprocal space function instead of the real space one. Further analysis of such extended data sets can be made by utilising modified versions of the techniques discussed previously in a sequential manner, starting from the chain conformation and moving towards the coarser length scales. Crystals can be introduced by arranging segments in specified ways while keeping them consistent with the chain conformation, and the overall effect on the scattering pattern can then be studied in an iterative manner [41–45].

PolymersPolymers20202020,,1212,, 0 2917 1818 of of 21 21 lengthlength scales scales associated associated with with di difffferenterent parts parts of of the the experimentally experimentally observed observed structure structure factor factor can can be be clearlyclearly seen seen and and the the di difffferenterent features features observed observed in in the the high-r high-r region region of of the the radial radial distribution distribution function function givesgives an an indication indication of of the the power power and and convenience convenience in in utilising utilising the the reciprocal reciprocal space space function function instead instead of of the the realreal space space one. one. Further Further analysis analysis of of such such extended extended data data sets sets can can be be made made by by utilising utilising modified modified versions versions ofof the the techniques techniques discussed discussed previously previously in in a a sequential sequential manner, manner, starting starting from from the the chain chain conformation conformation andand moving moving towards towards the the coarser coarser length length scales. scales. Crystals Crystals can can be be introduced introduced by by arranging arranging segments segments in in specifiedspecified ways ways while while keeping keeping them them consistent consistent with with the the chain chain conformation, conformation, and and the the overall overall e effffectect on on thethe scattering scattering pattern pattern can can then then be be studied studied in in an an iterative iterative manner manner [ [4141––4545].].

Figure 14. Experimentally observed structure factor (left) and radial distribution function (right) Figure 14. Experimentally observed structure factor (left) and radial distribution function (right) as obtained from a simultaneous small and wide angle neutron scattering experiment in NIMROD. as obtained from a simultaneous small and wide angle neutron scattering experiment in NIMROD. The different length scales associated with the different regions of the functions is highlighted. The inset The different length scales associated with the different regions of the functions is highlighted. The inset in the right figure indicates the time evolution of the radial distribution function with temperature. in the right figure indicates the time evolution of the radial distribution function with temperature. 9. Conclusions 9. Conclusions In this work, we have presented an overview of the possibilities for structural analysis In this work, we have presented an overview of the possibilities for structural analysis offered by intimately coupling high-quality broad Q neutron scattering data with atomistic models. offered by intimately coupling high-quality broad Q neutron scattering data with atomistic models. Taking advantage of the possibility to separate the structure factor into inter and intrachain contributions Taking advantage of the possibility to separate the structure factor into inter and intrachain contributions allows for the use of reciprocal-space functions instead of the traditional analysis in real space via the allows for the use of reciprocal-space functions instead of the traditional analysis in real space via the radial distribution function. This possibility makes the analysis of the diffraction data unambiguous radial distribution function. This possibility makes the analysis of the diffraction data unambiguous and eliminates any potential issues arising by terminations and resolution of the Fourier integral. and eliminates any potential issues arising by terminations and resolution of the Fourier integral. Atomistic models based on internal parameters that are associated with the polymer structure Atomistic models based on internal parameters that are associated with the polymer structure can be built with a high level of confidence, and the calculation of the relevant scattering function can be built with a high level of confidence, and the calculation of the relevant scattering function is trivial. Direct comparison of the calculated with the experimentally observed scattering using a is trivial. Direct comparison of the calculated with the experimentally observed scattering using a statistical test allows a large number of parameters and models to be tested in an iterative way within a statistical test allows a large number of parameters and models to be tested in an iterative way within a very short period of time. Studies on amorphous polyethylene, poly(tetrafluoroethylene), polystyrene, very short period of time. Studies on amorphous polyethylene, poly(tetrafluoroethylene), polystyrene, polypropylene and polybutadiene copolymers indicate the robustness of the methodology and offer polypropylene and polybutadiene copolymers indicate the robustness of the methodology and offer unique insights on the spatial arrangements of the polymer chains in the bulk. Except the traditional unique insights on the spatial arrangements of the polymer chains in the bulk. Except the traditional connectivity parameters, these methods allow for extrapolation and identification of large-scale connectivity parameters, these methods allow for extrapolation and identification of large-scale statistical properties like the orientation correlations, the characteristic ratio and the end-to-end length, statistical properties like the orientation correlations, the characteristic ratio and the end-to-end length, thus intimately linking the diffraction pattern with unique structural features and correlations as a thus intimately linking the diffraction pattern with unique structural features and correlations as a function of the local conformation and the segmental packing. function of the local conformation and the segmental packing. Decomposition of the scattering function into partial terms allows for the study of particularly Decomposition of the scattering function into partial terms allows for the study of particularly labelled parts of the backbone and bulky side groups, and the way their correlations affect the overall labelled parts of the backbone and bulky side groups, and the way their correlations affect the overall observed diffraction. Experiments in selectively labelled glassy polystyrene give particular insight in observed diffraction. Experiments in selectively labelled glassy polystyrene give particular insight in the different packing arrangements of the chains and the role played by the phenyl groups reinforcing the different packing arrangements of the chains and the role played by the phenyl groups reinforcing previously reported ideas based on X-ray diffraction on limited Q range. Similar ideas can be used previously reported ideas based on X-ray diffraction on limited Q range. Similar ideas can be used in the study of the extent of mixing in the local level as this is seen through partial structure factors Polymers 2020, 12, 2917 19 of 21 for isostructural blends. Initial studies on selectively labelled polybutadiene blends indicate the role played by the local chain conformation in the extent of mixing. These results are in agreement with previously reported suggestions on the role of the connectivity and local topology on the level of mixing. Using this approach, we can map in a qualitative and quantitative way the different correlations available in the system, and couple them together with the level of local order and see their effect on the overall miscibility, something not possible within the framework of the traditional theory where all these factors are condensed to a single interaction parameter.

Supplementary Materials: The following are available online at http://www.mdpi.com/2073-4360/12/12/2917/s1, Supplementary information contains: Table S1. Characterisation information for the 1,2 and 1,4 polybutadiene, Table S2 comparison of results of the analysis on vitreous Selenium with previously reported data, preparation methods for the polyethylene materials discussed in the text and the polybutadiene blends. Author Contributions: Conceptualization, T.G. and G.R.M.; methodology, T.G. and G.R.M.; software, T.G. and G.R.M.; validation, T.G. and G.R.M.; formal analysis, T.G.; investigation, T.G. and G.R.M.; resources, G.R.M.; writing—original draft preparation, T.G.; writing—review and editing, T.G. and G.R.M.; project administration, T.G. and G.R.M.; funding acquisition, G.R.M. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Polytechnic of Leiria and FCT (Portugal) through UC4EP PTDC/CTM-POL/7133/2014) and UID/Multi/04044/2019, the University of Reading and EPSRC (UK). Acknowledgments: The work described was made possible by the never-ending help of the SANDALS and NIMROD beam-line scientists, Alan Soper, Daniel Bowron, Tristan Youngs and Chris Benmore in ISIS Neutron Facility in the UK. We would like to thank Fred Davis, Robert Olley, Barbara Rosi-Schwartz, Clark Balague, David Lopez and Supatra Siripitayannanon for all their valuable comments, help and assistance during sample preparation and data analysis. Furthermore, we would like to thank Adrian Wright for providing us with the selenium diffraction data. Conflicts of Interest: The authors declare no conflict of interest.

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