Structure determination of liquid carbon tetrabromide via a combination of x-ray and neutron diffraction data and reverse Monte Carlo modeling
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l NOTICE: this is the author’s version of a work that was accepted for pub-lication in Journal of Molecular Liquids. Changes resulting from the publish-ing process, such as peer review, editing, corrections, structural formatting,and other quality control mechanisms may not be reflected in this document.Changes may have been made to this work since it was submitted for publica-tion. A definitive version was subsequently published in
Journal of MolecularLiquids , (2014), 204-210, DOI:10.1016/j.molliq.2014.05.0111 tructure determination of liquid carbon tetrabromidevia a combination of x-ray and neutron diffraction dataand reverse Monte Carlo modeling László Temleitner
Wigner Research Centre for Physics, Hungarian Academy of Sciences, Konkoly-Thege M.út 29-33, 1121, Budapest, Hungary
Abstract
In order to reveal the atomic level structure of liquid carbon tetrabromide, anew synchrotron x-ray diffraction measurement, over a wide momentum trans-fer (Q-)range, has been performed. These x-ray data have been interpretedtogether with a neutron diffraction dataset, measured earlier, using the reverseMonte Carlo method. The structure is analysed on the basis of partial radialdistribution functions and distance dependent orientational correlation func-tions. Orientational correlations behave similarly to other carbon tetrahalides.Moreover, the information content of the new x-ray diffraction data set, and inparticular, of the varying Q-range, is also discussed.Only very small differences have been found between results of calculationsthat apply one single experimental structure factor and the ones that use bothx-ray and neutron diffraction data: the latter showed slightly more orderedcarbon-carbon radial distribution function, which resulted in seemingly moreordered orientational correlations between pairs of molecules. Neither the ex-tended Q-range, nor the application of local invariance constraints yielded sig-nificant new information. For providing a simple reference system, a hard spheremodel has also been created that can describe most of the partial radial dis-tribution functions and orientational correlations of the real system at a semi-quantitative level.
Keywords: molecular liquid, x-ray diffraction, Reverse Monte Carlo,orientational correlations
PACS:
1. Introduction
The covalently bonded tetrahedral shape molecule, carbon tetrabromide(tetrabromomethane,
CBr ) has become a model system for studies on plastic Email address: [email protected] (László Temleitner)
Preprint submitted to Elsevier October 4, 2018 or orientationally disordered) crystalline phase among carbon tetrahalides. Itsdiffraction pattern shows only a few, not too intense Bragg-peaks; on the otherhand, a significant amount of diffuse scattering contributions appear[1]. To de-scribe this behaviour various models have been refined in the past [2, 3, 4, 5].Apart from its plastic phase,
CBr has several high pressure modifications, in-cluding phase III that appears to be also plastic crystalline[6]. Studying theseexotic variations by total scattering powder diffraction may only be possible inthe future, by instrumentation developed very recently for synchrotron x-raydiffraction and high-pressure techniques.The liquid phase of CBr is stable between 365 and 463 K[7]. In contrastto the solid phases of carbon tetrabromide, only few experiments were per-formed by neutron[1, 8, 5] and x-ray[4] diffraction on the liquid. Two of them[1, 4] (including the only published x-ray result until now) compared qualita-tively the pattern of different phases; however, detailed analyses of the liquidphase was not their subject. The first theoretical investigation[9] using diffrac-tion data was performed by the help of reference interaction site model[10],which estimated the atomic radius of bromine atoms. A more detailed analy-sis performed on measured diffraction datasets by Reverse Monte-Carlo (RMC)simulation was provided by Bakó et al.[8], who reported all of the three par-tial radial distribution functions and made a simple analysis of orientationalcorrelations. Their most important result was that they ruled out the for-merly proposed Apollo-model[11] from the most probable orientations; similar-ities between plastic crystalline and liquid phase were also surmised. A morerecent contribution[12] discussed similarities to XCl liquids by orientationalcorrelations based on molecular dynamics simulation and Rey’s classificationof mutual orientation of tetrahedral molecules[13]. This classification has beenproven to be useful by numerous recent studies for liquids containing tetrahe-dral molecules[14, 15, 16, 17, 18, 19]. Another recent examination[5] provideda systematic study of orientational correlations (again, by means of Rey’s clas-sification) in liquid and two crystalline phases by neutron diffraction, provingthe similarities of orientational correlations between the plastic crystalline andliquid phases.For the full determination of the two-particle level structure of a disorderedmaterial (containing n different kinds of atoms) purely from diffraction exper-iments, we must be able to perform n ( n + 1) / independent diffraction mea-surements, varying the scattering parameters of each element in the materialin question (wherever such a variation is feasible). However, by applying the-oretical considerations, or by performing some sort of structural modeling, therequired number of measurements might be reduced (see, e.g. [14, 15]).Throughout the present work, effects of the new measured x-ray data, its usetogether with earlier neutron diffraction results[5], and the applied momentumtransfer range will be discussed, in terms of atomic radial distribution functionsand orientational correlations between molecules. The feasibility of analysingthe structure of liquids of tetrahedral molecules based on only one single diffrac-tion experiment, using molecular constraints and RMC, have been discussed bymany authors[8, 20, 5]. This approach has been proven valid [17] for several3 Cl liquids experimentally. However, such a validation is lacking for the caseof CBr . Also, the latest publication on the subject[5] used an assumptionbased on earlier work[21], that the applied Q-range (up to only 8 Å − ) is suf-ficient for describing intermolecular correlations; this assumption also needs tobe verified.
2. Experiment
The sample, provided by Sigma-Aldrich, contained 99% pure
CBr poly-crystals at room temperature.The X-ray diffraction experiment has been carried out at the BL04B2 high-energy x-ray diffraction beamline[22] of the Japan Synchrotron Radiation Re-search Institute (JASRI/SPring-8, Hyogo, Japan). The incoming photon energywas chosen to be 37.65 keV (corresponding to a wavelength of 0.329Å). The cap-illary transmission geometry with single HPGe detector (in the horizontal plane)setup has been used.The powdered sample was filled into a 1 mm diameter, thin walled borosil-icate glass capillary (GLAS Müller, Germany) mounted in a Canberra vacuumfurnace, available at the beamline. The liquid phase measurement was per-formed at . ± . K, recording the intensities of scattered photons by agermanium detector and the incoming beam by monitor counter. To optimizethe performance of the experimental apparatus, the patterns have been recordedin four, slightly overlapping, segments, differing by the incoming beam widthand height. After the measurement on the sample, scattered intensities of theempty capillary were also recorded by the same conditions.Raw intensities were normalized by the monitor counter, corrected for atten-uation, polarization, and empty capillary intensities. Then, the whole patternwas reconstructed from the segments, scaled in electron units and corrected forCompton-scattering contributions following the standard procedure[23].
3. Reverse Monte Carlo modeling
Series of Reverse Monte Carlo simulations (
RMC_POT [24, 25]) have been per-formed, in order to reveal the information content of the diffraction datasetsbeyond the already known evidences, such as molecular parameters and density(see Table 1). Taking into account solely the molecular geometry and the den-sity, two hard sphere Monte Carlo simulations have been started: the atomicparameters of
HS0 were identical to RMC calculations with experimental data(see below). In contrast, atomic radii closer to reality have been applied inthe
HS1 model. Of the experimentally constrained simulations, three runswere performed using both experimental (x-ray and neutron diffraction) data:
NXl and
NXlli made use of the entire measured Q-range of both datasets,whereas in
NXsh the maximum momentum transfer values were identical forthe two experimental datasets. For the ’neutron-only’ case the earlier simulatedconfigurations[5] ( N ), as well as a new simulation ( Nli ) were considered.4 more detailed description of the Reverse Monte Carlo method can befound elsewhere[26, 24], here only the way of calculation of the (total scattering)structure factor ( F ( Q ) ) is shown. During the RMC calculation, the partial radialdistribution functions (prdf, g xy ( r ) ) are calculated from atomic coordinates.The prdf’s are then Fourier-transformed to obtain the corresponding partialstructure factors ( S xy ( Q ) ): S xy ( Q ) − π̺ Z ∞ r ( g xy ( r ) −
1) sin( Qr ) Qr dr, (1)where ̺ denotes the number density. To calculate the total scattering struc-ture factor belonging to a given experiment, the partial structure factors aresummed by proper coefficients ( w ij ). Coefficients depend on concentration ( c i )and atomic form factors[27] for x-rays ( f i ( Q ) ), and Q-independent scatteringlengths[28] for thermal neutrons as follows: w ij = c i c j f i ( Q ) f j ( Q )( P c i f i ( Q )) (2 − δ ij ) , (2)where δ ij is the Kronecker symbol. The applied coefficients are shown in Table2 for both experiments.Concerning the density of liquid CBr , earlier works performed calculationsusing atomic number densities of 0.031 Å − [9, 8] and 0.02688 Å − ([12], refer-ring to [29]); these values correspond to bulk densities of 3.41 and 2.961 g/cm ,respectively. Interestingly, the former value is identical to that of liquid CCl (0.031 Å − [9] or 0.0319 Å − [17]). Since the volume of the CBr moleculeis larger than that of the CCl molecule, the number density of liquid CBr should be lower than that of liquid CCl , unless there are some special ori-entations that would enable denser packing. Since no long-range orientationalcorrelations have been found earlier[8] in liquid CBr (which might result in amore densely packed structure), it can be concluded that some of the earlierworks [8, 9] actually used a number density that is far too high. Also, a RMCtrial run gave worse agreement (with an R wp about 12% for the neutron dataset)with unphysical features in r space, in comparison with the lower density case(see Table 1). For these reasons, in each calculation reported here, the lowerdensity value was taken.For each model, the simulation box contained 6912 molecules with an atomicnumber density of 0.026888 Å − , which provided 54.36 Å half box length. Tomaintain the geometry of the molecules during the series of single atomic moves,fixed neighbour constraints[30] have been applied between carbon and bromine( . ± . Å) and between bromine and bromine atoms ( . ± . Å) within themolecule. For pairs of atoms belonging to different molecules, closest distanceconstraints (cut-offs) have been used between carbon-carbon (3.5 Å), carbon-bromine (2.5 Å) and bromine-bromine (2.8 Å) pairs. At
HS1 run, these limitswere 5.0 Å, 3.8 Å and 3.5 Å, respectively.The RMC procedure provides sets of configurations that agree with diffrac-5ion results within the experimental error. In order to improve the statisticalaccuracy of the results, series of independent particle configurations have beensaved; the configurations have been taken so that they were separated by atleast one successful move for each atom.In order to favour as uniform environments of atoms as possible, the localinvariance[31, 25] constraint has been applied for each prdf’s of
Nli and
NXlli runs up to 10.88 Å in each distance bin. However, the simulation has becomeprohibitively slow, thus only one configuration has been saved for these tworuns.
4. Results and discussion
When RMC simulations were completed, very good agreement between modeland measured structure factors were obtained: the R wp -factors are low in eachcase (see Table 1). The residuum (Figure 1) seems to be structureless in thecase of neutron-, but somewhat more structured for the x-ray diffraction dataset.This might be the effect of small errors at the normalisation of different seg-ments of raw x-ray diffraction data. Even though numerous runs were per-formed, the residuum follows similar characteristics for each of them. Thus,only the residuum of NXsh and
NXl models are shown in Figure 1. Althoughthere is a small difference between them, they do not differ significantly. Also,the intermolecular contributions calculated from
NXl model (figure 1) are neg-ligible (or in the order of experimental errors) beyond 6-8 Å − . These suggestthat the information provided by the x-ray dataset between 8.0 and 14.0 Å − does not contribute much to the primary results of the simulation, i.e. to theunderstanding of intermolecular correlations.Moving to the real space analysis, the intermolecular part of the RMC simu-lated prdf’s are shown in figure 2. Having noted the only very small differencesbetween results of simulations that apply experimental data, we can conclude,that they do not differ much from each other. Thus, the assumption on the fea-sibility of structure determination based on limited Q range and on one singleexperimental dataset looks adequate for liquid carbon tetrabromide, too.Going into somewhat more details, we can found slight differences betweenthe N and NX runs concerning the CC partial: the latter shows a more signifi-cant first maximum (around 6.0 Å) and minimum (about 8.4 Å). The coefficientbelonging to this partial is very small for both types of data; on the other hand,there is a good contrast between x-rays and neutrons for the other two partialcontributions. Note that as the CBr and
BrBr partials are well determined(i.e., have high contributions to the total scattering structure factor), the CC partial must adjust to them, via the applied molecular constraints.A trend similar to what was found for the CC prdf can be observed forthe CBr partial, but not for the
BrBr one (whose coefficient is the highest).Concerning the
BrBr prdf, (the simple) calculation N reveals this function asprecisely as the more sophisticated NXsh model. In contrast, a slight difference6ppears (the maximum is shifted from 3.7 to 3.9 Å), if we compare the prdf’sbelonging to
NXsh and
NXl runs. That is, the applied Q-range seems to playa (very) minor role in determining prdf’s of liquid carbon tetrabromide whenRMC modeling is used.The outcome of simulations using the local invariance constraint (
Nli and
NXlli ) are almost identical to the corresponding unconstrained runs ( N and NXl , respectively); these are therefore not shown in figure 2. This finding showsthat the molecular geometry has been taken into account well by fixed neighbourconstraints.In order to be able to separate the structural consequences originating tothe packing fraction (density and molecular shape) from those of experimentaldata, two hard sphere simulations have been conducted (see table 1). The
HS0 results, whose intermolecular cut-off’s were identical to those calculations thatuse experimental datasets, differ much from results based on experimental data.This suggests that effective atomic (’hard sphere’) radii are larger than thoseapplied in our RMC simulations. Also, a hard sphere model is not able todescribe intermolecular prdf’s in every detail. In run
HS1 , the cutoff’s wereincreased, to get more realistic results. Although agreement with results ofexperimentally constrained runs are not perfect, due to arbitrary selection ofintermolecular cut-offs, the
BrBr prdf is approximated well at distances longerthan the first neighbour maximum (except around the following minimum andmaximum distances). The applied (’hard sphere’) cut-off is in agreement withan earlier RISM calculation[9] (where the intermolecular
BrBr distance was . ± . Å). This suggests that pairwise correlations are largely governed bythe close packing of bromine atoms, in agreement with Bakó et. al.[8]. In thecase of the CC prdf the situation is qualitatively similar, i.e., an overall goodqualitative agreement was achieved: the HS1 model is able to describe maximumand minimum positions, although estimating exact intensities is beyond thecapability of the model.
In the present analysis of the mutual orientational correlations between pairsmolecules, the classification of Rey[13] has been applied (figures 3 and 4). Inshort, each class is described by the number of atoms belonging to each moleculebetween two planes containing the centres and perpendicular to the connectingline. This way, 1:1 (corner-corner), 1:2 (corner-edge), 1:3 (corner-face), 2:2(edge-edge), 2:3(edge-face), 3:3 (face-face) classes can be formed.By examining the resulting orientational correlations for liquid carbon tetra-bromide(figures 3 and 4), we can conclude that nearly independently of the ac-tual model, almost all curves for a given class follow similar characteristics. The(most of the time, marginal) differences that can be found between them willbe discussed below.Generally speaking, distance-dependent probabilities of each class follow theregular characteristics shown for perfect tetrahedral[12, 17] and for
CBr − x Cl x type molecules[18, 32, 19] (neglecting the differences between Cl and Br atoms).The first maxima for the classes follow each other from the closely contacting7:3 pairs at the intermolecular CC cutoff distance through 2:3 (5.4 Å), 2:2(6 Å), 1:3 (6.2 Å) and 2:1 (7.2 Å) to the 1:1 (about 7.4-7.7 Å) correlations.The most probable orientation at the asymptotic limit is the edge-edge (2:2)one, for which the correlation function becomes structureless beyond 10Å. Thecorner-face (1:3) class, which would describe the ’Apollo’ kind of orientationalcorrelations, is less abundant, in agreement with earlier suggestions[8, 12], butproduce remarkable oscillations at higher distances, up to 15 Å.In the cases of the two classes just mentioned, 4 atoms are positioned betweenthe planes, same as the average for random orientations. For classes corner-edge(1:2) and edge-face (2:3), containing 3 and 5 atoms respectively, the deviationfrom the average is compensated by alternating these two orientations (see parts ( c ) of figures 3 and 4). The behaviour observed for liquid CBr here agreeswith the generic case[12] and does not show long range oscillations as foundfor liquid CCl [17]. Finally, the egde-edge (1:1) and corner-corner (3:3) typearrangements also occur as usual among tetrahalides.Focusing at the differences between the outcome of various models, first thelocal invariance runs Nli and
NXlli are examined. We can conclude that theirorientational correlations do not differ from the corresponding ’standard RMC’models ( N and NXl ) (apart from the statistical noise). This reflects the factthat the ’local invariance’ constraints influence primarily the intra molecularparts of the prdf’s and not the inter molecular parts; for this reason, these(computationally expensive) constraints are not considered any further.The next question may be whether the available/applied Q -range causes anydifference in terms of orientational correlations. Comparing NXl and
NXsh re-sults in Figure 4 immediately shows that the differences are negligible. Thismakes sense because increasing the Q -range contributes mainly to fixing themolecular geometry more accurately (which is already done by fnc constraints).Thus the applicability of the original Q-range[5] for revealing orientational cor-relations in liquid carbon tetrabromide could be confirmed.The additional information content of the x-ray diffraction dataset may beexamined and clarified by analysing differences in the probabilities of the orien-tational classes belonging to N and NXl models (figure 3). Several small, butdistinct features may be observed. As we can expect based on the more ordered CC prdf around the first maximum in the combined model, we can find differ-ences at this distance range for 2:2, 3:3 and 1:3 orientations. As the intensitiesof the former ones become lower, the probability of the last one increases incomparison with the corresponding N model. However, since the BrBr prdfdoes not change when both diffraction data are modeled together, the more or-dered g CC ( r ) influences other orientation correlations beyond the first maximaof CC prdf. The most significant change is that the intensities of the first max-ima of 1:2 and minima of 2:3 classes become sharper and the probabilities getcloser to the asymptotic values over the 8.5-10 Å distance range. To compensatethis behaviour, the minimum of 1:3 is shifted to 8.8 Å and probabilities of 2:2arrangements also become smaller over the 7.2-8.0 Å range. This suggests thatthe first coordination shell is slightly more ordered than found when using onlyone diffraction dataset. Nevertheless, the majority of orientational correlations8ould be captured within errors by using one single experimental result. Thatis, recent RMC studies on the CBr − x Cl x family of molecular liquids[32, 19]were most certainly able to capture most structural details of the real systems.Finally, we turn to the comparison with hard sphere results. As discussed insection 4.1, only the HS1 model has been used for comparison, whose cutoffs hadbeen set according to experimental data. As this model differs from experiment-constrained runs mainly in first coordination shell range, this kind of model isa very good approximation above 11Å, where the second maxima of g CC ( r ) isfound. It is important to note that the N model is closer to HS1 , in comparisonwith
NXl , which contains finer details of pair correlations. In general, the
HS1 model overestimates probabilities of the first maxima of 2:3, 2:2, 1:2, 1:1, whileunderestimates the occurrence of the 1:3 class. The conclusion from such ahigh level of agreement is that orientational correlations are nearly completelydetermined by close packing of molecules in liquid
CBr as suggested earlier[8].
5. Conclusions
For determining the structure of liquid carbon tetrabromide, high-energyx-ray diffraction experiments were performed, followed by an extensive seriesof reverse Monte Carlo calculations. Previously determined neutron diffractiondata have also been made use of. Orientational correlations in the resultingstructural models agree with general characteristics found for liquids of tetra-halides. During the calculations, effects of the varying Q-range and of the num-ber and type (x-ray and/or neutron diffraction) of applied experimental datasets have been tested. It has been concluded that one single experimental dataset measured over a limited Q -range may be sufficient for producing adequatestructural models of this molecular liquid. However, using two experimentaldatasets yielded a better determination of the carbon-carbon prdf and gaveslightly more ordered orientational correlations than found before. Increasingthe upper limit of the Q -range from 8 to 14 Å − has not provided significantadditional insight in terms of orientational correlations.For the purpose of reference, hard sphere Monte Carlo runs have also beenperformed. The calculation that operated with realistic cut-off values was com-pared in detail with models based on experimental data at the level of orien-tational correlations. An overall semi-quantitative agreement was found, withcompletely matching extrema positions, while some probability differences couldbe detected mainly around the first maxima of each kind of arrangement. Thissuggested that a great deal of structural details in liquid CBr is due to theclose packing of bromine atoms, including the nearly perfect determination ofthe CC prdf. Acknowledgement
This work was supported by the Hungarian Basic Research Found (OTKA)under Grant No. 083529. The synchrotron radiation experiment were performed9t the BL04B2 beamline of SPring-8 with the approval of the Japan SynchrotronRadiation Research Institute (JASRI) under the Proposal No. 2010A1303. Theauthor would like to acknowledge the Japan Society for the Promotion of Science(JSPS) for a Postdoctoral Fellowship in the period 2009-2011 and Dr. S. Kohara(JASRI) both as host researcher during the JSPS fellowship and as beamlinescientist for teaching him the complete beam alignment of BL04B2. The help ofL. Pusztai (Wigner RCP) is acknowledged for carefully reading the manuscript.
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CBr . Upper panel: neutron diffraction[5]; lower panel: x-ray diffraction(this work). Circles: experimental data; red solid lines: RMC model ( N and NXl ); blacksolid lines: intermolecular contributions calculated from the
NXl model; blue lines: residuumcalculated for the
NXl model (magnified by 5); green line: residuum for the
NXsh model(magnified by 5), respectively. The residuals are shifted along the y-axis by 0.5 in the upper,and by 1.2 in the lower panel.
10 15 20 2500.511.522.5 5 10 15 20 2500.20.40.60.811.21.41.6 g xy (r) r [Å] Figure 2: RMC refined intermolecular partial radial distribution functions of liquid
CBr ascalculated from the various structural models. Upper panel: CC ; middle panel: CBr ; lowerpanel:
BrBr . Black solid line:
HS1 ; red solid line: N ; blue solid line: NXsh ; cyan dashedline:
NXl . r CC [Å] P [ % ] (a) (b)(c) (d) Figure 3: Representation of orientational correlations by the distance-dependent probabilitiesof the classes of Rey, as calculated from the various model structures. (a) solid lines 2:2; (b)solid lines 1:3; (c) solid lines: 1:2, thin solid lines with circles: 2:3; (d) solid lines: 1:1, thinsolid lines with circles: 3:3. Colors of different models: black lines:
HS1 ; red lines: N ; cyanlines: Nli ; blue lines:
NXl . Green lines: g CC ( r ) (shifted along the y -axis). r CC [Å] P [ % ] (a) (b)(c) (d) Figure 4: Representation of orientational correlations by the distance-dependent probabilitiesof the classes of Rey, as calculated from the various model structures. (a) solid lines 2:2; (b)solid lines 1:3; (c) solid lines: 1:2, thin solid lines with circles: 2:3; (d) solid lines: 1:1, thinsolid lines with circles: 3:3. Colors of different models: black lines:
HS1 ; red lines: NX ; cyanlines: NXlli ; blue lines:
NXl . Green lines: g CC ( r ) (shifted along the y -axis). able 1: Parameters of the Reverse Monte Carlo simulations performed. For combined runsthat apply both neutron and x-ray diffraction data, the Q-ranges and R-factors are also shown. Run Experimental data Q-range [Å − ] R wp [%] Saved cfgHS0 none — — 195HS1 none — — 184N[5] neutron 0.39-8.0 4.84 50Nli neutron 0.39-8.0 5.12 1NXsh neutron, x-ray 0.39-8.0; 0.65-8.0 4.82; 4.89 68NXl neutron, x-ray 0.39-8.0; 0.65-14.0 4.84; 5.12 84NXlli neutron, x-ray 0.39-8.0; 0.65-14.0 5.29; 5.44 1 Table 2: Contribution of each partial structure factor to the experimental total scatteringstructure factors.
Experiment