Ordered array of ω particles in β -Ti matrix studied by small-angle X-ray scattering
Jana Šmilauerová, Petr Harcuba, Josef Stráský, Jitka Stráská, Miloš Janeček, Jiří Pospíšil, Radomír Kužel, Tereza Brunátová, Václav Holý, Jan Ilavský
OOrdered array of ω particles in β -Ti matrix studied bysmall-angle x-ray scattering J. ˇSmilauerov´a ∗ , P. Harcuba, J. Str´ask´y, J. Str´ask´a, M. Janeˇcek, J.Posp´ıˇsil, R. Kuˇzel, T. Brun´atov´a, V. Hol´y Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic
J. Ilavsk´y
Argonne Nat. Laboratory, Argonne Ill., USA
Abstract
Nano-sized particles of ω phase in a β -Ti alloy were investigated by small-angle x-ray scattering using synchrotron radiation. We demonstrated thatthe particles are spontaneously weakly ordered in a three-dimensional cubicarray along the (cid:104) (cid:105) -directions in the β -Ti matrix. The small-angle scatter-ing data fit well to a three-dimensional short-range-order model; from the fitwe determined the evolution of the mean particle size and mean distance be-tween particles during ageing. The self-ordering of the particles is explainedby elastic interaction between the particles, since the relative positions of theparticles coincide with local minima of the interaction energy. We performednumerical Monte-Carlo simulation of the particle ordering and we obtaineda good agreement with the experimental data. Keywords: ω -Ti phase, Ti alloys, small-angle x-ray scattering, self-ordering ∗ Corresponding author
Email addresses: [email protected] (J. ˇSmilauerov´a),
[email protected] (P. Harcuba), [email protected] (J. Str´ask´y), [email protected] (J. Str´ask´a), [email protected] (M. Janeˇcek), [email protected] (J. Posp´ıˇsil), [email protected] (R. Kuˇzel), [email protected] (T. Brun´atov´a), [email protected] (V. Hol´y), [email protected] (J. Ilavsk´y)
Preprint submitted to Acta Materialia November 9, 2018 a r X i v : . [ c ond - m a t . m t r l - s c i ] M a r . Introduction Metastable β -Ti alloys are increasingly used in aerospace and automo-tive industry mainly due to excellent corrosion resistance and high specificstrength. The high strength is achieved through ageing treatment involvingseveral phase transformations [1]. Therefore, investigation of these phasetransformations is of significant importance.Above 883 ◦ C, pure titanium crystallizes in a body-centered cubic struc-ture ( β phase). When cooled below this temperature ( β -transus) it marten-sitically transforms to a hexagonal close-packed structure ( α phase). Metastable β -Ti alloys contain a sufficient amount of β -stabilizing elements (Mo, V, Nb,Fe) so that the martensitic β → α transformation is suppressed and the β phase is retained after quenching to room temperature [2].Several metastable phases can emerge during ageing of these alloys de-pending on the content of β -stabilizing elements. The present study focuseson hexagonal ω phase. Tiny and uniformly distributed particles of the ω phase serve as precursors for a subsequent precipitation of the α -phase par-ticles that are responsible for significant strengthening.The ω phase is formed upon quenching by a diffusionless displacive trans-formation as first proposed by Hatt et al. [3] and lucidly described byde Fontaine [4]. The transformation can be described as a collapse of twoneighbouring (111) β planes into one plane. More formally, these two planesare displaced by ± [111] along the body diagonal of the cubic unit cell. One(111) β plane between two pairs of collapsed planes remains unchanged. Thisproduces a hexagonal structure with two differently populated alternating’basal’ planes. Such hexagonal structure is coherent with the parent β phase[5]. It was shown experimentally that this displacive β → ω transformationis completely reversible at low temperatures at which diffusion does not playa role [6]. The ω phase forms fine, a few nanometers large particles uniformlydispersed throughout the β matrix. Due to its formation mechanism, the ω phase can exist only in certain orientations with respect to the β matrix.The topotactical relationship between the β and ω lattices can be describedas [7] (0001) ω (cid:107) (111) β , [11¯20] ω (cid:107) [011] β . (1)The particles of the ω phase further evolve and grow during ageing througha diffusion controlled reaction [8]. This process is irreversible and is accom-panied by rejection of β stabilizing elements from the ω phase.2t has been observed that the shape of the ω particles can be eitherellipsoidal or cuboidal. According to Blackburn et al. [9], the shape is relatedto the lattice misfit strain. They suggested that the ellipsoidal shape of the ω precipitates arises from an anisotropy of the strain energy of the precipitaterather than the matrix, whereas the cuboidal shape is determined by thestrain energy in the matrix.Despite countless studies describing the ω phase, there is still not muchknown about the causes of formation of the ω -phase particles and their spatialordering. Some argue that the ω particles formation follows from a spinodalchemical separation of the β phase [10, 11]. This conclusion is based onobservations of the spatial ordering and/or chemical inhomogeneity of theparticles and the host material. Others suggest that the formation of theseparticles can be attributed to elastic instabilities of the parent β matrix [12].In the last decades, kinetics of the phase separation in alloys has beenintensively studied both theoretically and by various experimental methods.For the theoretical description of the phase separation both macroscopic andmicroscopic approaches have been published, the former describes the phasesas elastic continua divided by ideally sharp interfaces, the latter takes intoaccount movement of individual atoms. The description is based on classi-cal works of Cahn and Hilliard [13, 14] and Lifshitz, Slyozov and Wagner[15, 16] (LSW theory) and includes various processes denoted spinodal de-composition, coarsening or Ostwald ripening (see also the review in [17]).From numerous numerical simulations and experimental data it follows thatthe time-dependent structure function S ( Q, t ) = (cid:42)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d r c ( r )e − i Q . r (cid:12)(cid:12)(cid:12)(cid:12) (cid:43) , (2)i.e., the square of the modulus of the Fourier transformation of the concen-tration function c ( r ) of a given phase averaged over a statistical ensemble,obeys a universal scaling law [18] S ( Q, t ) ∼ Q max ( t ) − F ( Q/Q max ( t )) , (3)where Q max ( t ) is the position of the maximum of the structure function S ( Q, t ) and F is a time-independent universal function. From the time-dependence of Q max we can deduce the evolution of the characteristic length L ( t ) ∼ π/Q max ( t ) during the coarsening process; from the LSW theory theasymptotic behavior Q max ( t ) − ∼ A + Bt follows.3 phase particles in metastable β -Ti alloys are nanometers in size andafter ageing have slightly different electron density than the parent β ma-trix (i.e. the x-ray indexes of refraction of the particles and the matrix aredifferent). Small-angle x-ray scattering (SAXS) is an ideal technique to de-termine the structure function, since the reciprocal-space distribution of thescattered intensity is proportional to the structure function multiplied bythe square of the difference in the electron densities of the β and ω phases.SAXS is a nondestructive technique based on elastic scattering from electrondensity inhomogeneities within the sample. The SAXS instrument recordsscattered intensity at small scattering angles, i.e., close to the direction of theincident beam. Obtained SAXS data contain information about importantmicrostructural parameters such as size, shape, volume and space correla-tions of the scatterers [19].To our knowledge, only a limited number of experiments employing SAXSon ω particles in titanium alloys has been performed and published. Fratzlet al. [20] investigated the growth of ω -phase particles in single crystals ofTi–20 at.%Mo. They found that the radii of the ω particles increased withageing time as ∼ t / and then stabilized at the value of approximately 75 ˚A.The same group of authors later investigated ω and α phase precipitationin Ti–12 at.%Mo single crystals by the means of SAXS [21]. In their work,the authors determined the shape of the ω particles and then observed thenucleation and coarsening of α plates which destroyed the ω structure.In our previous paper [22] we studied the structure of the ω particles ina single-crystalline β matrix by x-ray diffraction (XRD). We confirmed thevalidity of the topotactical relations in Eq. (1) and found that the β lat-tice is locally compressed around the particles. In this paper we perform asystematic SAXS study of the evolution of sizes of the ω particles in single-crystalline β -Ti alloy samples during ageing. We demonstrate that the ω particles are self-ordered and create a three-dimensional cubic array alongthe crystallographic axes (cid:104) (cid:105) β of the β matrix and that the mean distancebetween the particles is proportional to their mean size. We explain theordering mechanism by considering the energy of the elastic interaction be-tween the particles caused by the local deformation of the lattice. Finally,we compare the measured data with the scaling behavior in Eq. (3).The paper is written as follows. The next section contains a brief descrip-tion of the growth of the Ti-alloy single crystals and the SAXS experiments.Section 3 contains a phenomenological model of the arrays of particles thatmakes it possible to fit the experimental data and to determine basic struc-4ural parameters. The results of the SRO model are also presented in Sec-tion 3. The driving force of the self-ordering process is discussed in Section4. The Section 5 provides a discussion of the obtained results.
2. Experiments
Single crystals of one of the metastable β titanium alloys, TIMETALLCB were grown in a commercial optical floating zone furnace (model FZ-T-4000-VPM-PC, Crystal Systems Corp., Japan) with four 1000 W halogenlamps. The growth process was carried out in a protective Ar atmospherewith Ar flow of 0.25 l/min and pressure of 2.5 bar. The growth speed was10 mm/h. The grown ingots had roughly circular cross section with diameterin the range of 8 – 10 mm. The length of the single crystals was typicallyaround 9 cm. The details of the single crystal growth process and charac-terization of the resulting ingot can be found elsewhere [23]. During singlecrystal growth, a shift in chemical composition of the material is possible.In order to assess this change, concentrations of individual elements weredetermined both for precursor material and grown ingot. Two experimen-tal methods were used. Concentrations of the main alloying elements (Ti,Mo, Fe, Al) were determined by energy dispersive x-ray spectroscopy (EDX)using scanning electron microscope FEI Quanta 200FEG. Because titaniumalloys are prone to contamination by interstitial N and O, the concentrationsof these elements were checked by an automatic analyzer LECO TC 500C.The chemical composition of the precursor and resulting single crystal aresummarized in Tab. 1. sample Ti Mo Fe Al N Oprecursor 88 . ± . . ± . . ± . . ± . . ± .
004 0 . ± . . ± . . ± . . ± . . ± . . ± .
03 0 . ± . Table 1: Chemical composition of the precursor material and the resulting single crystalin at. %
Each crystal was solution treated at 860 ◦ C for 4 h in an evacuated quartztube and water quenched in order to homogenize the structure and ensurethe retention of β phase. Subsequently, the crystals were cut into 1.2 mmthick slices perpendicular to the length of the crystal. The slices were thenaged in salt bath at three temperatures 300 ◦ C, 335 ◦ C, and 370 ◦ C for 2, 4,8, 16, 32, 64, 128 and 256 h. At these ageing temperatures, the ω phase5articles grow, but at least at the lowest temperatures in the series, theprecipitation of the α -Ti phase is not expected [24, 25]. The samples werethen ground and polished from both sides utilizing 500, 800, 1200, 2400 and4000 grit SiC papers. Final polishing was carried out on a vibratory polisherusing 0.3 µ m and 0.05 µ m aqueous alumina (Al O ) suspensions and 0.05 µ mcolloidal silica. The final thickness of the samples for SAXS measurementwas approximately 200 µ m. The crystallographic orientation of the slices wasdetermined by standard Laue diffraction with the accuracy better than 1 deg.Small-angle x-ray scattering (SAXS) experiments have been carried outat the beamline 15-ID at APS, Argonne National Laboratory (USA). Weused the photon energy of 25 keV, the width of the primary x-ray beamwas set to 150 × µ m. The scattered beam was detected by a large two-dimensional detector with 2048 × β , [110] β and [111] β of the primary beam with respectto the β -Ti lattice. The SAXS data were calibrated by a standard procedure[19], using Ag-behenate and glassy carbon samples for angular and intensitycalibrations, respectively. After the calibration, the scattered intensities wereexpressed as photon fluxes scattered from unit sample volume into unit solidangle of one sr normalized to unit flux density of the primary radiation andcorrected to the sample absorption.In figure 1 we plotted the SAXS intensity maps of the samples after var-ious ageing times (ageing temperature 300 ◦ C) measured in the orientation[001] β of the primary x-ray beam. The arrow denotes the orientation of the[100] β direction in the host lattice determined by independent x-ray diffrac-tion (the Laue method). The maps clearly exhibit side maxima in directions[100] β , [010] β indicating that the particles are arranged in a disordered three-dimensional cubic array with the axes along (cid:104) (cid:105) β . With increasing ageingtime the side maxima move closer to the origin and they become stronger andnarrower; this development can be explained by increasing mean distance L of the neighboring particles and increasing particle size. Figure 2 shows theSAXS maps of the sample after 300 ◦ C/256 h ageing measured for the threecrystallographic orientations of the primary beam; the positions of the sidemaxima again support the hypothesis of the ordering of the particles along (cid:104) (cid:105) β axes. 6 z ( / n m ) Q y (1/nm) Q z ( / n m ) Q y (1/nm)128−2 −1 0 1 2 [ ] Q y (1/nm)256−2 −1 0 1 2 Figure 1: The SAXS maps of samples after ageing at 300 ◦ C; the time of ageing in hours isindicated in the upper left corners of the panels. The measurements have been carried outin the [001] β -orientation of the primary x-ray beam. The line scans plotted in Figure 5were extracted from these maps along the dashed lines. Q y (1/nm) Q z ( / n m ) (001) [100][010]−2 −1 0 1 2−2−1012 Q y (1/nm)(110) [001][1−10]−2 −1 0 1 2 Q y (1/nm)(111) [1−10][11−2]−2 −1 0 1 2 Figure 2: The SAXS maps of the sample after 300 ◦ C/256 h ageing measured in threeorientations of the primary x-ray beam. The arrows denote the crystallographic directionsin the β host lattice determined by the Laue method. z ( / n m ) Q y (1/nm) Q z ( / n m ) [ ] Q y (1/nm)32−2 −1 0 1 2 Figure 3: The same situation as in Figure 1, ageing temperature 370 ◦ C. The SAXS intensity maps of samples aged at 335 ◦ C are quite similar toFig. 1, therefore, we do not present them here. In Fig. 3 we show the mapsof the samples aged at 370 ◦ C, since their appearance obviously differs fromfigure 1.Theoretical description of the small-angle x-ray scattering from an or-dered three-dimensional array of particles along with numerical simulationand fitting to the experimental data will be described in the next section.
3. Short-range-order model of the ordering of particles
In this section we present a model for the simulation and fitting of theSAXS data. The model is purely phenomenological, but its simplicity makesit possible to fit numerically the measured data. A more physically substan-tiated simulation approach will be presented in Sect. 4.The signal measured in a small-angle-scattering experiment (SAXS) in agiven pixel of a two-dimensional detector J = I inc d σ dΩ ∆Ω (4)8s proportional to the intensity of the primary beam I inc , the solid angle ∆Ωdetermined by the angular aperture of the detector pixel and the differentialcross-section d σ/ dΩ of the scattering process.The differential scattering cross-section is simulated using a standard ap-proach [26] including the kinematical approximation (i.e. neglecting multiplescattering from the particles) and the far-field limit. The explicit formula forthe differential scattering cross-section reads:d σ dΩ = K π | ∆ n | e − µT (cid:42)(cid:88) n (cid:88) m Ω FT n ( Q )Ω FT ∗ m ( Q )e − i Q . ( r n − r m ) (cid:43) . (5)Here we denoted K = 2 π/λ , µ is the linear absorption coefficient, T is thesample thickness measured along the primary x-ray beam, and ∆ n is the dif-ference of the refraction indexes of the particle and the host material (causedby a slight difference in their chemical compositions). It is therefore assumedthat the refraction index (which is proportional to the electron density) ishomogeneous within the particle and also the electron density of the hostmaterial is assumed homogeneous. The electron density is determined bychemical composition (i.e. concentration of impurity atoms Mo, Fe and Al),as well as by the specific volume per one atom, which is affected by elastic de-formation of the lattice and/or by the presence of structure defects. Ω FT n ( Q )is the Fourier transformation of the shape function of the n -th particle:Ω FT n ( Q ) = (cid:90) d r Ω n ( r )e − i Q . r ;the shape function Ω n ( r ) is unity inside and zero outside the particle. Fur-ther, r n in Eq. (5) is the position vector of the n -th particle, the double sum (cid:80) n (cid:80) m runs over all particles in the irradiated sample volume, and the av-eraging (cid:104) (cid:105) is performed over random positions and sizes of the particles. Thescattering vector Q = K f − K i is considered in vacuum, since the refractioncorrection is unimportant in our transmission geometry and the absorptioneffect is included in the absorption term exp( − µT ) in Eq. (5). It is worthy tonote that the measured signal in Eqs. (4,5) is proportional to the structurefunction defined in Eq. 2: J = I inc | ∆ n | K π Q F (cid:18) Q Q max (cid:19) ∆Ω , (6)where F is the universal scaling function from Eq. (3).9 ° C ° C ° C ° C ° C ° C Q / Q max I / I m a x (a) 1 2 3 4 ° C ° C ° C ° C ° C Q / Q max (b) 1 2 3 4 ° C ° C ° C ° C ° C Q / Q max (c) Figure 4: Line scans across the satellite maxima measured on samples after ageing at300 ◦ C (a), 335 ◦ C (b), and 370 ◦ C (c). The scans are normalized to the same height andposition of the first satellite maximum. The red arrow in panel (c) denotes the position ofthe secondary satellite maximum.
In the literature, several models can be found describing a possible cor-relation of the particle sizes with their positions [26]. In the following weassume the local-monodisperse approximation (LMA). Using this approach,the irradiated sample volume consists of many domains, one domain containsparticles of a given size and given mean distance between nearest particles.The differential scattering cross-section is thend σ dΩ = K π | ∆ n | (cid:68)(cid:12)(cid:12) Ω FT R ( Q ) (cid:12)(cid:12) G R ( Q ) (cid:69) sizes , (7)here we have denoted G R ( Q ) = (cid:42)(cid:88) n (cid:88) m e − i Q . ( r n − r m ) (cid:43) positions (8)the correlation function of the positions of the particles with a given radius R . In the following we omit the subscript R for simplicity.Since the position of a particle is only affected by the positions of theparticles in few nearest coordination shells, the ordering of the particles canbe described by a short-range order model (SRO). In this model we assumethat the distances of a given particle from their neighbors are random witha given statistical distribution. As the ω particles create a cubic array, thethree-dimensional correlation function G ( Q ) can be expressed as a directproduct of three one-dimensional correlation functions as shown by Eads10t al. [27]. The one-dimensional correlation function G (1) ( Q ) can be calcu-lated directly [27, 28]: G (1) ( Q ) = N (cid:26) (cid:20) ξ ( Q )1 − ξ ( Q ) (cid:18) N − ( ξ ( Q )) N − ξ ( Q ) − (cid:19)(cid:21)(cid:27) . (9)Here we denoted N the number of coherently irradiated particles in onedimension and ξ ( Q ) = (cid:10) e − i Q . L (cid:11) , where L is the random vector connecting the actual centers of neighboringparticles lying in the same one-dimensional chain. In the following we assumethat N is very large and we use the limiting expression for G (1) ( Q ): G (1) ( Q ) → N (cid:20) (cid:18) ξ ( Q )1 − ξ ( Q ) (cid:19)(cid:21) . The three-dimensional correlation function G ( Q ) can be expressed as aproduct of three one-dimensional correlation functions along the orthogonalaxes parallel to the axes of the cubic array of particles. Since the coordinate Q of the intensity line scans in Fig. 4 is parallel to the array axis and theother two coordinates along the scans are approximately zero (we neglectthe curvature of the Ewald sphere), the correlation function used for thesimulation of the line scans is G ( Q, ,
0) = N (cid:107) N ⊥ (cid:18) σ L L (cid:19) (cid:20) (cid:18) ξ ( Q )1 − ξ ( Q ) (cid:19)(cid:21) , (10)where we used the limiting value N ( σ L /L ) of the one-dimensional correla-tion function for the zero argument. N (cid:107) and N ⊥ denote the numbers of thecoherently irradiated particles in the directions parallel and perpendicular tothe primary beam, respectively. L is the mean inter-particle distance and σ L is its root-mean square (rms) deviation. The total number of the coherentlyirradiated particles can be expressed by the irradiated sample volume V : N (cid:107) N ⊥ = VL . In the simulation we assume that the particles are spherical, and calcu-lating the function ξ ( Q ), we apply the condition that neighboring particles11ust not intersect. Therefore, in the averaging over all possible L ’s we ex-cluded the case | L | < R , where R is a random particle radius (assumedfixed in the averaging over L ’s). Therefore, we assumed a truncated normaldistribution of the random vectors L with the mean value L = (cid:104)| L |(cid:105) and rmsdeviation σ L . Further, we assumed the Gamma distribution of the particleradii R with the mean value R and the rms dispersion σ R ; for each radiuswe considered the correlation function calculated by Eq. (9) assuming themean particle distance L proportional to the actual value of R : L = ζR .The averaging in Eq. (7) is then performed numerically by integrating over R -values, keeping ζ constant. It can be proved by a direct calculation thatthe short-range order model presented here obeys the scaling law in Eq. (3),if the mean values R and L are proportional ( L = ζR ) and the relative rms deviations are constant, i.e. σ L ∼ L and σ R ∼ R .We have checked the validity of the scaling law in Eq. (3) by rescaling theline scan to the same position and height of the satellite maxima, the resultsare displayed in figure 4. From the figure it follows that at the two lowerageing temperatures the line scans are scaled according to Eq. (3) (panelsa and b). The large spread of intensity values for higher Q in Fig. 4 is dueto high noise due to background subtraction. However, obvious deviationsfrom the scaling behavior can be observed for the highest ageing temperatureof 370 ◦ C, see Fig. 4(c). In particular, the line scan of the sample after 2 hageing exhibits secondary side maximum (denoted by arrow), which indicatesa better ordering of the particle positions and/or smaller rms deviation ofthe particle sizes. We do not observe this secondary maximum for longerageing times for the other temperatures. The secondary maximum graduallydisappears during annealing.From the SAXS maps we extracted line scans along the [100] β direction(dashed lines in figures 1 and 3) and fitted them with SRO model. Formulasin Eqs. (7, 9, 10) yield absolute flux densities therefore from the fit wewere able to determine the contrast | ∆ n | in the refraction indexes of theparticle material and the host phase, appearing in the multiplicative pre-factor. Nevertheless, in order to determine both | ∆ n | and the mean particlesizes, we had to assume that the mean inter-particle distance L and themean radius R are proportional, i.e., L = ζR , and the proportional factor ζ is the same in all samples in the same ageing series.Figure 5 compares the measured (grey points) and fitted (lines) line scans.It is obvious that the agreement of the theory with experimental data is quitegood. Figure 6 shows the measured and fitted line scans of the sample after12 Q (1/nm) i n t e n s it y ( a r b . un it s ) ° C (a) −2 −1 0 1 2 Q (1/nm) ° C (b) −2 −1 0 1 2 Q (1/nm) ° C (c)
Figure 5: The line scans extracted from the SAXS maps (grey dots) for samples aged at300 ◦ C (a), 335 ◦ C (b) and 370 ◦ C (c) and their fits by the SRO model (lines). The numbersdenote the ageing time, the scans are shifted vertically for clarity. ◦ C/8 h ageing in more detail; we plotted by dotted and dashed lines thecontributions of the particle shape (function (cid:104) (cid:12)(cid:12) Ω FT ( Q ) (cid:12)(cid:12) (cid:105) ) and the correlationfunction G ( Q ) of the particle positions, respectively. From the figure it isobvious that the shape factor slightly shifts the side maxima at Q max towardssmaller | Q | so that it would be misleading to determine the mean particledistance L just from the formula L = 2 π/Q max .Parameters of the particle ordering determined from the fits of the linescans are summarized in Fig. 7. In panel (a) we plotted the time-dependenceof the mean particle radius R determined from the SAXS data. In this panel,we compare these radii with the particle radii determined by XRD using themethod described in our previous paper [22]. For the sample series aged at300 ◦ C and 335 ◦ C both radii coincide within the error limits and their time-dependence roughly agrees with the prediction of the LSW theory [ R ( t )] ∼ A + Bt ( A and B are constants). The third series aged at 370 ◦ C behavesin a different way. With increasing ageing time the particle radii determinedfrom XRD decrease, however, the error bars of these radii are larger than forthe other ageing temperatures. In order to obtain a reasonably good fit ofthe data from the third sample series, we had to fit the parameters L and R independently , not considering the proportionality factor ζ . This fact makesthe fitting results less reliable than for the other two series, however, it isobvious that the particles sizes determined from SAXS are larger than thosefrom XRD. Figure 7(b) displays the time dependence of the mean particledistance L determined from SAXS. Again, for sample series at 300 ◦ C and335 ◦ C L increases with the ageing time and follows the polynomial formula13 Q (1/nm) i n t e n s it y ( a r b . un it s ) measuredfittedshapecorrelation Figure 6: The line scan extracted from the SAXS map of sample after 300 ◦ C/8 h age-ing (points) and its fit by the SRO model (full line). The contributions of the particleshape and correlation to the simulated line scan are displayed by dotted and dashed lines,respectively. [ L ( t )] ∼ A (cid:48) + B (cid:48) t following from the LSW theory. For the highest ageingtemperature, no distinct evolution of the L values during ageing can beestablished.In figure 7(c) we have plotted the time dependence of the relative rmsdeviation σ R /R of the particle radii. In the first two sample series aged at300 ◦ C and 335 ◦ C the relative rms deviation does not change significantlyduring ageing, while at the highest ageing temperature of 370 ◦ C we observea distinct increase of this value, i.e. the width of the size distribution of theparticles increases during ageing. The relevance of this result is somewhatlimited by the fact that the fit of the SAXS data of the last sample series isless reliable than for the other two temperatures (see the discussion in Sect.5), however the qualitative tendency is obvious.Figure 7(d) shows the time dependence of the relative rms deviation σ L /L of the inter-particle distances. At 300 ◦ C and 335 ◦ C these rms devia-tions remain nearly constant, while at 370 ◦ C they slightly decrease with theageing time, however the errors of these parameters are quite large. There-fore, the second-order maxima in the line scans depicted in Fig. 4(c) canbe ascribed to the form-factor of a single particle and not to the correla-tion function of the particle positions. Finally, in Fig. 7(e) we demonstratethat the mean inter-particle distance L scales linearly with the mean par-ticle radius R determined from SAXS, obeying the approximative formula14 R ( n m ) ° C335 ° C370 ° C/XRD370 ° C/SAXS (a) 0510152025 L ( n m ) ° C335 ° C370 ° C (b)10 t (h) σ R / R ° C335 ° C 370 ° C (c) 10 t (h) σ L / L ° C 335 ° C370 ° C (d)0 5 1001020 R (nm) L ( n m ) ° C 335 ° C370 ° C/XRD 370 ° C/SAXS (e) . × R Figure 7: Parameters of the particles determined from the fit of the line scans to the short-range order model. See the text for a detailed description. The full lines in panels (a) and(b) are the graphs of fitted functions ( A + Bt ) / ( A and B are suitable constants), followingfrom the LSW theory. The straight line in (e) represents the dependence L = 2 . R . L ≈ . × R .Finally, from the fits we determined the contrast of the refraction index | ∆ n | . The difference of the refraction indexes of the particle material andthe matrix is proportional to the difference ∆ ρ el in the electron densities:∆ n = − λ r el π ∆ ρ el , where λ is the x-ray wavelength, r el ≈ .
818 ˚A − is the classical electronradius, and we neglected the dispersion corrections. In Fig. 8 we plottedthe contrast of the electron densities relatively to the electron density of the15 t (h) | ∆ ρ e l / ρ e l | ° C335 ° C 370 ° C Figure 8: The difference of the electron density ∆ ρ el of the ω particles and the β -Ti matrixrelatively to the electron density of the matrix ρ el vs ageing time. nominal Ti alloy (according to Tab. 1) as functions of the ageing time. Inall sample series, the contrast | ∆ ρ el | increases with ageing time. During theageing at 300 ◦ C and 335 ◦ C, the contrast values are smaller or around 10%of the nominal value, these changes can be explained by the changes in thechemical composition of the particles by several at. % of Mo, Fe, and/orAl. Of course, one single value of | ∆ ρ el | for a given sample does not allow todetermine complete chemical composition of the particles. For the highestageing temperature of 370 ◦ C the contrast values following from the fit aremuch larger and do not correspond to any physically relevant value. Thisresult will be discussed in Sect. 5.
4. Driving force of the ordering
In the previous section we demonstrated that the ordering of the particlesagrees well with the short-range order model. In this section we show thatthe driving force of the ordering can be attributed to the minimization of theelastic interaction energy of the particles.We have verified in our previous paper [22] that the crystal lattice arounda particle is elastically deformed, the reason of the deformation is a differencebetween the actual lattice parameters a ω , c ω of the ω lattice of the particleand their ideal values a (id) ω , c (id) ω following from the topotaxy relation of the β and ω lattices [7, 20]. Most likely, this lattice mismatch is caused by adifference in the chemical composition; during the formation and growth of16 particles the β -stabilizing impurities (Mo and Fe in our case) are expelledfrom the particle. Since the β -Ti matrix is highly elastically anisotropic, thelocal deformation field around a particle is anisotropic, too. The interactionenergy of a particle pair is given by the formula [29, 30, 31] E int = − (cid:90) Ω (B) d r σ (A) jk ( r ) (cid:15) (B)0 jk ( r ) , j, k = x, y, z. (11)The integral in this formula is calculated over the volume Ω (B) of particleB, ˆ σ (A) ( r ) is the stress tensor in the matrix in the points belonging to Ω (B) ,caused by another particle A, and ˆ (cid:15) (B)0 is the mismatch of the lattice of particleB with respect to the host lattice. Using the mismatch values f a = ( a ω − a (id) ω ) /a (id) ω , f c = ( c ω − c (id) ω ) /c (id) ω defined in our previous paper and using the coordinate axes across and alongthe c -axis [0001] ω of the hexagonal ω lattice, the matrix ˆ (cid:15) (B)0 has the formˆ (cid:15) (B)0 = f a f a
00 0 f c . (12)The stress tensor ˆ σ (A) ( r ) caused by the particle A was calculated taking intoaccount the elastic anisotropy of the host lattice and the mismatch matrixˆ (cid:15) (A)0 analogous to that in Eq. (12) using the continuum elasticity approachbriefly described in the Appendix of our previous paper [22].Assuming the typical mismatch values f a = 0 .
002 and f c = 0 .
01 found in[22] and the particle radius R = 3 nm we calculated the dependence of theinteraction energy on the relative position r of particles (Fig. 9) in the (1¯10) β plane in the cubic β -Ti lattice. In this figure, the center of one particle is inthe graph origin; since Eq. (11) is valid only for non-intersecting particles,the excluded region | r | ≤ R is shaded (the grey area). The simulationswere performed for all 16 combinations of the orientations of the hexagonal c -axes [0001] ω of particles A and B with respect to the cubic β lattice, theinteraction energy plotted in this figure is averaged over all orientations.The figure clearly indicates that minima of the interaction energy occurin six equivalent directions (cid:104) (cid:105) β from the particle center in the distanceof about 2 . × R . On the other hand, maxima of the interaction energy oc-cur along eight equivalent directions (cid:104) (cid:105) β . Numerical simulations demon-strated that the anisotropy in the distribution of E int is determined entirely17 /R → [110] β z / R → [ ] β [ ] β positivenegative ~ 2.2 × R −4 −2 0 2 4−4−2024 Figure 9: Dependence of the interaction energy of a particle pair on the relative positionof the particles. The simulation was performed for spherical particles with the radiusof 3.6 nm and with the mismatch values f a = 0 . , f c = 0 .
01, taking into account allpossible orientations of the hexagonal [0001] ω -axes in both particles. The grey area denotesthe region, where the particles intersect. The contour step is 0.1 eV. by the elastic anisotropy of the host lattice and it is only very slightly affectedby the anisotropy of the mismatch according to Eq. (12).The (cid:104) (cid:105) β directions in which the minima of E int occur agree with theorientations of the basis vectors of the disordered array of particles deter-mined from the SAXS data in the previous section. Therefore the anisotropyin the distribution of interaction energy indicates that the interaction en-ergy plays a role in the self-ordering mechanism of the particles. In orderto support this hypothesis we performed a simple Monte-Carlo (MC) sim-ulation of the distribution of particles. MC simulations are widely used inthe simulation of x-ray diffuse scattering and small-angle scattering. OurMC simulation program is similar to the MC simulation program for small-angle neutron scattering (SANS) [32], however, it takes into account elasticinteraction between the particles.The simulation procedure consists of the following steps:1. we determine randomly the particle radius R using a random numbergenerator, assuming the Gamma distribution of the radii with the meanvalue R and order m R ,2. we choose randomly the position of the first particle in the simulationcube D × D × D ,3. we choose randomly the position of a next particle and one of four18ossible orientations of its hexagonal [0001] ω -axis,4. we calculate the total interaction energy E int of this particle with otherparticles seated in the previous steps,5. we generate a random number p ∈ [0 ,
1] and we settle the particle inthe position chosen in the previous step if p < K exp[ − E int / ( k B T )],6. we repeat items 3-5 N times, where N is the number of attempts toplace a particle,7. we repeat items 1-6 M times, where M is the number of simulationcubes,8. we calculate the scattered intensity using the formula J ( Q ) = const . M (cid:88) k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω FT R k ( Q ) N k (cid:88) n =1 e − i Q . r ( k ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (13)where r ( k ) n are the particle position vectors generated in items 1-5, and N k ≤ N is the actual number of settled particle for given k -th simula-tion cube.Therefore, the simulation procedure has the following parameters: R isthe mean radius of the particles, m R = ( R /σ R ) is the order of the Gammadistribution of the radii. D is the size of the simulation domain and itis comparable to the coherence width and/or length of the primary x-raybeam; we took D = 50 nm. The constant K was chosen so that the values K exp[ − E int / ( k B T )] lie between 0 and 1, i.e. K ≈ exp[min( E int ) / ( k B T )]and we found that the simulation results do not depend much on K . Thesimulation temperature T is not directly connected to the ageing temperatureand we choose the value of T to obtain the best match of the simulationresults to the experimental data, namely k B T = 0 . N of thetrials to set the particle positions was chosen much larger than the expectednumber of the particles in the D × D × D cube; we used N = 10 . The number M of the simulation cubes is determined by the ratio of the total irradiatedsample volume to the coherently irradiated volume. This ratio is roughly10 for our experimental conditions, however we used M = 10 to keep thecalculation time in reasonable limits. The MC simulation procedure is onlyqualitative, since it describes properly neither the microscopic mechanism ofthe β → ω transition, nor the growth of nucleated ω particles.Figure 10 shows the examples of the simulated SAXS maps in planes(001) , (110) and (111) perpendicular to the primary x-ray beam. In the19 y (1/nm) Q z ( / n m ) (001) [100][010]−2 −1 0 1 2−2−1012 Q y (1/nm)(110) [011][1−10]−2 −1 0 1 2 Q y (1/nm)(111) [1−10][11−2]−2 −1 0 1 2 Figure 10: SAXS maps simulated in three reciprocal planes (001) , (110) and (111) by theMonte-Carlo method described in text. simulations we took R = 3 . m R = 20. The maps exhibit distinctside maxima, the positions of which very well coincide with the maxima inthe measured maps in Fig. 2. The distance of the simulated maxima from theorigin Q = 0 is inversely proportional to the mean radius R of the particles;from the simulation we found that the position Q max of the maximum at the[100] axis obeys the formula2 πQ max = L = ζR ; ζ = 2 . ± . . (14)The factor ζ ≈ . ζ = 2 . ± . ◦ C/256 h ageing in figure 1 withthe MC simulation performed for R = 3 . m R = 20; the simulatedintensities were multiplied by a suitable constant to obtain the same heightsof the side maxima. The shapes of the intensity distributions coincide well.20 Q (1/nm) i n t e n s it y ( a r b . un it s ) measuredMC simulation Figure 11: Comparison of the line scan extracted from the measured SAXS map in Figure 1of sample after 300 ◦ C/256 h ageing taken in direction [001] β (points) with the result ofthe Monte-Carlo simulations performed for R = 3 . m R = 20 (line).
5. Discussion
The SAXS data were compared with simulations based on a phenomeno-logical SRO model and we found a reasonably good agreement (see figure 5).From the fit we determined the mean particle radius R and inter-particledistance L and their dependence on the ageing time t . In samples aged at300 ◦ C and 335 ◦ C the mean particle radii determined from SAXS and XRDcoincide within the error limits. Furthermore, in agreement with the LSWmodel, the radius R and the distance L increase roughly as t / , i.e. thetotal number N of particles decreases as 1 /t in these samples. The samescaling laws were also demonstrated from the XRD data in our previous pa-per, so that both XRD and SAXS data are consistent and they confirm thevalidity of the LSW model for the ageing temperatures 300 ◦ C and 335 ◦ C.The samples aged at the highest temperature of 370 ◦ C behave differ-ently, namely, the mean particle distance L and the mean radius R deter-mined from SAXS remained nearly constant during ageing, while the XRD-determined particle sizes are much smaller. The main reason might be that370 ◦ C is temperature sufficient for α phase particles precipitation. It is well-known that the α particles have the form of platelets [20, 21] parallel tothe (0001) α basal planes perpendicular to (cid:104) (cid:105) β directions. In SAXS, suchplatelets give rise to intensity streaks along (cid:104) (cid:105) β ; these streaks should bevisible in the SAXS intensity maps in the orientation (110) β . In Fig. 1221 y (1/nm) Q z ( / n m ) ° C [001][1−10]−2 −1 0 1 2−2−1012 Q y (1/nm)335 ° C [001][1−10]−2 −1 0 1 2 Q y (1/nm)370 ° C [001][1−10] [1−11] −2 −1 0 1 2
Figure 12: The SAXS intensity maps of the last samples of all three ageing series measuredin the (110) β plane. The [1¯11] β -oriented streak is clearly visible in the map of the sampleaged at the highest temperature. we compare the SAXS maps of the last samples of all ageing series 300 ◦ C,335 ◦ C, and 370 ◦ C. A [1¯11] β -oriented streak is clearly visible indeed only inthe map of the sample aged at the highest temperature of 370 ◦ C. Full de-scription of these streaks and the evaluation of size of α platelets are beyondthe scope of this paper.In the structure model used for the fitting of the SAXS data [Fig. 5(c)]we did not include the α platelets, which increase the scattered intensity forsmall Q ’s. Consequently, the parameters resulting from the fit of this dataseries are less reliable. This affects mainly the values ∆ ρ el of the contrastof the electron density in Fig. 8. The ∆ ρ el values of the 370 ◦ C series arestrongly overestimated, since the scattered intensity was ascribed only to the ω particles, and a part of the intensity stems also from the α platelets.At the highest temperature (370 ◦ C), the size of the ω particles seen byXRD is smaller than the size detected by SAXS. This temperature may behigh enough for the ω particles to grow quickly at the beginning of ageingand then start to dissolve at longer ageing times (or to transform to the α phase). As the ω structure disappears, XRD detects smaller size of the ω particles. On the other hand, SAXS detects inhomogeneities in the electrondensity (i.e. chemical composition), which may remain the same even afterthe ω phase dissolves. However, this hypothesis would need more thoroughinvestigation.A gradual change in the mean chemical composition of the ω particlesduring ageing at 300 ◦ C and 335 ◦ C is the reason for the slight increase of the∆ ρ el values in Fig. 8. The increase of the chemical contrast during ageing22ould be ascribed to a gradual ejection of the β -stabilizing elements (Mo andFe in our case) from the volumes of the ω particles during the ageing process.In the 370 ◦ C sample series, the ∆ ρ el values are much larger and they cannotbe explained by mere chemical changes. Most likely, aforementioned shellstructures are the reason for these values, however this effect requires furtherinvestigation.From the SAXS data it also follows that for ageing temperatures of 300 ◦ Cand 335 ◦ C the mean particle distance L is proportional to their mean radius R . This finding indicates that a particle-particle interaction is the reasonof the ordering. Nevertheless, the phenomenological SRO model used here cannot explain fully the SAXS data. In this model the position of a givenparticle is affected only by the positions of neighboring particles. On theother hand, the inter-particle interaction mediated by elastic deformation ofthe host lattice is long-ranged and the position of a given particle is thereforeaffected by more distant particles as well. The SRO model fails especiallybetween the central peak and lateral maxima, where the measured intensityexhibits a deeper dip than the simulated curve for ageing temperatures of300 ◦ C and 335 ◦ C. The shape of the intensity distribution in this region canbe affected by the asymmetry of the statistical distribution of the randomvectors L connecting neighboring particles. Another reason of the discrep-ancy between the measured and simulated data for small | Q | could be theabove-mentioned core-shell structure of the particles modifying the radialprofile of the refraction index.From the SAXS data shown above it clearly follows that the ω particlesare self-ordered in a three-dimensional cubic array with the axes along (cid:104) (cid:105) β directions. This finding differs from the conclusions in Ref. [33], wherethe authors claim that the particle ordering occurs along directions (cid:104) (cid:105) β .The authors support this statement by a transmission electron micrograph(TEM), where only few particles are depicted. The ordering along three (cid:104) (cid:105) β directions may in certain cases appear as (cid:104) (cid:105) β ordering in TEM, butthe statistical relevance of SAXS data is much higher, since the number ofirradiated particles in a typical SAXS experiment is several 10 , i.e. by manydecades larger than in TEM. The (cid:104) (cid:105) β -oriented ordering of particles canbe explained by the following simple argument. As we have shown above,the arrangement of the particles is close to a thermodynamic equilibrium,i.e. the particle positions correspond to the minima of the interaction energyof particles. As stated by Shneck et al. [31], the sign of the hydrostaticstress, i.e. the sign of the trace Tr(ˆ σ ) of the stress tensor, is decisive for the23rdering. Namely, if a particle compresses the surrounding lattice (which isthe case of our samples) and Tr(ˆ σ ) < (cid:104) (cid:105) β in our case.We performed a series of Monte-Carlo simulations explaining qualita-tively the ordering mechanism. The positions of the SAXS maxima and thelinear dependence of the mean particle distance on the size of the particlesfollowing from the simulations agrees well with the SAXS data. However, adetailed comparison of the experimental data with the simulation results isnot possible, since the simulation model is not fully atomistic. It does nottake into account both the atomistic mechanism of the β → ω transition andthe kinetics of the particle formation and growth.
6. Summary
We have studied the sizes and positions of hexagonal ω Ti particles insingle crystals of cubic β -Ti alloy by small-angle x-ray scattering. We deter-mined the dependence of the particle size and distance on the ageing time anddemonstrated that the particle growth can be described by the LSW model[15, 16]. We found that the particles spontaneously order creating a cubicthree-dimensional array with the axes along the cubic axes (cid:104) (cid:105) β of the hostlattice. The structure of the array can be described by a phenomenologicalshort-range order model and we demonstrated by a Monte-Carlo simulationthat the driving force of the ordering is the minimization of the elastic energyof inter-particle interactions. Acknowledgements
The authors gratefully acknowledge prof. Henry J. Rack for helpful com-ments on phase transformations in Ti alloys and for the idea of their inves-tigation by the means of SAXS. The work was supported by the Ministry ofEducation, Youth and Sports of Czech Republic (Project LH13005), by theCzech Science Foundation (Projects P204/11/0785 and 14-08124S), and bythe Grant Agency of Charles University in Prague (Project 106-10/251403).24he single-crystal growth was performed in MLTL (http://mltl.eu/) withinthe program of Czech Research Infrastructures (Project No. LM2011025).The ChemMatCARS Sector 15 of the synchrotron source APS is principallysupported by the National Science Foundation/Department of Energy undergrant number NSF/CHE-0822838. Use of the Advanced Photon Source, anOffice of Science User Facility operated for the U.S. Department of Energy(DOE) Office of Science by Argonne National Laboratory, was supported bythe U.S. DOE under Contract No. DE-AC02-06CH11357.