摘要:

Vibrational Properties of Vitreous Germania by InelasticCold Neutron ScatteringBY A. J. LEADBETTER AND D. LITCHINSKYSchool of Chemistry, University of BristolReceived 16th June, 1970Inelastic cold neutron scattering experiments have been carried out on polycrystalline (hexagonal)and vitreous GeOz. Coherence effects are observed for both materials for o 2 300 cm-' and it hasbeen possible to estimate the approximate form of two average acoustic branches of the spectrumfor the polycrystal. For the glass the coherence effects may be interpreted in terms of wave vectorconservation conditions operating in a similar manner to that in polycrystals. The approximateshape of two average acoustic branches has been estimated together with the frequency of a furtherbranch near the equivalent of the Brillouin zone boundary.The data have also been examined onthe basis of the incoherent approximation to determine the main features of the frequency distributionG(o) for the glass at w< 600 cm-', in which region the most prominent peak is near 300 cm-'.The low frequency peaks in G(w) account extremely well for the low temperature heat capacity.The neutron scattering results have been compared with the infra-red and Raman spectra of vitreousGeOz showing that the relation between G(w) and the spectra is different in each case but that theinfra-red absorption coefficient probably approximates reasonably well to G(w), except below about200 cm-'. A comparison has also been made with a theoretical frequency distribution for a modelof vitreous GeOz.The neutron spectra for the glass show a peak at 13 cm-', estimated to containabout 0.05 % of the total modes and attributed to resonance modes associated with some particular,but so far unidentified, structural defects.1. INTRODUCTIONIn the harmonic approximation a rigid glass has proper normal modes of vibrationexactly as does a crystal or a molecule. The main object of this work is to investigatethe nature and frequency distribution of these modes for a number of simple glassesby means of inelastic cold neutron scattering experiments. In the long-wave low-frequency limit, glasses like crystals behave as elastic continua and well-definedfrequency (a), wave vector (4) relations exist for both longitudinal and transversewaves. The lack of periodic symmetry characteristic of all glasses will result in abroadening of these dispersion [ c o ( ~ ) ] curves with increasing q, corresponding todecreasing " phonon " lifetimes.One of our objects is to investigate the extent towhich a phonon-like description of the vibrational excitations is meaningful or usefulfor glasses. Previous neutron scattering experiments on simple liquids and onvitreous Si02 and BeF, have revealed coherence effects into the equivalent of ~the second Brillouin zone which are most simply interpretable, at least for the acousticbranches of the spectrum, in terms of the existence of phonons of rather short life-times. We report here similar experiments and conclusions for vitreous GeO,.At higher frequencies a phonon-like description of the excitations is probablymuch less useful. Even the simplest network glasses have fairly complex structures ;their crystal analogues typically have of order 10 atoms per primitive cell, so thatmost of the vibrational modes are in optical branches.However, even if it weretheoretically possible to describe the higher frequency vibrations in glasses in similarterms, the directional averaging, disorder broadening and large number of branches6A . J . LEADBETTER AND D. LITCHINSKY 63make it highly unlikely that these branches could be defined experimentally. It maybe, therefore, that at high frequencies the only simplifications possible in describingthe nature of the vibrational modes is a description of the principal atomic motionsinvolved (e.g., Si-0 stretching) as suggested by Bell et aL4 from the results of theirdetailed model calculations and used empirically by infra-red spectroscopists.In addition to information about the nature of the vibrational modes, inelasticneutron scattering experiments can also give information about their frequencydistribution G(co).Experiments on simple network glasses cannot give G(o) itself,however, for a number of reasons. First, the atoms in these glasses are almostpurely coherent scatterers, so that wave vector as well as energy conservation condi-tions control the scattering, making possible the determination of dispersion curvesbut not directly of G(m) itself. Secondly, even at high momentum transfer wherethe scattering approximates to incoherent behaviour, the observed cross-section is aweighted sum of contributions from the chemically different atoms and the weightingfactors depend on energy and momentum transfer.The principal unknown quantitiesin these factors are the displacement vectors of the atoms in the individual normalmodes. Nevertheless, it is still possible to obtain much useful information about thegeneral shape of G(co) and the positions of its main peaks.In addition to the general effects of lack of periodic symmetry on the vibrationalproperties, which occur for all glasses, there is the further possibility of more localizedvibrational excitations associated with specific structural features in a particularglass. Bell and Dean have discussed evidence for localized vibrations of non-bridging oxygen atoms in vitreous silica and there is also fairly strong evidence for theexistence of resonance vibrations of very low frequency (z20 cm-l), probablyassociated with oxygen vacancies in the vitreous silica For both vitreoussilica and beryllium fluoride the inelastic neutron scattering cross-section has asignificant value down to very low frequencies (- 10 cm-l) consistent with thepresence of such localized excitations but no discrete peaks were found.Data havebeen obtained for vitreous germania down to low frequencies to search for localizedlow frequency vibrational excitations in this material.2. EXPERIMENTALThe nieasurements were made using the cold neutron, chopper time-of-flight apparatuson the Herald reactor at A.W.R.E., Aldermaston.The experimental details have beendescribed el~ewhere.~. Incident neutrons of energy about 4.5 meV (-35 cm-' opticalwave number) were used so that only energy gain spectra are seen. Most of the experimentswere carried out using a coarse resolution chopper operated with its passband centred on thetransmission edge of the beryllium filter at about 41$ to give a time of flight resolution ofaround 10 %. One experiment was also run on vitreous GeO, using a higher resolutionchopper with its passband completely inside the beryllium edge and giving a time of flightresolution of around 8 % but a much lower count rate. The main purpose of this experi-ment, however, was to avoid interference from the step in the incident spectrum at 4A(-870ps/m) obtained with the coarser chopper (see fig.2) and so to determine the cross-section down to as low energy as possible.Two samples of vitreous GeO, were investigated : one supplied by C. R. Kurkjian ofBell Telephone Labs. Inc. and the other made by us from 99.999 % pure crystalline powdermelted at -1400°C in a platinum-lined container for about 24 h. The specimens were inthe form of flat slabs 2-3 mm thick. The -OH content of the two specimens was measuredusing the optical density of the absorption band at 3550cm-'. The sample made by uscontained about ten times more -OH than the Bell Labs. specimen : 70 and 700 -OHgroups per. million Ge atoms assuming an extinction coefficient identical to that for -OHin Si02.10 The higher resolution experiments were done on the specimen with the highe64 VIBRATIONAL PROPERTIES OF GERMANIA-OH content but there were no significant differences between the neutron spectra of thetwo specimens determined in the other experiments.The crystalline GeOz specimen was 99.999 % pure powder and was heated for 24 h atabout 1070°C to remove traces of the tetragonal form and give pure hexagonal (quartz-like)material of good crystal quality.Large single crystals of hexagonal GeOz have not so farbeen made, nor do they occur naturally, so only polycrystal experiments were possible.On the other hand, a direct comparison with the results from polycrystal specimens is ofgreat help in interpreting coherence effects observed in scattering from the glass.In allcases the specimens were held between 0.05 mm thick aluminium sheets at ambient tempera-tures (T-296 K) and mounted at 45" to the beam in an evacuated or helium-filled chamber.The specimen thicknesses were always such that less than 17 % of the incident neutrons werescattered .The specimen scattering was corrected for background and container scattering and forenergy variation of detector efficiency and converted to absolute cross-section per steradiantime of flight interval (d2a/dQdt) by comparison with the elastic scattering from a vanadiumslab. No corrections were made for multiple scattering, although rough estimates suggestthat the only significant contribution to the inelastic scattering from this source arises froma Combination of an elastic and an inelastic process in which the observed energy transferwill still be a direct measure of a single vibrational quantum but the momentum transferfor the inelastic event will be indeterminate.Multiphonon contributions should be smallas the specimen temperature was always below the Debye temperature for the material.3. COHERENT SCATTERING FROM GeO,(i) BASIS FOR INTERPRETATION OF DATAThe neutron scattering process is characterized by the energy (hw) and momentumThese are related according to the energy and ( h e ) change of the neutrons.momentum conservation conditionsQ2 = k2 i- kg -2kko cos 8; fiw = (h2/2rn)(k2 - kg),where k and ko are the scattered and incident neutron wave vectors, 8 is the scatteringangle and m the neutron mass.Eqn (3.1) together describe the locus of experimentalobservation in the (qQ) plane for a given ko and 8.For a crystal the energy and wave vector conservation conditions take the formwhere w(q,s) is the frequency of a phonon of wave vector q in spectral branch s and zis a reciprocal lattice vector. These conditions permit the determination of dispersionrelations in a relatively straightforward manner from single crystal experiments. Forpolycrystal experiments Q takes all orientations with respect to the lattice and condi-hm(neutron) = ho(q,s) Q+q = 27cz (3.2)tion (2) becomesQ + q > 2 ~ ~ 2 Q-q (3.3)with O<q<q,,,, where qmax is determined by the size of the first Brillouin zone.Furthermore, for a given Q (and w) more than one reciprocal lattice point maycontribute to the scattering according to (3.3).The observed spectra may thereforebe rather complex and difficult to interpret in detail but in favourable cases at low Q(and o) where only one or two reciprocal lattice points can contribute to the scatteringit should be possible to obtain some information about directionally averagedacoustic branches of the spectrum in a simple manner. Thus, when the experimentalu(Q) locus crosses a dispersion curve [~(q)] scattering is allowed. The sharpnesswith which the scattering cross-section rises (the scattering edge) and the positionof the subsequent peak in the scattered intensity will depend on the anisotropy ofthe w(q) and particularly on a complicated function g(Q,o) which is essentially thA .J . LEADBETTER AND D . LITCHINSKY 65orientationally averaged inelastic structure factor Finel. Thus, the double differentialcross-section per cell for one-phonon scattering from a polycrystal with p atoms perunit cell may be writtend20 d20 t3 k nQ2 dQ, 4 = --= --dQdE dRdtrn k , 2wwhereandFinel = 6, exp (iQ p l ) a a,MF3 exp (- w).(3.4)1The delta function is an expression of the energy conservation condition and thewave vector conservation condition (3.2) determines the allowed values of Q. 0 isa unit vector along Q, I labels the atoms in the unit cell, b is the scattering length,p a position vector, M an atomic mass and 2W the Debye-Waller exponent, btI isthe displacement vector for a given atom in a particular normal mode o(q,s); nis the phonon occupation number [n = (exp hco/kT- l)-l].The average (. . .) istaken over all orientations of the crystal with respect to Q. Apart from the simplefactor (k/ko)n Q2/20 the intensity of one phonon scattering is therefore determinedby g(Q,co) for the evaluation of which a detailed lattice Pynamical model is required.The most important part of Finel is the scalar product Q e a , . In the simple limitingcase of a monatomic crystal this is a maximum at Q+q = 2z4Q and q colinear)and zero at Q-2m (Q and q perpendicular) for pure longitudinal modes, and zeroand maximum at the same values of Q respectively for pure transverse modes. Forlongitudinal modes the peak positions in the polycrystal scattering cross-section thenindicate the position of the average dispersion curve while the average transversebranch is defined by the position of the scattering edges.I* l 2 The peak positionin the latter case is not strongly dependent on scattering angle and gives the positionof the top of the transverse branch near the zone boundary.For fairly complex crystals like hexagonal Si02, BeF, and GeO, with nine atomsin the primitive cell the effect of g(Q,co) in determining the intensity of the differentbranches will be less sharp and clear-cut than for the simple case outlined above andwe expect to observe scattering with reasonable (though unknown) intensity fromall branches at all relative orientations of Q and q.The results for silica and berylliumfluoride were consistent with this view and it was possible to determine some of thedetails of average dispersion curves by examination of peaks and scattering edges inthe spectrum, and for silica good agreement was obtained with the results of singlecrystal experiments.This simple approach has therefore been adopted in analyzingthe data for hexagonal GeO,.For the glass there is no simple theoretical starting point in treating coherenceeffects although these should certainly be important at low Q. Following our workon vitreous SO, and BeF, we study the coherence effects on an empirical basis,examining peak and edge positions in the spectra for angle dependence, to investigatewhether any pattern emerges and how any such pattern compares with that foundfor the polycrystal and whether it is consistent with a phonon-like interpretation.In principle, also we would wish to determine phonon liketimes but it has so far notbeen possible to do this quantitatively, at least with present experimental resolution.66 VIBRATIONAL PROPERTIES OF GERMANIA(ii) RESULTSSome typical primary data for hexagonal and vitreous GeO, at three scatteringangles are shown in fig.1 in the form of the reduced double differential cross-section(d20/dQdt) 471/oGeO2 against time of flight. The function g(Q,co) for these samew/cm-l600400 200 lQ0806D 401 1 I I l l I T 1 IE 2 “18c10Ut/ps m-lFIG. 1.-Scattered neutron spectra for GeOz at three scattering angles. 0 , vitreous ; 0, polycrystal(hexagonal) ; I, statistical errors ; 7, step in incident spectrum for coarse resolution rotor ; -1 , centreof incident spectrum.angles, calculated from the data using eqn (3.4) is also shown in fig.4 and 5.Coherence effects are evident at a11 energy transfers below about 300 cm-I in thatpeak positions and relative intensities are dependent on scattering angle. The resultsare only amenable to simple interpretation as discussed in Q 3(i) for c o r l O O cm-lso we confine the discussion here to the very low energy region of the spectrum.Energy, momentum transfer w(Q) data obtained from peak and edge positions inthe scattered neutron spectra are shown in fig. 2 and 3.For the polycrystal data tentative estimates of the shapes of two average acousticbranches have been made : the lower of these is markedly broadened due to crystalanisotropy (cf.Si02 quartz ’). Near the zone boundary it extends from about 50to 65 cm-l. The higher of these branches appears to be less broadened and near thezone boundary its frequency is 75-80 cm-’. Almost all the low frequency featuresof fig. 2 can be understood on the basis of scattering involving the first two reciprocaA . J . LEADBETTER AND D . LlTCHINSKY 67lattice points as shown in the figure. The lowest energy point at 90" scatter anglepresumably arises from scattering involving higher 2 values. The peak positions atu) - 100 cm-I are largely angle-independent and probably arise from the highestenergy acoustic branch near the zone boundary.The peak at 130 cm-l is attributedto optical modes and there is an infra-red absorption peak l3 at 123 cm-I correspond-ing to an optical branch at the zone centre.I 5 O rIE3 \QlA-'FIG. 2.-(o,Q) plot for hexagonal GeOz. Squares represent peak positions and bars extend topeak edges at half height. Estimates are shown of two average acoustic branches based on scatteringinvolving two reciprocal lattice vectors (2~7).The results for vitreous GeO, may be interpreted in terms of average dispersioncurves by assuming that the first maximum in the diffraction pattern of the glass atQ - 1.6 acts as a (directionally averaged) reciprocal lattice point for the inelasticscattering process. There is some dependence of peak positions on scattering anglecompatible with this assumption and with the sound velocities 14* l5 and tentativeestimates of the position of two average acoustic branches are shown in figure 3.The frequencies of these branches near the " zone boundary " are - 26 and - 38 cm-l.The maxima in the cross-section near Q - 1.6 k1 occur at these frequencies suggestingan appreciable transverse character for these branches.Slightly angle-dependentpeaks are observed near 50 and 70 cm-l at most angles and are tentatively assignedrespectively to an acoustic branch near the zone boundary and an optic branch.Regardless of the detailed accuracy of these interpretations it seems clear that acousticbranches do exist in the glass. Furthermore, although at very small wave vectorsthe transverse branch is doubly degenerate because the glass is macroscopicallyisotropic this degeneracy is lifted as q increases due to the microscopic anistropyand indeed the modes will no longer be purely transverse or longitudinal.It is notpossible to draw any quantitative conclusions about the width of the acoustic branchesalthough qualitatively the peaks in the spectrum (and in g(Q,w)) are broader for theglass than for the polycrystal and the width also appears to increase with Q.The low temperature heat capacity is a sensitive probe of low energy excitations.The Cp results 16* I7 clearly indicate that the first peak in the frequency distributio68 VIBRATIONAL PROPERTIES OF GERMANIAis at about 26cm-' in excellent agreement with the neutron spectra.16 A moredetailed analysis shows that the heat capacity can be fitted at least for Ts5 K witha G(w) containing three maxima at 26, 38-40 and 50-55 cm-l containing respectivelyabout 1.2, 1.3 and 1.5 % of the total modes.The Cp data below 5 K indicate thepresence in some specimens of other excitations additional to the disperse acousticwaves (see § 5). The heat capacity and neutron data are thus in good agreementand show that, if the thee peaks in G(m) at N 26, - 38 and - 50 cm-' indeed containall the acoustic modes, then these comprise about 4 % of the total, which suggestsan apparent primitive cell for the glass containing about 8 GeO, units, which doesnot seem unreasonable.QklFIG. 3.-(o,Q) plot for vitreous GeO,. Squares represent peak positions and bars extend to peakedges at half-height.Estimates are shown of two average acoustic branches, with transverse andlongitudinal sound velocities,l"* l5 based on the position of the fist diffraction maximum atQ-1.6 A-l.4. FREQUENCY DISTRIBUTION FOR GeO,Although it is not possible to obtain the frequency distribution G(u) directlyfrom coherent scattering experiments on a complex polyatomic solid it is possible toobtain much useful information about its main features. Thus, the low energyscattering discussed in § 3, although markedly influenced by coherence effects, givesat least the positions of peaks in G(cu) corresponding to the tops of the acousticbranches and rather flat optical branches. As Q (and u) increase, coherence effectsdiminish in importance and when Q is much larger than the size of the first Brillouinzone, i.e., for a polycrystal when many reciprocal lattice points contribute to thescattering at a given Q, then the form of the cross section and of g (Q,co) as definedby eqn (3.4) approaches that for incoherent scattering, viz.,(4.1) ginc(Q, GO) = <c &(o aJ2M; ' exp (- 2 W,)(N)- '6(w - cu(q, s))),q,s A .J . LEADBETTER AND D . LITCHINSKY 69whereandJ G(m)dw = 1.0If we separate the contributions from the different atoms and carry out the averagingover all orientations of Q assuming chemically identical atoms to be dynamicallyequivalent we get, for GeO,Substituting known values for b and M gives%e -g(Q, a) = [age exp ( -2WGe)+4.2z exp ( -2W0)J---G(o),MGe(4.5)In this approximation the frequency distribution is related to the experimental functiong(Q,w) by a weighted sum of displacement vector times Debye Waller factor for thedifferent atoms.The Debye Waller factors have been obtained using estimates - ofthe directionally averaged mean square amplitudes 2 and the relation 2W = Q2u2.FIG. 30", B, 60"; C , 90"70 VIBRATIONAL PROPERTIES OF GERMANIAIsotropic values of u2for the glass were assumed to be the same as the average valuesfor hexagonal Ge0,,18 viz., 0.007 A2 and 0.013 A2 for Ge and 0 respectively atT- 300 K. The Debye Waller factors, exp (- 2W), for Ge and 0 are always greaterthan 0.77 and 0.62 respectively for these experiments. Their effect is not importantbut offsets to a slight extent the factor of 4.2 in eqn (4.5).The values of the displacement vectors are much more difficult to estimate and adetailed model is required.Some eigenvector calculations have been made by Belland Dean l9 and using their results rough estimates suggest thatfN(co) varies by lessthan 50 % over the whole frequency range, being largest above about 300cm-lwhere the oxygen displacements are larger than those of germanium.w jcm-1FIG. 5.-A, B, C : g,(Q,o),w plot for vitreous GeOz at 30°(A) 60"(B) and 9O"(C) scattering angles.Units arbitrary but internally consistent. D : Debye spectrum corresponding to measured soundvelocities and normalized to curve C. E : infra-red absorption coefficient of vitreous GeOa, arbitraryunits. F : smoothed theoretical frequency distribution G(w) for vitreous GeOz calculated by Bell,Bird and Dean : approximately normalized to g(Q,w) at 90" scattering angle (C).Plots of the experimental function g(Q,w) for polycrystalline and vitreous GeO,are shown in fig.4 and 5 for three scattering angles. The dependence of the resultson scattering angle shows that in both cases and particularly for the crystal, coherenceeffects are important below about 300 cm-l. Hence, because of the detailed in-adequacy of the incoherent approximation and also because we regard our estimatesoffN(w) as of only semi-quantitative significance we make'no attempt to give G(w)itself. Nevertheless, peaks which appear at all angles in g(Q,w) are undoubtedlyreal peaks in G(w) and the overall shape of G(co) for the glass will not be very differentfrom that of g(Q,o) particularly at the highest scattering angle. Above about200 cm-l g(Q,o) is broadly similar for crystal and glass ; similar results were obtainedfor SO2 and BeF,.For vitreous GeO, peaks in G(w) are at 26, 38, 50, 70, 100, 190A . J . LEADBETTER AND D. LITCHINSKY 71220, 270-300, -400 and 550-600 cm-l. Some of these peaks are broad and notwell-defined as shown in fig. 5. Between about 70 and 200 cm-l, coherence effectsare marked but not easily interpretable, but there is still a relatively high number ofmodes in this region.Also shown in fig. 5 are the theoretical G(w) calculated by Bell, Bird and Deanand the Debye spectrum corresponding to the measured sound ~elocities,~~"normalized to g(Q,co) by assuming the areas of g(Q,co) and G(m) to be the same and80 % of the total modes to be at o < 600 ~ r n - l .~ Since G(w) must be considerablyhigher than the limiting Debye spectrum at the peaks arising from dispersion of theacoustic branches, this comparison, plus that of the relative area under the g(Q,co)function up to -60 cm-I (-4 %) with the results of the Cr, analysis, shows thatg(Q,w) is probably a reasonable approximation to G(w) in this range. This showsalso that the theoretical spectrum is inadequate below about 100 em-l, presumablydue to the nature of the model used. Elsewhere the theoretical spectrum appearsto be in moderately good agreement with experiment. The large peak in thetheoretical G(w) at 350 cm-l presumably corresponds to the experimental peak near300 cm-l and the 500 cm-l peak in the model calculation must correspond to thebroad peak around 550 cm-I observed experimentally.The upper side of this peakis not well-defined experimentally because 600 cm-l represents about the practicalupper limit of the present cold neutron experiments.Finally, we compare the neutron scattering results with infra-red and Ramanspectra of vitreous GeO,. The normal crystal selection rules no longer apply to theglass and the infra-red absorption coefficient cc and the Raman intensity I may bewritten in a similar form to equation (4.6) aswhere n is the phonon occupation number and the upper sign refers to Stokes andthe lower to anti-Stokes scattering. Little is known about fr(o) or &(a) so it is notyet possible to determine G(w) from the experimental spectra.In fig.5 the infra-red absorption coefficient of vitreous GeO, in nujol mullsdetermined in this laboratory l3 is shown on an abritrary scale. There are twobroad peaks with maximum absorption at 560-570 and -290-300 cni-l and a broadshoulder at 400-500 cm-l. These results are in excellent accord with those in g(Q,w)and suggest that in this frequency rangefI(w) is not strongly dependent on frequency.At frequencies below 150 cm-l the spectrum was measured on a plate of GeO,using a Fourier transform interferometric spectrometer. The absorption coefficientdecreases smoothly with u3 and if the absorption is indeed a disorder-induced one-phonon process then fr(co) at u) 2 200 cm-l must be strongly frequency-dependent.Simple theories due to Bagdade and Stolen ** (fr(co) cc const +w2 depending on co)and Whalley [fI(co)ccco2] in neither case produce results which much resemble g(Q,w)so these simple theories appear not to be valid at least for GeO, for 25 < cu < 200 cm-I.A determination of the Raman spectrum of vitreous GeO, 22 shows that &(a)is different from fI(cu) : although there are peaks at 320 and 560 cm-l much thestrongest peak in the spectrum is at 412 cm-l which presumably corresponds to theshoulder in g(Q,co) and the infra-red spectrum at 400-450 cm-l.A broad peak isalso observed in the Raman spectrum (I) around 60 cm-' at -300 K although, byanalogy with silica, I/(n + 1) is presumably only a rather broad shoulder.Sufficientdetails were not given to permit a proper comparison with g(Q,co) at low frequencies72 VIBRATIONAL PROPERTIES OF GERMANIA5. LOW-FREQUENCY RESONANCE MODES I N VITREOUS GeO,A number of properties of vitreous silica imply the existence of resonance modes offrequency less than about 20 cm-1.6'8 In particular, the heat capacity, thermalexpansivity, thermal conductivity and spin lattice relaxation show temperaturedependence which may be interpreted in these terms. In addition, the differentialneutron scattering cross-section is significant in this energy region although nodetailed structure has been observed in the cross-section. The inelastic neutronspectrum of vitreous germania has been determined to low frequencies in order toseek evidence for such modes.The cross-section in undoubtedly significant down toat least 10 cm-l and at most scattering angles there appears to be some structure in thespectrum for w<20 cm-l but the counting statistics are poor. The data for alleight scattering angles have therefore been summed to improve the statistics and theresultant spectrum is shown in fig. 6. The peak at -26 cm-I is due to the lowestenergy acoustic branch of the spectrum. There is, in addition, a peak at about13 cm-l which does not arise from a broadened acoustic branch whereas the structurenear 20 cm-l is, mostly at least, due to coherent peaks at particular scattering angles.The simplest explanation of the peak at 13 cm-1 is that it arises from resonance modesassociated with particular defects in the structure. Using the incoherent approxi-mation the fraction of modes in t h s peak has been estimated from its relative areain the g(Q,w) function.Assuming a constant fN(co) in eqn (4.6) gives -0.05 % ofthe total modes in this peak but if the scattering were due to H atoms this fractionwould be 2 or 3 orders of magnitude smaller due to the effects of the high cross-sectionand small mass of H onfN(m).o/cm-FIG. 6.-,(Q,w),w plot for vitreous GeOl at very low frequencies.Low temperature heat capacity determinations 16* also suggest the possibilityof this type of excitation in vitreous GeO,. Experiments were performed between1.3 and 25 K on two specimen~,~' one of which was oxygen deficient, and no significantdifference in the results was found.The heat capacity of both specimens was,however, markedly higher than the earlier results of Morrison and Antonioufor T"<6 K. This difference is real and may be empirically described by a differencA . J . LEADBETTER AND D. LITCHINSKY 73in the frequency distributions amounting to about 0.04 % of the total modes locatednear 12 cm-l.The correlation with the neutron data is so good as to suggest that the samephenomena are being observed and that they are vibrational niotions associatedwith some particular structural feature (or impurity) not present in all samples ofgermania. Some unpublished spin-lattice relaxation work on defects in germaniais mentioned in ref.(1 6) as indicating an average vibrational frequency associatedwith the defects in the region of 7 cm-l. However, no details have yet been published.To sum up : there is little doubt that low frequency excitations near 13 cm-l existand that they are probably rather localized, they are probably not due to impuritiesor oxygen deficiency but we can offer no positive suggestions at this stage on thebasis of the available evidence.6. CONCLUSIONSInelastic cold neutron scattering experiments on GeO, have revealed coherenceeffects for the vitreous form most simply interpretable in terms of w(Q) relationssimilar to those in poly-crystals. The first diffraction maximum in vitreous GeO,appears to be acting like a reciprocal lattice point and the shape of two averageacoustic branches of the spectrum has been estimated.The main features of thefrequency distributions have been determined for both crystalline and vitreous GeO,and compared, for the glass, with thermal properties, infra-red and Raman spectraand a theoretical distribution. A peak in the frequency distribution has been foundat 13 cm-l containing about 0.05 % of the total modes which is attributed to resonancevibrations associated with structural defects as yet unidentified.We are grateful to the Science Research Council and Pilkington Bros., Ltd. forfinancial support and to the University Support Group at A.W.R.E., Aldermastonfor much experimental help.S. J. Cocking, Adu. Phys., 1967, 16, 189.A. J. Leadbetter, J. Chem. Phys., 1969, 51, 779.A. J. Leadbetter and A. C. Wright, J. Non-Cryst. Solids, in press.R. J. Bell, N. F. Bird and P. Dean, J. Phys. C., 1968, [2] 1, 299.R. J. Bell and P. Dean, Proc. Int. CunJ Locaked Excitations in Solids, Irvine, Calif, 1967, ed.R. F. Wallis (Plenum Press, New York, 1968), p. 124.A. J. Leadbetter, Phys. Chem. Glasses, 1968, 9, 1.J. G. Castle, Jnr. and D. W. Feldman, J. Appl. Phys., 1965,36, 124.F. Davies et al. Neutron Inelastic Scattering (I.A.E.A., Vienna, 1968), vol. 2, p. 341.lo G. Hetherington and K. H. Jack, Phys. Chem. Glasses, 1962,3, 129.W. M. Lomer and G. G. Low, in Thermal Neutron Scattering, ed. P. A. Egelstaff (AcademicPress, New York, 1965), p. 1.l2 S. J. Cocking and Z . Guner, Inelastic Scattering of Neutrons in Solids and Liquids, (I.A.E.A.Vienna, 1963), vol. 1, p. 237.l3 A. J. Leadbetter and M. f. Wood, unpublished work.l4 R. E. Strakna, unpublished results, quoted in ref. (16).lS C. R. Kurkjian, private communication.l6 A. A. Antoniou and J. A. Morrison, J. Appl. Phys., 1965,36,1873.l7 K. E. Wycherley, Ph.D. Thesis, (University of Bristol, 1969).l8 G. S. Smith and P. B. Isaacs, Acta Cryst., 1964,17, 842.2o W. Bagdade and R. Stolen, J. Phys. Chem. Solids, 1968, 29, 2001." E. Whalley and J. E. Bertie, J. Chem. Phys., 1967,46, 1264.22 M. Hass, J. Phys. Chem. Solids, 1970, 31,415.' P. G. Klemens, J. G. Castle, Jnr. and D. W. Feldman, Phys. Chem. Glasses, 1964,5,104.R. J. Bell and P. Dean, private communication

ISSN:0366-9033    

DOI:10.1039/DF9705000062

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

The Investigation of Atomic Motions in Crystalline, Amorphousand Liquid Selenium, and in Crystalline and LiquidTellurium by Neutron SpectroscopyB Y A. AXMA”,* W. GISSLER,? A. KOLLMAR AND T. SPRINGERInst it ut fur Fest korperforschung der Kernforschungsanlage JulichReceived 1st June, 1970The energy spectra of monochromatic neutrons scattered on selenium and tellurium have beeninvestigated by means of a rotating crystal spectrometer. In polycrystalline selenium the highestspectral line can be identified as an intrachain bond stretching mode. Transforming selenium fromits crystalline to its amorphous and liquid state, the shift of the line energy is only a few percent.The other lines in the spectra (from bond shearing and torsional modes of the chains) are influencedmore strongly.For tellurium, on the other hand, the spectral lines of the solid disappear completelyin the liquid phase. The experimental results are compared with the results of optical spectroscopy.Furthermore, the diffusive motion of the atom has been studied by quasi-elastic neutron scatteringfor liquid selenium containing various admixtures of iodine (0-6 %), which change the average chainlength of the polymeric melt, and for liquid tellurium. For Se, one obtains an effective diffusionconstant D which describes the motion of small chain segments over atomistic distances. D is ofthe order of cm2/s. Its activation energy was about 5 kcallmol.The crystalline structure of selenium and tellurium can be visualized as a hexagonalarrangement of helical chains as shown in fig.1. The atoms within the chains areconnected by covalent bonds whereas the interatomic forces between neighbouringchains are more of the van-der-Waals type. In selenium, the covalent character ofthe binding is more pronounced than in tellurium, as can be inferred from the latticestructure.l The chains are partly conserved in the amorphous and also in the liquidphase of selenium ; this has been concluded from diffraction experiments 2-6 and inthe liquid from its high viscosity.’~ The constitution of amorphous selenium isill-definedy depending on the preparati~n.~ In addition to the helical chains, thereexists a considerable amount of small ring molecules, probably of the type Se8 in theamorphous and in the liquid phase.*-l* The viscosity of liquid selenium is stronglyreduced by the admixture of small amounts of iodine or ch10rine.l~~ l2 For higherconcentrations this can be attributed to a breaking of the chains by the formation,e.g., of I--Se .. . Se-I complexes. For liquid tellurium a polymeric structure hasalso been suggested 13-16 ; this is discussed later in connection with our experiments.The aim of this work was to study by means of neutron spectroscopy the wholespectrum of atomic vibrations in the crystalline phase of Se and Te, and its changeif the crystal is transformed into the amorphous or into the liquid state. So far, anumber of investigations by means of optical spectroscopy have been carried out onsolid Se and Te.17-22 With neutrons it is easy to perform experiments also in theliquid phase and at elevated temperatures.A further objective of this work withneutrons was the investigation of scattering at very small energy transfers whichreveals information on the diffusive motion of the scattering atoms. For a har-monically-bound atom in a solid, the energy spectrum of the scattered neutrons* present address : Inst. v. Laue-Langevin, Grenoble.1 present address : Research Laboratory, Petten, Ndlds.7A . A X M A N N , W . GISSLER, A . KOLLMAR A N D T . SPRINGER 75contains a sharp line at energy transfer hw = 0 (“ Mossbauer line ”), whereas fordiffusing atoms this line has a finite width. For the diffusive motion of a freeparticle 23 this so-called quasi-elastic line has a Lorentzian shape with a full width athalf maximumwhere D is the self diffusion constant of the scattering atom and K = ko-k is thescattering vector which can be varied by changing the scattering angle (hlc =momentum transfer, ko, k = wave vector of the incident and the scattered neutron).AE = 2hK2D (1)wa-FIG.1.4rysta.l structure of selenium and tellurium (from ref. (1)).If the scattering atom is part of a large molecule or of a polymeric chain, the quantityD as defined by eqn (1) is not necessarily the true self-diffusion constant of the laigemolecule. Under these circumstances, D can be considered as an efective diffusronconstant which describes the diffusive motion of the atom over atomistic distances.Eqn (1) has been derived for incoherent scattering ; for coherent scattering, whichdominates in Se and Te, it can be considered as a reasonable approximation as longas K does not reach the region of the first diffraction peak of theThe investigation of quasi-elastic scattering on liquid Se and Te, and of Se withvarious iodine concentrations seemed to be of special interest, particularly in com-paring the results with the viscosity.Part of our work has been published earlier 2 5including experiments on Te single crystals 26 which are not discussed here.EXPERIMENTALSAMPLESX-ray diffraction photographs of the Se and Te powder used for the experiments showedsharp Debye-Scherrer lines which were compatible with the trigonal crystalline structure.Amorphous Se was produced by quenching the molten material in cold water.Only diffuserings and no sharp reflections were observable in the X-ray photographs. The selenium+iodine mixtures were prepared by heating of the substances over 24 h at 500°C in sealedV2A steel tubes which later served as sample containers for the scattering experiments. Th76 ATOMIC MOTIONS BY NEUTRON SPECTROSCOPYviscosity of liquid Se with 0 and 6 at. % iodine was measured after this treatment and theresults agreed with those from Krebs and Morsch.ll A chemical analysis showed that theimpurity content of the liquid samples was <lo0 p.p.m. for iron metals and <50 p.p.m.for other elements, in spite of the fact that there was some chemical reaction in the tube walls.NEUTRON SPECTROSCOPYThe intensity distribution of the scattered neutrons was measured as a function of energyE for an incident energy Eo = 4.8meV by means of the time-of-flight method, using therotating crystal spectrometer at the FRJ-2 reactor in Julich.The resolution (full width athalf-maximum) at Eo was 0.27 meV. The scattering probability in the samples was between12 and 16 %. Fig. 2 and 3 show typical time-of-flight spectra. The high peak at Eo isdue to elastic or quasi-elastic scattering, the other peaks are produced by inelastic scatteringprocesses with energy transfer hm = E- Eo where the neutrons have gained energy from theatomic vibrations.Fig. 4 shows representative plots of the quasi-elastic line width A E as a function of themomenturn transfer hlc. The values are corrected with respect to energy resolution by meansof the diagrams of Melk~nian.~'RESULTS AND DISCUSSIONATOMIC VIBRATIONSThe time-of-flight spectra in fig.2 and 3 show a number of well-defined peaks.Their positions are represented in table 1 and 2, together with optical measurementsand theoretical calculations.(a) polycrystx 5<b) amorph Ptx 2 0 rn 0 .flight time (ms)I I2 3I I I I30 5 0 20 10 5 10050 2 0 10 5energy E (mewFIG. 2.-Typical time-of-flight spectra of selenium. Incident energy Eo = 0.0048 eV, scatteringangle 87". (a) polycrystalline, 20°C ; (b) amorphous, 20°C ; (c) liquid, 340°C ; (d) liquid with 4 at.% iodine, 340°CA. AXMANN, W. GISSLER, A . KOLLMAR AND T. SPRINGER 77I000500 nv) Y.C( Eid2 Wc, h0 2 0d u.M5 0 00energy E (meV)FIG.3.-Time-of-flight spectra of tellurium. (a) polyc rystalline, 20°C ; (6) liquid, 470°C.Before discussing these data the fundamental differences between light and neutronspectroscopy should be considered. Optical spectroscopy observes vibrational modeswith a wave vector q N 0 whereas neutrons can interact with modes throughout theBrillouin zone because of the large accessible wave numbers. For coherent neutronscattering on a polycrystal, peaks should be observable in the scattering spectra atTABLE 1.-SPECTROSCOPIC RESULTS FOR SELENIUM (ENERGIES hm in eV)01 0203 O i 0;o;cryst. 11.7 1 6 . 2 31.5neutron spectr. amorph. 15.3 32.4this work liquid (350°C) 12.8 34cryst.17 17.5 34amorph. l7 14.6 31optical spectr.cryst.20 12.8 17.3 28.8cryst.l*, l9 1 2 . 6 17.7 27.7amorph.l*, l9 1 0 frequencies between 6.2 and 3 1 . 4TABLE 2.CPBCTROSCOPIC RESULTS FOR CRYSTALLINE TELLURIUM (ENERGIES hCr, in ev)ref. 01 020 3 Oi 0;o;neutron spectr. this work 10.0 10.5 18.51 7 1 1 1 3optical spectr. 2 1 10.7 11.41 8 11.1 11.3theory 29 9.2 1 5 . 217.617.720.78 ATOMIC MOTIONS BY NEUTRON SPECTROSCOPYfrequencies where the density of normal modes is high, i.e., at extrema in the dispersionbranches ~ ( q ) , ~ ~ especially for " flat " optical branches (" flat " means ~ ( q ) doesnot change strongly from q = 0 to the Brillouin zone boundary). Such a spectrummight be modified by the conservation law for the quasi-momentum (K = G-q)and by the dynamical structure factor which depends on the orientation between Kand the polarization vector of a mode.For a polycrystal this has, however, nogreat importance because many reciprocal lattice vectors G and orientations of Gcome into play,25 and the peaks in the spectra actually correspond to singularities inthe density of the modes. This argument holds equally well for the non-crystallinestate.Se and Te have three atoms per unit cell and therefore 6 optical and 3 acousticdispersion branches. According to theoretical calculations by H u h and Geicket aL20 there is a bundle of three optical branches (O;,2,3) which are quite flat andwell separated from the other branches. These three branches correspond to stretch-ing modes of the covalent bonds. The high energy peak (no.5) for crystalline Seand Te must be attributed to these modes. Two of the lower optical branches(O,,O,) have the character of a bond-shearing mode (for small q-values) and one isprimarily a torsional mode of one chain against the neighbouring chains (01).20 Itis highly probable that the peaks 3 and 4 correspond to the flat section of thesebranches. The infra-red vibrational energies shown in table 1 and 2 coincide withthe energies of the neutron peaks which means that the branches are relatively flatas predicted by theory. Comparing the frequencies of Se and Te one can estimatethat the coupling constants for the bonds within the chains are about two times higherfor Se than for Te. It is assumed that the low energy peaks 1 and 2 are due toacoustic vibrations.The neutron spectroscopic results for the solid phases of Seand Te by Kotov et aL30 are consistent with our data, showing one peak at theposition of our peak 5, and one close to our peaks 3 and 4.If crystalline selenium is transformed into the amorphous and into the liquid state(melting temperature -217°C) the position of peak 5 in fig. 2 shifts only slightly tohigher energy. Its width remains essentially the same. This peak corresponds to thebond stretching mode in the chains. It could contain contributions from bondvibrations in the Ses molecules where the forces between nearest neighbours areabout the same as in the chains. Also the lower peaks 3 and 4 (torsional and bondshearing modes) are observable in the amorphous and in the liquid phase with a smallshift to lower energy.These results are compatible with the idea that the chainstructure is conserved in the non-crystalline phases of Se with a certain degree ofshort range order between the chains. The observed energy shift can be explainedby the increase of the inter-chain distance^.^'^ For amorphous Se, a band ofscattered intensity appears in the region of the acoustic modes (at h < 7 meV)which is not visible in the crystalline state.For tellurium there is a large difference in the spectra of the solid and the liquidphase at 470°C (fig. 3) where the spectral peaks disappear completely (melting temp.,450°C). This cannot be explained by the higher temperature (which could enhancemulti-phonon processes) because these peaks were also clearly visible in a spectrumwhich was measured at 350°C. Diffraction experiments on Te 13* l5 show that thearrangement of the nearest neighbours is approximately the same in the solid and inthe liquid (between 2 and 3 atoms at a mutual distance of 3 A).Peaks in the distri-bution function at large distances have been interpreted as an indication of polymericchains. However, the observed disappearance of the peaks in the energy spectrum(in particular of the stretching mode) suggests that the chain structure has brokendown to a large extent. As a result, the disordered neighbourhood of an atom causeA , AXMANN, W. GISSLER, A . KOLLMAR AND T . SPRINGER 79an extensive smearing of the peaks in the spectrum? even if the bonds between nearestneighbours are conserved.DIFFUSIVE MOTIONSFor liquid selenium, pure and with various iodine concentrations, the quantityD as defined by eqn (1) was determined from the slope of straight lines drawn throughthe experimental points as in fig.4. Within the experimental errors one cannot seeany deviations from eqn (1) which should be expected for about KZ 2 A-2 due tocoherence effects. Fig. 5 shows that this quantity D depends only slightly on theiodine concentration and therefore on the chainlength which varies from about lo3(pure Se) to about 20 (with 6 % I) atoms per chain.7 Furthermore, in fig. 5 thequantity D is compared with the viscosity q ; the latter is plotted as kT/& (arbitrarilytaking R = 2.3 A), a quantity which should be proportional to the " true '' diffusionconstant.0'3 -n 3 Eu WQ 0.2-0 I i iK2 (A-2)FIG.4.-Representative plot (T = 350°C) of the width of the quasi-elastic line for pure molten Se,as a function of the square of the scattering vector 1c. 0 , pure; A, 3 at. %; w, 6 at. % iodine.Solid line AE = 2h1c2D with D = 0 . 6 ~ cm*/s.From the temperature dependence measured at 5 temperatures between 250 and450°C the activation energy ED for pure liquid Se was determined, according to theequation D = Do exp (-E,IkT). The result was ED = 5.0 kcal/mol. From Done can extract a rest-time between successive jumps of the randomly moving atomsin the chains. s at 450°C (with s = 2.3 A).The structural relaxation time from ultrasonic attenuation 31 was 5 x lo-' s at 400°C.Such a discrepancy has been discussed also for various organic liquids.32 So far,no theoretical model of the quasi-elastic scattering on the Se atoms in the polymericchains is available.Existing theories 33 are not applicable because they hold foranother K-region than was used here. Nevertheless, certain qualitative conclusionscan be drawn. The quasi-elastic line is composed of two parts, one from atomsOne finds z N s2/6D = 1.5 80 ATOMIC MOTIONS BY NUETRON SPECTROSCOPYbound in the chains (- 60 %), and the other from atoms in the Sea molecules (- 40 %).Both contribute about the same amount to the-total scattering intensity. The diffusivemotion of the atoms in the Se8 molecules is certainly as fast or faster than in thechains.From this one can conclude that the measured (effective) line width is, ingood approximation, representative for the motion of those atoms which are boundin the chains (the effective width of the composed line is predominantly determined by0 3 6iodine concentration (%)FIG. 5.--Solid lines with experimental points : effective diffusion constant of Se atonls, D = AE/2hK2as a function of iodine concentration and temperature, as measured with neutrons. Thin lines :kT/qR with E = 2.3 A (q = viscosity from (1 1)). (a) 450°C (b) 400"C, (c) 350°C ; (d) 300°C,(e) 250°C.that contribution which has the smaller width, i.e., the smaller 0). This motion ofthe atoms in the chains has, over short distances (i.e., several A), an effective diffusionconstant D and a jump rate l/z which are much higher as one could expect from thelow viscosity, and which is nearly independent on the chain length.The activationenergy for this diffusion process is much lower than the activation energy for theviscosity E,, = 18 kcal/mol (heat of evaporation, 6.3 kcal/mol). It is not yetclear what kind of motion is described by these parameters. In any case, it concernsunits of the chain which are several times smaller than those responsible for theviscosity (see. ref. (34)).For liquid tellurium which has a relatively low viscosity, the quasi-elastic widthgives a diffusion constant D = 2.6 x cm2/s at 450°C with an error of about25 % because the basis of the quasi-elastic line is not well defined.This agrees withthe true diffusion constant which is 2.9 x cm2/s at 470"C.35 This indicates thatthe atoms in the liquid are quite mobile and that the corresponding life-time of theinteratomic bonds is only about z = 6 x 10-l2 s.The authors are grateful to Dr. G. Wolff and Dr. H. W. Nurnberg from theLaboratory for Chemical Analysis for their valuable support and to Dr. B. Alefeldfor his help and for many discussionsA . AXMANN, W. GISSLER, A . KOLLMAR AND T . SPRINGER 81’ P. Grosse, Springer Tracts in Modern Physics (review), Ergebn. der Exakt. Naturwiss. (SpringerVerlag, 1969).K. Lark-Horovitz and E. P. Miller, Phys. Rev., 1937,51, 380.J. A. Prim, Trans. Faraday SOC., 1937,33, 110.H. Richter and F. Herre, 2. Naturforsch. A, 1958, 13, 874.E.H. Henninger, R. C. Buschert and L. Heaton, J. Chem. Phys., 1967,46,586.H. Krebs, 2. unorg, Chem., 1951, 265, 156.A. Eisenberg and A. V. Tobolski, J. Polymer Sci., 1960,46,19.G. Briegleb, Z.phys. Chem. A, 1929,144,321.’ R. Chang and P. Romo, Acta Cryst., 1967,23, 700.lo H. Krebs, 2. Naturforsch., 1957, 12b, 795.‘ I H. Krebs and W. Morsch, 2. anorg. Chern., 1950, 263, 305.l 2 S. Hamada, N. Yoshida and T. Shirai, Bull. Chem. SOC. Japan, 1969,42, 1025.l 3 R. C. Buschert, I. G. Geib and K. Lark-Horovitz, Phys. Rev. A, 1955,98, 1157.l4 A. S. Epstein, H. Fritzsche and K. Lark-Horovitz, Phys. Rev., 1957, 107,412.l6 Y. Tieche and A. Zareba, Phys. Kondens. Materie, 1963, 1,402.l 8 G. Lucovsky and R. C. Keezer, Solid State Corn., 1967,5,439.l9 G.Lucovsky, A. Mooradian, W. Taylor, G. B. Wright and R. C. Keezer, Sol. State. Comm.,2o R. Geick and U. Schroder, Proc. Int. Symp. Physics of Selenium and Tellurium (Pergamon Press,21 P. Grosse, M. Lutz and W. Richter, Solid State Corn., 1967, 5, 99.22 K. J. Siemsen and H. D. Riccius, J. Phys. Chem. Solids, 1969, 30, 1897.23 K. S. Singwi and A. Sjdander, Phys. Rev., 1960, 119, 863 ; G. H. Vineyard, Phys. Reu., 1958,24 K. S. Singwi, Phys. Rev. A, 1964,136,969.25A. Axmano, W. Gissler and T. Springer, Proc. Int. Symp. Physics of Selenium and26 W. Gissler, A. Axmann and T. Springer, Proc. Copenhagen Con$ Neutron Inelastic Scattering27 Melkonian, Proc. In?. ConJ Peaceful Uses of Atomic Energy (New York, 1956), 4, 340.28 L. van Hove, Phys. Reu., 1953, 89, 1189.29 M. H u b , Ann. Phys., 1963, 8, 647.30 B. A. Kotov, N. M. Okuneva and A. L. Shak-Budagow, Soviet Phys., Solid State, 1968, 9,31 M. B. Gitis, I. G. Mikhailov and S. Niyazov, Soviet Phys., Acoustics, 1969,15,259.32 K.-E. Larsson and U. Dahlborg, Physica, 1964, 30, 1561.33 G. Jannink and P.-G. deGennes, J. Chem. Phys., 1968,48,2260.34 S. Glasstone, K. J. Laidler and H. Eying, The Theory of Rate Processes (McGraw Hill, London,35 Potard, unpublished, personal communication by Dr. B. Cabane.G. Tourand and M. Breuil, Compt. rend., 1970, 270,109.R. C. Caldwell and H. J. Fan, Phys. Rev., 1959,114,664.1967,5,113.1969), p. 277 ; R. Geick, U. Schroder and J. Stuke, Phys. Stat. Sol., 1967,24,99.110, 999.Tellurium (Pergamon Press, 1969), p. 299 ; A. Axmann, Thesis, (Aachen).(Int. Atomic Energy Agency, Vienna, 1968), 1, 245.201 1 ; B. A. Kotov, N. M. Okuneva, A. R. Regel, Soviet Phys., Solid State, 1967,9,955.1941), chap. 9

ISSN:0366-9033    

DOI:10.1039/DF9705000074

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Vibrational Spectra of Simple Silicate GlassesBY P. H. GASKELLResearch and Development Laboratories, Pilkington Brothers Limited, Lathom,Ormskirk, Lancashire, England.Received 16th June, 1970An attempt has been made to describe the vibrational spectrum of a vitreous material in terms ofa simpIe model based on the unit cell of the crystal. The model has been adapted to include broaddistributions of intertetrahedral angles and configurations which are salient features of the structuresof silicate glasses. Normal coordinate calculations of the spectra of models representing vitreouschain silicates and vitreous silica have been performed. The results show that some detailed structuralinformation is accessible using this approach and it has been possible to provide explanations for theshape of the infra-red spectrum of vitreous silica and sodium metasilicate at high frequencies and tooffer a reason for the low-frequency Raman continuum and excess infra-red absorption which arefeatures of the spectra of all silicate glasses.Infra-red and Raman spectroscopy, although potentially fruitful sources ofstructural information, have contributed relatively little to our knowledge of thevitreous state.One considerable obstacle to successful implementation is thedifficulty inherent in any quantitative or semiquantitative treatment of experimentaldata. While the spectrum of a crystal or a simple liquid can, in principle, be assignedto the vibrations of a structural unit, such as the molecule, or the unit cell of a crystal,no such simple model can be adequate for a glass.It is, however, desirable to be able to relate the experimental spectrum of a glassto the vibrations of a small model structure.In previous work, attempts havebeen made to understand the spectrum of a glass by comparing it with the observedor calculated spectrum for the crystalline phase. There is no reason for supposingthat this should be an adequate approximation, but in many cases, notably in one-component glasses, the major features of the spectra of the crystalline and vitreousphases of a material do show notable similarities and it has thus proved possible toelucidate some of the more important structural features of the glass. For example,the symmetry and coordination of network-forming atoms can be unambiguouslydetermined and in some glasses, evidence for the existence of " supermolecular "structural modules has been obtained.Some limited success with this approach encourages the hope that a more detailedanalysis of the spectra of glasses may lead to deeper knowledge of their structures.One possible approach which we shall attempt in this paper, would be to modifythe crystal model by introducing some of the features characteristic of the disorderof the vitreous state and examine the effects which these have on the spectrum.We begin with the assumption that a silicate glass is a disordered arrangement ofessentially undistorted silicon-oxygen tetrahedra.This situation has been found invitreous silica by Warren and Mozzi who were unable to detect any larger " super-structural unit " containing more than one tetrahedron.Essential to the structureof vitreous silica is a wide variation in the intertetrahedral angles Si-0-23 and asimilar range of angles would not be unexpected in other glasses.8P. H . GASKELL 83Crystalline silicates also exhibit a remarkable variety of arrangements of tetrahedra.Even within a single class of minerals, such as the single-chain silicates, structureshave been found with from two to seven tetrahedra in their repeat units. In theliquid or vitreous state it seems reasonable that disorder should appear as the refusalof a structure to conform to a single configuration and we should therefore considerthe possibility of a glass exhibiting more than one type of configuration, possiblywithin the space of a few interatomic distances.A multicomponent glass may also contain several structural types, such as chainand sheet structures, existing side by side or in interpenetrating lattices. During phase-separation the range of structural types may vary as the phases become distinct and itshould be possible in principle to follow this demixing by vibrational spectroscopy.In this paper we shall describe methods for calculating the effects introduced bythe first two of these disorder parameters on the vibrational spectrum of a crystal.First, we attempt to assess the magnitude of changes in the spectrum as the anglebetween adjacent tetrahedra is varied and, secondy, calculate the effect of alteringthe configuration of neighbouring tetrahedra.We then argue that a reasonablerepresentation of the spectrum of the glass is obtained by taking an average over arange of bond angles and configurations.DEPENDENCE OF THE VIBRATIONAL SPECTRUM OF SIMPLE SILICATES ONTHE OXYGEN BOND ANGLEWarren and Mozzi's analysis of the X-ray scattering from vitreous silica indicatesa broad distribution of the angles Si-0-Si which link adjacent tetrahedra.Fig. 1, reproduced from this paper, shows that the angles E at oxygen range from 120to 180" with a most probable value of 144". Variations of this magnitude will alterthe vibration frequencies of a lattice, irrespective of whether this lattice is periodicor disordered, although the sensitivity of individual modes to changes in E will vary.E (degrees)FIG.1.-Distribution of angles at oxygen in vitreous silica proposed by Warren and Mozzi.'Changes in the oxygen bond-angle affect the spectrum in two ways, first, throughthe geometry of the lattice and secondly, through the force constant,& for bendingof the Si-0-Si bond. In terms of the Wilson GF matrix method, which is usedhere to calculate the spectrum, the first affects the G matrix and the second, the Fmatrix. Both may be calculated using the geometry and force field of the undistortedunit cell of the crystal as a basis.84 SPECTRA OF SILICATE GLASSESElements of the G matrix involving bent bonds are dependent on the orientationof the plane containing the bent bond. This introduces unnecessary complicationsand consequently, G matrix elements have been calculated using the so-called " rotat-ing bent bond " (RBB) modelY4 in which the oxygen atom is considered to occupypoints on a circle around Si-Si axis, fig.2b, and G matrix elements are averaged overthis " orbit ". Using this method, a structure involving non-linear Si-0-Sibonds may be treated using the much simpler geometry and higher symmetry of alinear model. Furthermore, since an average over all configurations of the bentbond is involved, the method is especially useful in the present context of disorderedsystems.Although calculations using RBB models give approximate eigenvalues, theaccuracy is good ; the spectrum for sodium metasilicate has been calculated using theFIG.2 4 a ) exact structure ; (b) R.B.B. model ; (c) linear bond model ; (d) hypothetical TI structurewith one SiOB group per unit cell.exact structure (fig. 2a) and also by the RBB approximation, (fig. 2b). Forfrequencies greater than 250 cm-l, the RBB calculation shows an r.m.s. error ofless than 3 % ; at lower frequencies the accuracy decreases since spurious eigenvaluesare introduced.The spectrum of a linear bond model of cristobalite of symmetry Oh has previouslybeen calculated by the Wilson GF matrix method.3* This calculation was repeatedwith RBB G matrix elements for values of E ranging from 120 to 180", using thefollowing diagonal valence force field :Si-0 stretch,f,, 4.2. m dyn/A0-Si-0 bend,&, 0.3 m dyn/ASi-0-Si bend,f,, 0.03 m dyn/AP .H. GASKELL3 3w 2 h.U9n 3Q3w851 I I I I1/ ;; \; ;; ' I 'II\IiIIIIIi I! I 4 I , 1E (degrees)FIG. 3.-Dependence of the spectrum of cristobalite on values of the oxygen bond angle E. Fullline G matrix effects. Dotted line F matrix effects. (Effects on the highest frequency Flu modedue to force constant changes are very small and have been omitted.)Results are shown in fig. 3. High-frequency modes are strongly dependent on E,whereas the lower frequency bending modes are almost unchanged. Furthermore, theinfra-red active modes of species Flu, decrease in frequency as E decreases, whereasthe frequency of the Raman active mode F,, increases. Using the values given infig. 1 for the distribution of values of E in silica glass, an estimate can be made o86 SPECTRA OF SILICATE GLASSESthe width of each band in the spectrum.The result of this calculation is shownin fig. 4.The band shapes of the Flu modes are directly comparable to the spectrum of theimaginary part of the complex dielectric constant, 2 n 2 ~ . This quantity has beenobtained by a Kramers-Kronig transform of the experimental reflection spectrum ofvitreous silica and is shown, together with similar data for quartz, from 1200 cm-Ito 900 cm-1 in fig. 5. The distribution of oscillators for the Flu mode closely corres-ponds to the curve for n21c for vitreous silica. The half-width of the Raman activemode F2g is 100 cm-l which is of the same order as that found in the Raman spectrumof vitreous silica (70 cm-l).I C6XE42I200 1100frequency (cm- l)FIG.5.4maginary part of the dielectric constant for vitreous silica (full line) and quartz (dotted)compared with the breadth of the Flu mode for cristobalite (dashed line) computed using the distri-bution of oxygen bond angles shown in fig. 1.The second infra-red active mode shown in fig. 4 is narrow (half-width -20 cm-l),whereas the half-width of this band obtained from the optical constant data of Mileris about 45 cm-l. The half-width of the corresponding band in a commercial soda-lime-silica glass (containing 73 % silica) is 75 cm-l, compared with 125 cm-l for theband at 1030 cm-l.Both Flu modes and the inactive FZu mode are affected by changes in the forceconstantf, (the F matrices for the Aau and FZg modes do not containf,). Changesin the frequency of these bands are easily calculated by altering the values off, inthe F matrices : the difficulty lies in estimating the magnitude of changes in fe whichare likely to occur as E varies, since no adequate force fields exist for a completerange of silicon-oxygen compounds.The value o f f , will increase as E decreases,and comparison with carbon compounds suggests that this increase is likely to bP. H. GASKELL 87about one order of magnitude as E changes from 180 to 120". In the linear moleculesC3 and C302 bending force constants are 0.005 and 0.05mdyn/A respectively.'*Several other similar molecules have force constants around 0.1 mdyn/A? In11-paraffins with tetrahedral bond angles the = C-CH2-C = bending force constantis taken to be 0.5 rndyn/A.l0 A plausible guess of the situation is shown in fig. 6,E (degrees)FIG.6.-Suggested relationship between the force constant fE and E.and using these data, the frequencies of modes Flu and F2,, have been recalculatedand are shown in fig. 3 and 4.A decrease in E and an associated increase in fe have opposite effects on thefrequencies of u modes, resulting in the band at 450 cm-l becoming even narrower.For the FZu mode, changes in& are dominant and the band is considerably broadened.The broadening of this band is not, however, large enough to offer a completeexplanation for the low-frequency continuum which is a major feature of the Ramanspectra of most silicate glasses.Raman scattering is still apparent at 10 cm-l andfrequencies of this magnitude would require fe to become improbably small.Calculations for the effect of E on other silicate glasses have also been made. Aninteresting effect occurs in the metasilicate system (symmetry D2,,). The two bandsfrequency, cm-'1000 8 00 6 0 0 4 0 0 2 0 0I I I 1 I 1IIIIP, 1 I?--$ p,I \ h J I \ 1 1I l l \ / I \ \ 1Y 2 I rip\ I I3i- I I \ I I \h5 I I 1Ag B!" B,u BlU Ap 819 B3g 820 k BJO BFIG. 7.-Computed spectrum for sodium metasilicate with a distribution of bond angles similar tofig. 188 SPECTRA OF SILICATE GLASSESof highest frequency (B1, and BZu) are broadened while the third B3,,, which involvesa silicon non-bridging oxygen stretching motion, is completely unaffected.Assuminga range of bond angles similar to those in fig. 1, but with a maximum at 133 insteadof 144", the spectrum has been calculated and is shown in fig. 7. (Broadening throughthe F matrix is less important in this system since the matrices for gerade modesdo not containf,). Broadening is sufficient to cause the highest B,, and Bzu modesto overlap, leaving two infra-red active bands rather than the three which are foundin the crystal. This is in fact observed in the glass as shown in fig. 8.frequency (cm-')FIG. 8.-Spectra of sodium metasilicate in vitreous (full line) and crystalline (dashed line) phases.CONFIGURATIONAL CHANGES IN CHAIN SILICATESWe now examine the sensitivity of the spectra of single chain silicate lattices tochanges in the arrangement of silicon-oxygen tetrahedra. We begin with the simpleststructure, consisting of a chain with a repeat unit containing only one tetrahedralSO$- group.We refer to this as a T, structure (Fig. 2d). The TI structure wouldrequire a bond angle of 109" so that while some germanates exist in this configurationthere are no representative among silicates. The spectrum has been calculated and isshown in fig. 9. Since there is only one tetrahedron per unit cell, optical activityis restricted to modes of vibration in which equivalent atoms in each tetrahedronmove in phase.The sodium metasilicate structure, fig. 2a contains two SO:- groups in therepeat unit, T2, and is derived from the T, structure by rotating alternate tetrahedraI I 6 II I I I I I I I IIII II I I I I II1 1 I I8 0 0 6 00 4 00 200frequency (cm- l)FIG.9.-Computed spectra for (a), TI ; and (b) T2 structures c = 109P. H. GASKELL 89through 180". In this lattice, in addition to the 8 in-phase motions, there are 12modes in which equivalent atoms in adjacent tetrahedra vibrate in antiphase. Theirdistribution is shown in fig. 9. This diagram shows modes with almost identicalfrequencies in either structure, and examination of the corresponding eigen-vectorsreveals a close similarity in the atomic motions. For example, near 900 cm-l allbut one of the modes in both spectra, involve stretching of silicon, non-bridgingoxygen bonds. There is another set of modes of the T2 structure which have distinctlydifferent frequencies from any in Tl.An examination of the eigenvectors showsthat these modes have no counterparts in T I ./ 3/ /($.'FIG. 10.-Four-membered helical structurefor the metasilicate chain.To examine this in more detail, another calculation was performed to investigatethe result of doubling the unit cell once more. The models are the linear sodiummetasilicate structure, shown in fig. 2c, and a helical chain with a repeat unit consistingof four SiOg- groups, with an angle of 90" in the xy plane between the orientationsI I I I I 1I I I 1 I II 2 0 0 1000 8 00 6 00 400 200 0frequency (cm-')FIG. ll.-Computed spectra for (a), four-membered helical chain ; (b) the linear metasilicate chain.Arrows indicate IR activity and circles Raman activity90 SPECTRA OF SILICATE GLASSESof neighbouring groups, fig.10. In order to simplify the calculations, linear Si-0-Sibonds were used throughout. Results are shown in fig. 11. This diagram againindicates the presence of modes with similar frequencies, which we call groupfrequencies. Examination of the eigenvectors for these modes reveals that theycorrespond to vibrations which are of the same type in both structures and for whichequivalent atoms have the same phase relationship with their neighbours, whenviewed with respect to the local geometry of the group.Fig. 12 shows atomic displacements in modes around 1000 cm-l. The B2u modein the metasilicate structure is one in which the silicon atom vibrates in antiphasewith neighbouring silicon atoms, when viewed from the non-bridging oxygen atoms,and this motion is carried over into the B1 mode of the helical structure.Also shownin fig. 12 is a mode of species E, in which the vibrations of a silicon atom is out ofphase with its neighbours by 90". This mode is a hybrid of B1, and B2, in the straightchain .t0 0Q QFIG. 12.-Displacement of atoms in some high frequency modes of the metasilicate chain and the4-membered helix.The eigenvectors of low frequency modes are complex and difficult to interpret,but at high frequencies the pattern appears to be clear. Conversion of TI to T2, by(conceptually) rotating alternate groups through 180" around the z-axis, affects thespectrum in two ways.First, the geometry of the structure changes and secondly,optical activity is conferred on modes, inactive in the TI structure, in which equivalentatoms in adjacent tetrahedra vibrate in antiphase. In-phase vibrations of T2 differin frequency from their counterparts in TI as a result of geometrical changes only,and since the in-phase, high frequency modes have almost identical frequencies inthe two spectra, it would seem that these changes in geometry are of secondaryimportance in this spectral region, at least. Out-of-phase vibrations representnew modes which are shifted from their " parents " by both geometrical and phasP . H. GASKELL 91factors. In the helical chain, doubly degenerate modes with a phase difference 6of 90" are now optically active.The activity of other modes also changes, e.g., theBzu mode which is infra-red active in the metasilicate structure, is Raman active inthe helix.a b C dFIG. 13.-(0,6) Diagrams for several chain structures (a) TI ; (b) Tz (linear metasilicate) ; (c) T4&membered helix) ; and (d), T8 (8-membered helix).The present results are inadequate to build-up a detailed quantitative pictureover the entire spectrum, but it is possible to draw an ( o J , ~ ) diagram which qualitativelyrepresents the high frequency modes, similar to (cu,k) dispersion diagrams in latticedynamics. Fig. 13 shows the (q6) diagram for TI and the derivation of the spectrumfor the T, and T4 structures. The unit cell is doubled in going from Tl to T,,effectively halving the (w,6) diagram as shown in fig.13b. In T4, the diagram isfolded once again, fig. 13c.We then argue, as in the previous section, that a good representation of thespectrum of a glass should be obtained by taking a weighted average over all possiblestructures with values of 6 ranging from 0 to 180°, which is equivalent to averagingover the (a,@ diagram. Care is needed to separate infra-red, Raman and inactivemodes, and this may be accomplished by examination of the group tables. For ann-membered helix, with m turns per repeat unit, the factor group may be denoted asC(2 mn/n) which is isomorphous with the point group D,. In the present example,additional two-fold axes are present at right-angles to the chain axis, so that theisomorphous point group is D,.In this group, optical activity is confined to singlydegenerate in-phase modes Al and A , and doubly degenerate El and E2 modes, inwhich 6 r27c/n and 4z/n respectively. El modes are infra-red and Raman active;E2 modes are Raman active only.An average over the (0~6) diagram, fig. 13a, leads to two strong infra-red bandscorresponding to the group modes B2 and Al. The BZu mode is infra-red activein the T2 structure only. In T4 it is Raman active B1 and inactive otherwise. Infra-red and Raman active modes El occur for all values of 6 together with Ramanactive modes E,. The result will be that the two infra-red bands B1, and BZu incrystalline sodium metasilicate merge in the glass, confirming the results of the previoussection. This result may be significant, since it could be argued that the largedifferences between the spectrum of the glass and the crystal shown in fig.8 indicatea structure in the glass which bears little resemblance to a chain92 SPECTRA OF SILICATE GLASSESThe Raman spectrum of crystalline sodium metasilicate has not yet been measuredso that assignments of the modes observed in this glass, fig. 14, is difficult. Thespectrum, is, however, consistent with the pattern shown in fig. 13.At low frequencies, modes of vibration become complex and it is difficult toanalyze the eigenvectors to give even a qualitative appraisal. One important pointis apparent, however. As each tetrahedron is added to the repeat unit, a total offour extra modes, which represent low frequency translational orI I iI I I I I I I I I1000 8 00 6 0 0 400 3frequency (cm-')FIG.14.-Raman spectrum of vitreous sodium metasilicate.rotational0modes of the Tl structure, become optically active. Although it is not easy to locatethese vibrations, it is likely that they will all lie below 200 cm-l and since they are theequivalent of acoustic modes in TI they will be strongly dependent on 6 andapproach zero frequency as 6 approaches zero. An average over the (m,6) diagramwould thus lead to optical activity extending down to zero frequency, and couldplausibly be responsible for the Raman continuum and the excess infra-red absorptionnoted in silicate glasses. These modes have not been created by disorder but aremodes which correspond to the acoustic modes of the crystal which have been pro-moted to optical activity by the disorder.CONCLUSIONSThe distribution of bond angles E which is an essential feature of all silicate glasses,is a major factor in determining the shape of the vibrational spectrum.Simplecalculations using models, based on the unit cell of the crystal, help to explain thedifferences observed in the spectra of crystals and glasses. In contrast, the calcula-tions of the previous section are tedious and application to two- or three-dimensionalstructures would be even more difficult. Some of the points which have emergedwill be quite general, however. Group frequencies, although difficult to calculatein three-dimensional structures, will still exist and may be identified in materialswhich crystallize in several structural forms.For instance, silica exists in five forms,several with high- and low-temperature modifications. Frequencies which recurin each structure are the equivalent of group frequencies in the linear chain and wilP . H. GASKELL 93v) *g 2 0 0 -E Es 3un 100- 3Q W'therefore also be present in the spectrum of the glass. A more quantitative guide isthe dispersion diagram of the crystal. This gives an indcation of the shape of the( q 6 ) hagram, but since it does not contain information from changes in geometry,it is not identical.Fig. 15, shows the density of states derived from the dispersions diagram forquartz computed by E1combe.l This displays the major features characteristicof the spectrum of silica glass, isolated high-frequency modes and a continuum ofstates at low frequency.t noI I I I I 1 I I IIL I I I II 2 0 0 1000 aoo 600 4 0 0 2 00frequency (cm-l)diagram of Elcombe.FIG. 15.-Density of vibrational states for wave vectors along the x-axis ; derived from the dispersionThe general conclusion from this work is that an uncritical interpretation of thespectrum of a glass using a model of the unit cell of the crystal as a basis is likely tobe misleading. With care it should be possible to derive some useful and detailedstructural information. For instance, an analysis of the shape of the infra-redspectrum of vitreous silica could be used to explore changes in the distribution of Ewith, say, temperature. Equally, an analysis of the shape of the Raman band of themetasilicate at 970 cm-l should give directly the distribution of values of S in meta-silicate chains.B. E. Warren and R. L. Mozzi, J. Appl. Cryst., 1969, 2, 164.F. Liebau, Acta Cryst., 1959, 12, 177.P. H. Gaskell, Trans. Faraday SOC., 1966, 62, 1493.P. H. Gaskell to be published.P. H. Gaskell, Phys. Chem. Glasses, 1967, 8, 68.M. Miler, Czech. J. Phys., 23, 1968, 18, 354.F. A. Miller, D. H. hmmon, and R. E. Witkowski, Spectrochim. Acta, 1965,21,1709.W. H. Smith and G. E. Leroi, J. Chern. Phys., 1966,45, 1778.' L. Gausset, G. Herzberg, A. Lagerqvist and B. Rozen, Astrophys. J., 1961,142,45.lo R. G. Snyder and J. H. Schactschneider, Spectrochim. Acta., 1965,21, 169.l 1 M. H. Elcombe, Proc. Phys. SOC., 1967,91,947

ISSN:0366-9033    

DOI:10.1039/DF9705000082

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Infra-red and Raman Spectra of GlassesPart 2.-Far Infra-red Spectrum of Vitreous Silica in the Range 100-15 cm-l.*BY P. T. T. WONG AND E. WHALLEYDivision of Chemistry, National Research Council of Canada, Ottawa, CanadaReceived 15th June, 1970The infra-red absorptivity of vitreous silica has been investigated in the region 100-15 cm-' at300 and 100 K. Above 18 cm-I the absorptivity is independent of temperature and is undoubtedlycaused by fundamental vibrations. A theory suggests that the absorptivity divided by the frequencysquared should be more closely related to the density of vibrational states than should the directspectrum. There is a maximum in this function at about 38 cm-' which is due to the vibrations thatcan be considered as derived from the transverse acoustic maximum of cristobalite when it is vitrified.There is no evidence of fine structure in the infra-red spectrum, and it seems likely that the greatlyincreased heat capacity of vitreous silica at low temperatures over that of cristobalite is partly causedby the broadening of the transverse acoustic maximum.The interaction of light with perfect crystals has been understood in principle formany years.The lattice excitations can be expanded as plane waves in the nucleardisplacements, each characterized by a wave vector. The wave vector of the excitationplus that of the light wave must be conserved when energy is transferred between thelattice excitation and the light wave. This means that out of vibrations ormore of a mole of a typical crystal only a few can interact with light.There are,however, no perfect crystals, and so these selection rules are never exact. They areuseful only because it is not difficult to make crystals pure enough and perfect enoughthat departures from wave-vector conservation are unimportant.The effect of dissolved impurities on the infra-red spectra of crystals was firstdiscussed by Lifshitz and recently the spectroscopy of slightly impure crystals hasbecome an important branch of solid-state physics.But many materials, and particularly many that are of great interest to chemists,are not crystalline and are often liquid. The lack of crystallinity has not inhibitedthe experimental spectroscopy of these substances, but it has prevented a detailedunderstanding of the spectra.The main difficulty is a theoretical one. Thetheory of perfect crystals is tractable because the translational symmetry allowsenormous simplifications of the algebra, and these cannot be made when the trans-lational symmetry is absent. The dynamics of a perfect crystal containing N unitcells each containing n atoms, so that there are 3nN normal vibrations, can be written,if surface effects are ignored, in terms of N 3n x 3n matrices, whose eigenvalues andeigenvectors are to be calculated. These N matrices are identical except for thevalue of the wave vector. An adequate description of the vibrations can be obtainedby calculating the roots of only a few of these N matrices. The dynamics of anon-crystalline material having the same number of atoms can be written only as* N.R.C.C.no. 115459P. T. T. WONG AND E . WHALLEY 95a 3nN x 3nN matrix, and no way has yet been found of calculating say, the distributionof frequencies, except by making N a small number and solving the matrix.There is, however, another approach to the spectroscopy of grossly disorderedsubstances and it is a purpose of this introduction to describe the progress made alongthis path in our laboratory. It is to start from the perfect crystal, which is wellunderstood both theoretically and experimentally, and to approach the vitreousstate gradually. An orientationally disordered crystal is intermediate between aperfect crystal and a glass in that, although there is long-range order in the molecularpositions, there is no long-range order in the molecular orientations.The connectionwith perfect crystals is even closer, in that some orientational order-disorder transitionsare gradual and might occur by a gradual decrease in the long-range correlationof the orientations.From the point of view of vibrational spectroscopy, there are two extreme kindsof orientationally disordered crystal. The first comprises those crystals whosemolecular reorientation time is longer than, say, 100-1OOOps so that the rate ofmolecular reorientation does not greatly affect the optical spectrum ; they can beconsidered as intermediate between perfect crystals to glasses. The second comprisesthose whose reorientation time is shorter than, say, 10-100 ps ; they can be consideredas intermediate between perfect crystals and liquids, in which there is not only nolong-range correlation of either positions or orientations, but also rapid reorientation.Only the first kind will concern us here.The vibrations of molecules in crystals call be approximately divided into twokinds, the internal vibrations that occur in the vapour phase and that are perturbedwhen the crystal is formed, and the external vibrations that can be considered asderived from the rotations and translations of the isolated molecule.The reasonthe division can be usefully made is that the internal and external vibrations are oftenfar enough apart in frequency that little coupling occurs between them. If moleculescannot be recognized in the crystal, as e.g., in solid silica, this separation cannot bemade.Some progress has been made in understandmg the spectra of orientationallydisordered molecular crystals.Three approximations have been made. The firstis that the translational vibrations are sufficiently separated in frequency from therotational and the internal vibrations that they can be treated in isolation. Thesecond is that the forces generated by the translational displacements of moleculesare independent of the molecular orientation. The third is that there is no correlationbetween the orientations of neighbouring molecules. The second and third may be asevere approximation for some crystals,* but they greatly simplify the theory. Thetheory leads to the prediction that not only do the zero-wave-vector vibrations thatare allowed by the factor group absorb light, but in addition all the lattice vibrationsabsorb, each very weakly, but with an important cumulative effect.On the basisof some reasonable approximations, the integrated intensity of these disorder-allowed vibrations is proportional to the square of their frequency. Consequently,the absorptivity divided by the frequency squared should be closely related to thedensity of vibrational states. The far-infra-red spectrum of ice, in which all thevibrations are disorder-allowed, can be interpreted both as to frequency and as tointensity in terms of intermolecular force constants and dipole moment derivatives,and indeed has yielded the first measurement of the dipole moment derivative forthe stretching of a hydrogen bond? At frequencies low enough that the densityof states can be described by the vibrations of a continuum, i.e., when the wavelengths are much longer than the lattice spacings, the density of states is proportionalin taking account of the correlation of orientations.* Some progress has recently been mad96 INFRA-RED AND RAMAN SPECTRA OF GLASSESto the square of the frequency, and so the absorptivity is proportional to the fourthpower of the frequency? This prediction has been verified for ice below about40 cm-l .6The theory for orientationally disordered crystals can be usefully generalized toglasses,' and the details will be published elsewhere. In the present paper, measure-ments of the absorptivity of vitreous silica are described and are interpreted in thelight of this theory.EXPERIMENTAL AND RESULTSThe vitreous silica was General Electric type 101 of stated purity 99.97-99.98 %.About one third of the total impurity is alumina, and the remaining is Na20, Fe,O,,CaO, Ti02, K20 and Li20.It was used as rods 30 mm diam. and 0.5, 1, 3, 5, 10and 30 mm long. The end faces were polished optically flat and the outer surfaceswere ground. The most bubble-free parts of the original bodes were used to makethe samples. The largest bubble in the samples used was 150 pm diam., and in the30mm sample the bubbles obscured only about of the area. There were5 bubbles bigger than 100 pm diam. in the 10-mm, 21 in the 30-mm, and none in theother samples.Bubbles can therefore have no significant effect on the transmission.Four samples of different thicknesses were loaded in a four-sample-holder low-frequency infra-red cell? Each of the sample holders could be rotated into thebeam in turn without otherwise disturbing the cell or the samples.The spectra were measured with a Perkin-Elmer model 301 infra-red spectro-photometer operated in the single-beam mode as described el~ewhere.~-~ Measure-ments were made at 300 and 100 K. A few confirmatory measurements were madeat room temperature through the courtesy of Dr. D. D. Klug with an R.I.I.C. LR-1lamellar grating spectrometer which was obtained towards the end of these measure-ments.In the range 100-31 cm-l the signal intensities were measured in the scanning modeusing recorder response 2 (half of full scale travel in 20 s) at a speed of 1 cm-l min-l.The values so obtained were checked with a Hewlett-Packard model 2401 C integratingdigital voltmeter at 4 cm-l intervals in the range 100-50 cm-l, at l-cm-l intervals inthe range 50-41 cm-l, and at 2-cm-1 intervals in the range 41-31 cm-l.Below31 cm-l the signal intensity was measured at intervals of 1 cm-l with the integratingvoltmeter using integration times of 5 min in the range 31-22 cm-l, 6 min in the range21-18 cm-l, and 10 min below 17 cm-l.To avoid the effects of reflection, samples of vitreous silica of the following thick-nesses were used as reference : 0.5 mm in the range 88-31 cm-l, 1 mm in the range31-17 cm-l, and 3 mm below 17 cm-l.Nevertheless, samples of different thicknesshad different apparent absorptivities. This is undoubtedly due to the effects ofhigher-order radiation reflected from the grating.* Above 16 cm-l, the higher-orderlight passing any sample was less than 1 % of the total light passing. Consequently(see eqn (2) below with a = 0), a graph of the apparent absorptivity against thereciprocal of the sample length gives the true absorptivity as the intercept and theratio of higher-order to first-order light in the reference beam as the slope. Below* When the measurements were essentially completed and the strong dependence of absorptivity onwave number particularly below 30 cm-l was found, it was realized that vitreous silica alone in suitablethickness eliminated completely the effects of high-order light in a grating spectrometer, and couldbe used to limit the band pass of a Michelson or a lamellar grating interferometer.Filters such ascrossed gratings or wire mesh can be replaced, with a gain in both filtering and transmission of thespectrometer. A detailed account of this aspect is in preparationP . T. T . WONG AND E . WHALLEY 9716 cm-l, the thinnest sample passed appreciable higher-order light, and a moreelaborate analysis was used. The apparent absorptivity is(2)11where Zo(v) and Z(v) are the intensities at wave number v in the reference and samplebeams, Z0(2v) and Z(2v) are the corresponding intensities at wave number 2v,= - In (1 +f)/(l +af),andf = Z O ( 2 V ) I ~ O ( V ) ,a = exp -[K(2v)-K(v)]Z,where K(2v) and K(v) are the absorptivities of the sample at wave number 2v and v.If K(2v) is known, then K andfcan be obtained from two values of the apparentabsorptivity by successive approximations.Throughout the range, the intensity of higher-order light in the reference spectrumobtained in this way agreed as well as could be expected with the intensity measuredby reflecting from a similar grating of half the line spacing at the same angle ofincidence.Exact agreement is not expected as the reflectivity of light in first andsecond order from two gratings is not the same at the same angle of incidence. Theintensity was always less than 7 % of the first-order light above 30 crn-l. Below30 cm-l it rose steeply with decreasing frequency to a maximum of 67 % at 15 cm-l.In the range 50-41 cm-l the early spectra showed signs of a change of slope.Unfortunately, this is near the interchange between the 5- and 2.5-lines-per-mm\ qnn-v II1 , , , , , , , , , I 40.5100 80 60 40 30 20 15FIG.1.-Log-log plot of the absorptivity of vitreous silica at 300 (upper curve, right-hand ordinate)and 100 K (lower curve, left-hand ordinate). The vertical lines indicate the estimated standard errorof the absorptivity at various frequencies, and the horizontal arrows indicate the spectral slit widthsused.0.1v/cm-'98 INFRA-RED AND RAMAN SPECTRA OF GLASSESgratings. When gratings and filters are interchanged, there is usually a change inthe transmission which is presumably due, at least in part to effects of higher-orderlight.With the techniques used here the absorptivity ought not to be affected byhigher-order light (see above). This is confirmed by the observation that identicalspectra were obtained with the 5-lines-per-mm grating using the filters normallyused with either the 5- or the 2.5-lines-per-mm grating. Furthermore, essentiallyidentical spectra were obtained with R.I.I.C. model LR-1 lamellar grating spectro-meter .The absorptivity is plotted as a function of wave number on a log-log scale infig. 1. The scanning and point-by-point measurements were averaged to obtainthe plotted curves. The vertical bars show the approximate standard errors atvarious frequencies. The spectral slit widths used throughout the range are alsoshown.100 80 60 40 20 0v/cm-'FIG.2.--Plot of K(v)/v2 against v for vitreous silica at 100K.Absorptivities have been reported by several workers but only Bagdadeand Stolen l2 made allowance for the reflected light. Since the refractive index is1.95 in the region 10 to 100 crn-l,l2-l4 the reflectance from one surface is 0.1 andthe transmission is appreciably affected by the reflectance. Consequently, detailedconiparison is possible only with Bagdade and Stolen's work. The absorptivitiesat " room temperature " read from Bagdade and Stolen's l2 fig. 2 are comparedwith ours in table 1.Theindependence of the transmission on the temperature has previously been observedby Filippov and Yaroslavskii lS in the range 250-50 em-l and 300 to 750 K, byHadni l6 in the range 100-220 cm-l at 300 and 4 K, by Wadni and others lo at thesame temperatures and a frequency range that is not explicitly stated, by Plendl andothers l1 in the range 300-67 cm-l at 300 K and " helium temperature ", and byBagdade and Stolen l2 who report without giving details that " the transmission ofAbove 18 cm-1 the absorptivity changes little between 100 and 300 KP .T . T . WONG AND E. WHALLEY 99fused quartz was also found to be roughly independent of temperature between 20 Kand room temperature ”, apparently at 34 cm-l.Below 18 cm-l the absorptivity decreases strongly with decreasing temperature.Measurements were made to 12 cm-l but below 15 cm-l the absorptivities obtainedwere unreliable for instrumental reasons.Nevertheless, the strong dependence ofthe transmission on the temperature was clear. This observation is new.TABLE 1.-ABSORPTIVITIES OF VITREOUS SILICA AT 300 K ACCORDING TOPRESENT WORK AND TO BAGDADE AND STOLEN l2vlcm-11 6 . 920304050607080Klcm-1present work ref. (12)0 . 2 20.441 . 8 03.315.056.988.801 0 . 90.260.461.472.884.226.858.319.9DISCUSSIONORIGIN OF THE ABSORPTIONThe fact that above 18 cm-l the absorptivity does not change between 100 and300 K accordmg to the present measurements, and over a wider temperature rangeaccording to others, undoubtedly means that difference bands from levels whoseexcitation changes appreciably from 100 to 300 K (750 K above 50 cm-l 15) do notcontribute appreciably.Since the corresponding sum bands have the same intensityapart from the Boltzmann factor, they do not contribute significantly either. Further-more, the absorption of the crystalline phases of silicon dioxide in this region 11* l 5is much smaller than that of vitreous silica and decreases strongly with decreasingtemperature ; presumably it is largely caused by difference bands which are allowedin a perfect crystal, apart from the effects of the factor-group selection rules, if thesum of the wave vectors of the combining levels is zero. The absorptivity of vitreoussilica is therefore largely caused by fundamental or one-quantum processes. Thefact that the absorption has few identifiable features implies that many vibrations,and presumably all of them, are contributing. This conclusion was first reached byHadni and others lo and has been elaborated by Bagdade and Stolen.12 The factthat the absorptivity depends little on the temperature implies that the vibrationscausing the absorption are nearly harmonic.Although only General Electric type 101 fused quartz was used in this work,Bagdade and Stolen l 2 also examined type 106 and found no appreciable differencesin spectroscopic properties.However, types 101 and 106 appear to differ mainly inthe sizes of bubbles and in the degree of internal strain.17 The few bubbles in thesamples used in the present experiments were not enough to affect the transmission,and clearly the strain is also unimportant.The absorption is no doubt caused by the disorder of the atomic positions 9 * lowhich prevents the strict selection rules of a perfect crystal and allows all vibrations,acoustic as well as optic, to interact with light as fundamentals.Not only are there no selection rules for absorption, but there is also none forRaman scattering.Consequently, all vibrations of a glass should occur in the Ramanspectrum also. The Raman spectrum of vitreous silica has been observed b100 INFRA-RED AND RAMAN SPECTRA OF GLASSESKrishnan l8 down to 30 cm-l from the exciting line and by Flubacher and others l9and by Hass ‘O to 10 cm-l. Krishnan l8 reported a weak continuum extendingfrom 30 to 500 cm-l, and Flubacher and others l 9 reported a continuum extendingfrom 10 to greater than 60 cm-l, but falling off in intensity below about 40 cm-l.This was ascribed to “ low-frequency optical modes occurring in the lattice frequencyspectrum ”.Hass ’O found the intensity of Raman scattering to be highly tempera-ture dependent below 100 cm-l in the range 300-12 K by an amount consistent withthe scattering being due to fundamentals. There can be no doubt that essentially thesame vibrations occur in the infra-red and the Raman spectra, and that they areall the vibrations of the glass having the appropriate frequency, both optic andacoustic. In fact as is discussed below the vibrations are probably closely relatedto the acoustic vibrations.We turn next to the detailed cause of the absorption. In an earlier paper theviewpoint has been taken that the disorder of the atomic positions in a vitreous solidwill cause absorption, and the absorption was calculated in terms of atomic para-meters.Bagdade and Stolen,12 on the other hand, adopt the view that the defects inthe vitreous silica like “ an oxygen tetrahedron without a silicon at the centre, or anoxygen which is not bonded to two silicons ”, cause the absorption. They usethe theories of Vinogradov 21 and of Schlomann 2 2 in which charges are distributedin an irregular manner in a continuous medium. The normal vibrations are planewaves, and because the charges are distributed irregularly, all the normal vibrationscan absorb energy from an electric field of the same frequency. No doubt bothcauses contribute, but their relative importance cannot be determined unambiguously.Since the structure of the ideal glass, that, without broken bonds and the like, issufficient to explain the observed absorption, and there is no evidence that Bagdadeand Stolen’s l2 defects are required, the absorptivity is interpreted in the next sectionon the basis of the theory in the earlier paper.7 Furthermore, it is likely (see below)that the vibrations of vitreous silica cannot be described by plane waves in this region,and so Vinogradov’s and Schlomann’s theories cannot apply in detail.Below about 18 cm-l, the absorptivity decreases strongly with decreasing tempera-ture.The absorption that disappears at low temperature is probably caused bydifference bands from the low-frequency vibrations that are active because of electricalor mechanical anharmonicity .Absorption having a similar origin is importanteven in perfect crystals and is understood in theory.23 In perfect crystals it givesrise to an absorption that is proportional to the temperature and to the frequencysquared. Thetemperature-dependent absorption in vitreous silica is also approximately proportionalto the frequency squared, and probably has a similar origin. However, the possibiltiythat it is caused by the tails of the higher frequency fundamental bands,6 whoseintensity will decrease with decreasing temperature if the life-time increases, cannotbe rigorously excluded.Absorption having probably this origin was observed in ice I.6FREQUENCY SPECTRUM OF VITREOUS SILICAAccording to ref.(7), the absorptivity K(o) at circular frequency co is given bywhere n is the refractive index at frequency co, c the speed of light, R is an effectivecharge or dipole moment derivative defined in ref. (7), and g(w) is the densityof vibrational states per unit volume. There is little dispersion of the refractivP. T. T. WONG AND E. WHALLEY 101index in this r e g i ~ n , l ~ - ' ~ and consequently all terms on the right side of eqn(3) except 02g(co) are independent of frequency.According to eqn (3) the absorptivity divided by the square of the frequency isrelated to the density of vibrational states. While the relation may not in fact beas simple as is implied in eqn (3) because of the assumptions made in the theory, wemight expect to recognize in the spectrum of K(v)/v2 some features of the density ofstates. Accordingly, a graph of K(v)/v2 against v at 100 K is given in fig.2.There is a broad maximum centred about 38 cm-l, which is presumably amaximum in the density of states. There is independent evidence for its existencefrom both the heat capacity and the inelastic neutron scattering spectrum of vitreoussilica. The apparent Debye temperature from the heat capacity l9 has a minimum(or alternatively the heat capacity divided by the cube of the temperature has amaximum) at 10K. If it is assumed24 to be due to a monochromatic group ofvibrations, the frequency is 37 cm-l and contains about 1.5 % of the modes.19The approximation of assuming a monochromatic peak makes the excellent agree-ment with the observed peak at about 38 cm-' coincidental.In fact, the heatcapacity also requires lower-frequency vibrations, but this does not affect the fore-going argument. The inelastic neutron scattering spectrum 2 5 shows what isdescribed as " an ill-defined shoulder " in this region.This peak is straightforwardly assigned to the transverse acoustic maximum,*an assignment also made by Leadbetter.25 The main reasons for this assignment arethat the apparent Debye temperature of cristobalite has a minimum at about 14 K,26which can be explained by a monochromatic peak at about 48 cm-1 containing about2 % of the vibrational modes, and the inelastic neutron scattering spectrum has amaximum at about 40 c111-I.~~ Since cristobalite is a perfect crystal, this maximumis undoubtedly caused by the transverse acoustic branch.t Above 40 cm-l K/v2 fallsoff slowly.There is a slight shoulder at about 80 cm-l that is presumably due tovibrations derived from the longitudinal acoustic maximum of cristobalite. Thisfeature occurs as a relatively strong peak in the neutron inelastic scattering spectrum.25The sound waves should absorb light at low enough frequencies, as wasfound in ice,6 and according to ref. (7) the absorptivity should be proportionalto the fourth power of the frequency. According to fig. 2, this is so belowabout 25cm-l at 100 K. This appears to agree with the prediction for soundwaves. However, the fourth-power region should not continue to the top of thetransverse acoustic maximum since the Debye approximation is very poor here.Furthermore, the heat capacity down to 2.3 K,19 1.3 or even 0.64 K 30 is muchhigher than expected for sound waves, clearly showing that other excitations arepresent.In addition, the heat capacity divided by the temperature cubed falls withdecreasing temperature down to about 2K,19 but then increases down to at least0.64 K.29* 30 There appears therefore to be a maximum in the density of states ata few crn-l. It is possible that this is caused by impurities, but both Spectrosil andVitreosil, which have different impurities, behave the same way.29The high heat capacity has been attributed "9 29 to low-frequency optical vibra-tions of defects. Neither the infra-red spectrum reported here nor the Ramanspectrum 1 9 9 2o shows signs of features that can be assigned to specific defects.It* Strictly speaking, the maximum derived from the transverse acoustic maximum of cristobalitewhen it is vitrified.t The heat capacity of cristobalite at low temperatures has been considered to be anomalouslyhigh 27-29 probably because it was not considered that the transverse acoustic maximum might beat much lower frequencies than in crystalline quartz. However, both the heat capacity and theinelastic neutron scattering spectrum 2 5 indicate that this is so. The reason is not known1 02 INFRA-RED AND RAMAN SPECTRA OF GLASSESseems more likely that the infra-red absorption down to 15 cm-1 and the Ramanscattering down to 10 cm-1 are caused by a transverse acoustic peak broadened by theeffects of the disorder, and the dependence of the absorptivity on the fourth powerof the frequency has no particular significance.It is quite possible that specificdefects will absorb in the infra-red below 15 cm-l. Experiments are underway toinvestigate this region.' I. M. Lifshitz, Zhur. Eksp. Teor. Fiz., 1942, 12, 117.E. Whalley and J. E. Bertie, J. Chem. Phys., 1967, 46, 1264.D. D. Klug and E. Whalley, to be published.J. E. Bertie and E. Whalley, J. Chem. Phys., 1967, 46, 1271.E. Whalley and H. J. LabbC, J. Chem. Phys., 1969, 51,3120.E. Whalley, to be published.A. Handi, G. Morlot, X. Gerbaux, D. Chanal, F. Brehat, and P. Strimer, Compt. rend., 1965,260,4973.J. Plendl, L. C. Mansur, A. Hadni, F. Brehat, P. Henry, G. Morlot, F. Naudin and P. Strimer,J. Phys. Chem. Solids., 1967, 28, 1589.W. Bagdade and R. Stolen, J. Phys. Chem. Solids, 1968,29, 2001.0. K. Filippov and N. G. Yaroslavskii, Opt. Spekt., 1963, 15,558 ; English trans. Opt. Spect.,1963, 15,299.A. Hadni, Ann. Phys., 1964,9,9.General Electric Specifications and Technical Data.R. S. Krishnan, Proc. Indian Acad. Sci. A , 1953, 37, 377.1959,12,53.' J. E. Bertie, H. J. LabbC and E. Whalley, J. Chem. Phys., 1969, 50, 4501.* P. T. T. Wong, J. E. Bertie and E. Whalley, Reu. Sci. Instr., 1970,41,283.lo A. Hadni, J. Claudel, X. Gerbaux, G. Morlot and J.-M. Munier, Appl. Optics, 1965,4,487.l 3 R. Geick, 2. Phys., 1961, 161, 116.l4 J. J. Taub and H. J. Hindin, Reu. Sci. Instr., 1963, 34, 1056.l9 P. Flubacher, A. J. Leadbetter, J. A. Morrison and B. P. Stoicheff, J. Phys. Chem. Solids,2o M. Hass, J. Phys. Chem. Solids, 1970,31,415.21 V. S. Vinogradov, Fiz. Tuerd. Tela, 1960, 2, 2622; English trans. Soviet Phys.-Solid State,22 E. Schlomann, Phys. Reu. A, 1964,135,413.23 B. Szigetti, Proc. Roy. SOC. A , 1960,258, 377.24 E. Katz, J. Chem. Phys., 1951, 19,488.25 A. J. Leadbetter, J. Chem. Phys., 1969, 51, 779.26 see 0. L. Anderson and G. J. Dienes, Non-crystalline Solids, ed. V. Frechette, (John Wiley and27 0. L. Anderson, J. Phys. Chem. Solids, 1959, 12,41.2s A. J. Leadbetter and J. A. Morrison, Phys. Chem. Glasses, 1963, 4, 188.29 G. K. White and J. A. Birch, Phys. Chem. Glasses, 1965,6,85.30 E. W. Hornung, R. A. Fisher, G. E. Brodale and W. F. Giauque, J. Chem. Phys., 1969,50,4878.1961, 2, 2338.Sons, New York, 1960), p. 449

ISSN:0366-9033    

DOI:10.1039/DF9705000094

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Raman Scattering and Far Infra-Red Absorption in NeutronCompacted SilicaBY R. H. STOLEN,* J. T. KRAUSE t AND C. R. KURKJIAN f-Received 17th June 1970In vitreous silica, which has been compacted by irradiation with fast neutrons, the ultrasonicabsorption, the excess heat capacity and the magnitude of the negative coefficient of thermal expansiondecrease. In this work we report measurements of the far infra-red absorption and Raman scatteringin such neutron compacted silica. These results are correlated with new measurements of ultrasonicabsorption and thermal expansion in the same material. The results are consistent with an inter-pretation based on a decrease in the number of low-frequency modes.Amorphous solids possess a variety of properties which differ from the propertiesof corresponding crystalline forms and which appear to be related to the presence oflow-frequency optical modes.Among these materials, neutron-irradiated silica isof particular interest. The reason for this interest is that structural differences ascompared with ordinary fused silica are fairly small while there is a sizeable decreasein low-temperature excess heat capacity,l ultrasonic absorption and in the magnitudeof the negative expansion ~oefficient.~ The normal mode frequencies should besensitive to small structural changes and in this work the changes in these frequenciesare studied both by Raman scattering and infra-red absorption. In order better torelate the observed frequency changes to other properties, ultrasonic absorption andexpansion coefficient are also measured.EXPERIMENTALThe samples used were Optosilz fused quartz.The irradiated samples received anintegrated flux of about 1020/cm2 of neutrons with energies greater than 10 KeV. The timelapse between irradiation and this study has been about 7 years during which time somesmall amount of annealing has doubtless taken place. Densities of the irradiated andunirradiated samples were 2.255 g and 2.201 g/cm3.The irradiated samples were a faint violet colour which arises from a broad absorptionband at about 62008, ( ~ ~ 0 . 3 4 cm-l). A weaker overlapping band ( ~ ~ 0 . 2 3 cm-l) iscentred at about 5250 A. At wavelengths shorter than 4000 8, the absorption rises rapidlydue to the many radiation induced centre^.^ The broad band at 5250A appears stronglyin fluorescence which complicates the taking of Raman spectra.The half-intensity pointsof the fluorescence band are roughly 5100 and 5500 A and are not particularly dependent onthe exciting line.The Cd+ 4416 A laser line was used for most of the Raman measurements since it fallsin the region of minimum absorption of the irradiated silica. Laser power was about50mW. A Spex double spectrometer was used to resolve the spectrum. The gratingblaze wavelength w+ 5000 A and the peak efficiency of the " S " cathode of the EM1 6256photomultiplier was 3800 A. The spectrum was not corrected for these frequency dependentfactors. Slit widths were 150 microns for most of the spectral region resulting in a resolution* Bell Telephone Laboratoties, Incorporated, Holmdel, New Jersey 07733$ Amersil Quartz Division of Engelhard Industries.Bell Telephone Laboratories, Incorporated, Murray Hill, New Jersey 07Y7410104 RAMAN SCATTERING IN NEUTRON COMPACTED SILICAof 5 cm-l.The scans were repeated for small frequency shifts with a resolution of 2 cm-l.Comparison of scattering intensities between different samples is difficult because of align-ment problems. In this case, however, the samples were the same size and shape, and somecare was taken to insure that the scans were made under similar conditions. The relativeintensities for different samples should be trustworthy to at least 15 %.The infra-red transmission was measured point by point between 10 and 80 cm-l usinga grating spectrometer.Resolution was about 1.5 cm-l over this range. Several sampleswere used ranging in thickness from 0.85 mm to 3.14 cm. The absorption per cm wasdetermined from the transmission using the relation,(1)which includes the effect of multiple reflections within the sample. a is the absorption percm, d is the thickness in cm and R is the reflectance from a single surface.T = [(l- R)' exp (-ad)]/[l- R2 exp (-2ad)J,t I I I I I I I I0 100 200 300 400 500 600 700 800 900frequency (cm-l)FIG. 1.-Room temperature Stokes Raman spectra for (A) unirradiated and (B) irradiated Optosilfused quartz. The scattered intensity in arbitrary units is plotted against the frequency shift in unitsof cm-l.The acoustic data were obtained at 20 MHz for shear waves using the pulse-echo andpulse-superposition techniques which have been described earlier.6 The thermal expansionmeasurements were made with a capacitance bridge modelled after that of White.'RESULTSThe Stokes Raman spectra of the irradiated and unirradiated Optosil samples areshown in fig.1. The weak features at 1060 and 1180 cm-' observed in ordinarysilica * appear to be obscured by the fluorescence; these frequencies are not shownhere. The gradual rise in intensity around 800 cm-' for the irradiated sample isprobably from the background fluorescence. Polarizers absorb somewhat at thesewavelengths and were not used with the result that the spectrum shown is a sum ofpolarized and depolarized components.A depolarized spectrum was obtained bydirecting the laser beam so that the direction of the electric field was along the scatter-ing direction. In the depolarized spectra (not shown here) only the low-frequencyband around 60 cm-l and the bands at 800 and 1100 cm-l appear. The sharp linesat 500 and 600 cm-' and the broad band from 200 to 500 cm-l are absent. For the60 and 800 cm-l bands, the depolarized intensity is less than 25 % of the polarizeR . H . STOLEN, J . T . KRAUSE AND C. R. KURKJIAN 105intensity. For the low-frequency depolaxized band the maxima in the Stokes andanti-Stokes intensities occur at 65 and 55 cm-1 in the irradiated silica as comparedwith 52 and 45 cm-l for the unirradiated material.The infra-red absorption for the irradiated and unirradiated glasses are plottedin fig. 2, as a/v2 where a is the absorption coefficient per cm and v is the frequency incm-l.This type of plot enables a better comparison between the infra-red absorp-tion and the Raman scattering at low frequencies. In correcting for reflection lossa refractive index of n = 1.95 was used for both irradiated and unirradiated silica.The error bars in fig. 2 show the error in a/v2 introduced by an error of 3 % in theindex. This error is about the same as the uncertainty fiom other sources.2.0 X I O - ~ t 1.8 -1.6 -1.4 -1.2 -N < 1.0 -0.8 -0.6 -0.4 -0.2 -01 I 1 I I 1 I I0 10 2 0 30 40 50 60 70 Efrequency (cm- l)FIG. 2.-Room temperature absorption coefficient per cm(ar) divided by frequency squared (v2)against frequency in units of cm-', for (A), unirradiated ; &d (B), irradiated Optosil fused quartz.The error bars represent the uncertainty caused by a 3 % error in the refractive index.The thermal expansion coefficients are in good agreement with the data of Whiteand Birch for both the irradiated and unirradiated samples below 10 K, with theexpansion coefficient becoming less negative by a factor of 3-5 upon irradiation.Above 10 K there are increasing discrepancies, presumably from a combination ofmaterial behaviour and cell corrections.As was found by Strakna,l irradiation causes a substantial decrease in the lowtemperature acoustic loss.While Strakna found the peak loss in his irradiatedsample decreased to 15 % of that in the unirradiated sample (Q-1 = 17.9 xto 2.7 x the loss in the irradiated sample in the present investigation decreasedto 30 % of the original sample (Q-l = 19.2 x The temperaturecoefficient of velocity decreased upon irradiation to essentially 30 % of its originalvalue, in substantial agreement with that found by Strakna.l0 The peak positionwas essentially 50 K for all three samples, although the precision in this determinationfor the irradiated sample i s poor because of the breadth of the peak.to 5.5 106 RAMAN SCATTERING IN NEUTRON COMPACTED SILICADISCUSSIONThe primary structural change in vitreous silica after neutron irradiation is adecrease in the average Si-0-Si bond angle from 142 to 138".There appears tobe no change in the Si-0 bond length.''.l2 The higher frequency features of bothRaman scattering and infra-red absorption are well understood in terms of such amodel. The band at 1120 cm-l which was obscured by fluorescence, decreases 12* l4to about 1llOcm-' on neutron compaction. This band corresponds to oxygenmotion along the Si-0-Si bond.13 Upon compaction, the band at about 800 cm-lincreases in frequency l4 by about 10 cm-l. This band also appears in both Ramanscattering and infra-red absorption and arises from a bond bending vibration in whichthe oxygen moves in the Si-0-Si plane? The observed frequency shifts areboth in quantitative agreement with a 4" change in the Si-0-Si bond angle.12* l4Certainly the most striking change in the Raman spectrum is the line which changesfrom a weak feature at 603 cm-l in ordinary fused silica to a strong relatively sharpline at 609cm-l in the irradiated material.Bands have been observed in somesilicates around 600cm-I but the relation between these bands and the featureobserved in fused silica is not obvious.16 It is clear, however, that either the smallstructural change has resulted in an enormous change in Raman cross-section or thatthe mode is associated with some sort of defect which is created in large numbers byneutron irradiation. There appears to be no direct relation between the sharp featureat about 500cm-l in the Raman spectrum and the strong infra-red active modereported at 470 cm--'.13 The line in the Raman spectrum shifts to higher frequencyupon neutron compaction while an infra-red reflection measurement (using a Perkin-Elmer 421 i.-r.Spectrometer) shows a shift of about 10 cm-l to lower frequencies.This shift of the infra-red band with increasing density is consistent with the resultsof Gaskell and Grove who found a shift to lower frequency as the density increaseswith higher fictive temperature.l'It is the broad feature at the lowest frequencies which is probably directly relatedto the various anomalous effects found in amorphous materials. The most noticeablechange is the shift of the band to higher frequency by about 15 cm-1 which occursin both infra-red absorption and Raman scattering. If both scattering and absorptionarise from the same band of low-lying modes it is not surprising that they should showsimilar behaviour.A similar correlation between Raman scattering and infra-redabsorption has been noted in vitreous GeO, and B203.' The factor of frequencysquared which connects the scattering intensity and the absorption coefficient isdiscussed in ref. (9). The important feature of the present work is that both thescattering and absorption show evidence for a reduction in the number of modesbelow 40cm-l. This reduction is required by those models which associate thelow-temperature anomalies with these low-frequency modes, since as indicated, theultrasonic absorption, excess heat capacity and magnitude of the negative thermalexpansion all decrease upon irradiation.Pine has shown that these modes may be related to the ultrasonic absorption byan Akhiezer process.18 Here, the sound wave modulates the frequency of themodes which reach equilibrium only after some relaxation time z.This leads to anabsorption of the form :C,Ty2 co%a--pv3 l+(m)2'C, is the heat capacity, p is the density, tr is the sound velocity, T the temperature, cothe angular frequency and y the Griineisen constant which is a measure of the anharR. H. STOLEN, J . T . KRAUSE AND C. R . KURKJIAN 107monicity. A complete analysis of the ultrasonic absorption as a function of frequencyand temperature would involve a sum over contributions of the form of eqn (2) wherey , z and Cv are known over the distribution of modes. However, an average y can beobtained from the volume coefficient of thermal expansion /3where Y is the volume and xT is the isothermal compre~sibility.~~ The averageGruneisen constant becomes large and negative at low temperatures which impliesthat y becomes larger for the lowest frequency modes.Present thermal expansiondata show a decrease at 20 K of about a factor of 3. A decrease of about 20 % incompressibility lo and 3 % in volume occur on irradiation. The heat capacity hasnot been measured but if the decrease in infra-red absorption reflects a decrease inthe density of states this would amount to a decrease of about 40 %. This is inagreement with the specific heat results of Strakna on similar material.2 From eqn(2) one might then expect a decrease in ultrasonic absorption in irradiated silica toabout 20 % of the unirradiated value.Present measurements show a decrease inabsorption to about 30 %, while Strakna obtained a decrease tq 15 % of the in-irradiated value. In view of the possible differences in radiation dosage and thepossibility of some annealing in our samples, this is not a large discrepancy.First,the frequency shifts resulting from compaction with neutrons are in agreement withfrequency shifts resulting from compaction due to heat treatment (i.e., fictive tempera-ture change). Secondly, the decrease in the number of low-frequency modes is inagreement with known decreases in heat capacity, ultrasonic absorption and thermalexpansion. Finally, the feature at 609 cm-1 remains a mystery, the resolution ofwhich may shed more light on the basic glass structure.Yave = PVXTC", (3)In conclusion it would appear that three general observations are obtained.We thank L.E. Cheesman, P. A. Fleury and J. F. Scott for their help with theRaman measurements, E. A. Sigety for help in measuring density and thermal expan-sion and R. E. Jaeger for supplying the samples.R. E. Strakna, Phys. Rev., 1961,123,2020.A. E. Clark and R. E. Strakna, Phys. Chern. Glasses, 1962,3, 121.G. K. White and J. A. Birch, Phys. Chem. Glasses, 1965, 65, 85.G. Hetherington and K. H. Jack, Phys. Chem. Glasses, 1962,3, 129.P. W. Levy, J. Phys. Chem. Solidr, 1960,13,287.C. R. Kurkjian and J. T. Krause, J . Amer. Cerurn. Soc., 1966,49, 139.G. K. White, Cryogenics, 1960, 1, 151. ' P. Flubacher, A. J. Leadbetter, J. A. Morrison and B. P. Stoicheff, J. Phys. Gem. Solids, 1960,12, 53.R. H. Stolen, Phys. Chem. Glasses, 1970, 13, 83.lo R. E. Strakna, A. E. Clark, D. L. Bradley and W. M. Slie, J . Appl. Phys., 1963.34, 1439.W. Primak, Phys. Reo., 1958,110, 1240.l I. Simon, J. Arner. Ceram. Soc., 1957,40, 150.l3 E. R. Lipphcott, A. V. Vallcenberg, C. E. Weir and E. N. Bunting, J . Res. Nat. Bur. Stand.,1958, 61, 61.l4 D. M. Dodd, unpublished data.l5 R. J. Bell, N. F. Bird and P. Dean, J. Phys. C. (Proc. Phys. Soc.,) 1968, 1, 299.l6 P. H. Gaskell, Phys. Chem. Glasses, 1967, 8, 69.l7 P. H. Gaskell and F. J. Grove, Proc. 7th Znt. Conf: Glass (Brussels, 1965).l8 A. S. Pine, Phys. Bea, 1969,185, 1187.l9 G. K. White, Cryogenics, 1964,4,2

ISSN:0366-9033    

DOI:10.1039/DF9705000103

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

GENERAL DISCUSSIONDr. N. I(. Hindley (Corning Glass Works, Corning, N . Y.) said: Callaerts et al.have discussed their data using equations which are valid for crystals. They arenot valid in amorphous semiconductors for at least two reasons. First, since crystalmomentum is not a good quantum number, the concept of effective mass is meaning-less. Secondly, near the band edge, even in the non-localized conducting states,the mean free path is so short that the Boltzmann equation is inapplicable. Thereare no exact equations like those we have for crystals, but I have derived some approxi-mate equations. They are based on the assumption that there is a mobility edge,and that the mean free path, in the conducting states near the edge, is very short,of the order of the interatomic spacing.This work is to be published in detail inJ. Non-Crystal. Polidi, but the results needed here can be summarized. Theconductivity CT is givenwhere ooe, (Toh are independent of temperature, and are proportional to the squareof the density of states at the mobility edge. Thus, the factor T3, usual for crystals,does not appear. re is the distance from the Fermi energy to the conduction bandmobility edge, and similarly for ch. The thermopower Q is the usual combination ofelectron and hole contributions,withThus, the constant term is + 1, instead of (5/2-s), with s depending on the scatteringmechanism, as in crystals.In analyzing experimental data, both electrons and holes must be taken intoaccount, in general, and it is also important to include the temperature dependenceof the energy gap.In the experimental results I have examined previously, I haveassumed re= c h = &Eg, but for some of the materials studied by Callaerts et al.it may be necessary to use the more general formulae. In spite of the approximationsthat go into this work, I would suggest that it is more appropriate to use these formulaethan the crystalline ones. However, this change would not have much effect on theconclusions drawn by Callaerts et al.= exp (-ce/kT)+aoh exp (-ChlkT), (1)OQ = ceQe+ahQh, (2)Qe = -(k/e>(L/kT+ 1) ; Qh = +(k/eK/kT+ f)* (3)Prof. A. R. Cooper (Case Western Reserve University, Ohio) said: With regardto the paper by Callaerts et al., the electrical conductivity Q depended markedly onthermal history.Does CT increase with annealing? and would they comment onthe effect of thermal history on the other properties, Hall mobility and thermoelectricpower ?Dr. P. Nagels (S.C.K./C.E.N., Belgium) said: In reply to Cooper, the electricalconductivity a decreased with annealing, e.g., for a CdGe, .0As2 quenched samplethe room temperature a-value decreased by a factor of 2, after it was annealed for2 h at about 200°C. The Hall mobility pH was measured on two samples of 0.3 molcomposition, one cooled in air and the other one quenched in ice-water. We observedthat the value of pH of the quenched sample was approximately 5 times higher than10GENERAL DISCUSSION 109that of the other one. So far the effect of annealing on the thermoelectric powerS has not beeh studied.We are planning to investigate thoroughly the effect ofthermal history on 0, pH and S.Prof. M. Shimoji (Hokkaido University, Supporo) (communicated) : I would liketo point out some characteristics of chemical bonds concerned with the transitionfrom the state of semi-conducting crystals to that of metallic (or semiconducting)liquids.(i) Element Semiconductor: As noted by Krebs and Fischer, the sp3 hybridizedbonds of group 4 element crystals (e.g., Ge), which are responsible for their tetra-hedrally coordinated structures, change into the metallic bonds on melting. Thedata of the Hall effect and other electronic properties indicate that the number ofthe conduction electrons in liquid Ge is nearly equal to that of the valence electrons.Further, the coordination number in the liquid state is also larger than in the solidstate.These facts show that in molten Ge the formation of the symmetrical sp3localized hybrid orbitals is unstable compared with that of delocalized (metallic)orbitals.However, in group 6 elements such as Te, the semiconducting properties stillremain on melting. The complicated chain structure of solid Te can be interpretedin terms of the sp3 hybrid formation containing two lone-pair orbitals. In solidTe the angle between the two covalent bonds is 102", which is smaller than thetetrahedral angle (109.5'). Such a (two-sided) directional bonding in Te, thesymmetry of which is low compared with that in Ge, is caused by the lone-pairelectrons, and the chain-like structure is retained also in the liquid state.Thedifference between Ge and Te in the melting process can thus be accounted for bythe effect of the lone-pair electrons, (e.g., symmetrical situation of bonding: thebreakdown of the covalent bonds through the free rotational melting for Ge and thepreservation of the localized (covalent) bonds due to the restricted orientation for Te).(ii) Compound Semiconductor: The III-V compounds (e.g., InSb) show a semi-conductor-metal transition on melting, as for Ge. The tetrahedron structure ofthese compounds can be described on the basis of the sp3 hybridization associatedwith the formation of the one donor-acceptor bond,2 in which the two electrons inthe one lone-pair bond, belonging originally to group 5 element atom (i.e., Sb),are shared to both of the I11 (In) and V atoms (Sb).This bonding may involve aweak ionicity, and the change in bonding of the III-V compounds at the meltingpoint resembles that of the IV element.On the other hand, the II-VI compounds with the zinc blende structures (e.g.,CdTe) undergo a semiconductor-semiconductor transition at the melting point.In this case the two lone-pair bonds of the Te atom yield the two donor-acceptorbonds. Then, the ionicity of the bonding in the II-VI compound may be largerthan that in the III-V compound. For example, recent calculations indicate thatthe new ionicity scale,h(AB), proposed by PhillipsY3 is about 0.3 for InSb and 0.6for CdTe.The increase in the number of lone-pair electrons (or the increase inionicity of chemical bonds) may be accompanied by the restricted orientationalmelting, like pure Te, which leads to the semiconducting state. In the compoundswith the stronger ionic bonds, such as the I-VII compounds, the semiconductingV. M. Glazov, S . N. Chizhevskaya and N. N. Glagoleva, trans. A. Tybulewicz, Liquid Semi-conductors, (Plenum Press, New York, 1969), p. 83.V. M. Glazov, S. N. Chizhevskaya and N. N. Glagoleva, trans. A. Tybulewicz, Liquid Semi-conductors, (Plenum Press, New York, 1969), p. 110.J. C . Phillips, Phys. Reu . Letters 1968,20,550 ; 1969,22,645110 GENERAL DISCUSSIONproperties cannot be observed in the liquid state even if their crystal structures areof zinc-blende type as in CuI.The adequate ionicity, which is closely related to theexistence of the two lone-pair bonds in the Te atom (or restricted orientation ofbonding), may be of basic importance to the semiconduction in the moltencompounds.Dr. E. Whalley (N.R.C., Canada) said: Bell and Dean have calculated the eigen-vectors, or normal coordinates, of a considerable number of vibrations. It wouldbe interesting to continue the calculations to give integrated infra-red intensities byassuming effective charges (which might be the same) for the central and non-centraldisplacements. A histogram of the infra-red absorptivity could then be calculated,and a comparison of experimental and theoretical absorptivities would give valuesfor the effective charges.The Raman intensity could also be calculated, and therelation between infra-red absorption and Raman scattering, and the relation ofboth to the density of states, could be explored. This would be of great help inunderstanding the infra-red and Raman spectra of glasses in general, as well as ofvitreous silica.Dr. R. J. Bell (N.P.L., Teddington) said: In reply to Whalley, we have in factalready made a calculation of the intensities of selected modes, using an effectivecharge model for the infra-red activity and a bond polarizability approximation forthe Raman case. Although our analysis of the results is not yet complete, I canmake some preliminary observations about the infra-red calculation. The firstthing to note is that the calculated intensity tends to be smallest for the low-frequencyor acoustic type of mode, and largest for the high-frequency vibrations: this onewould expect from elementary considerations.It is important to realize, however,that the intensity does not show an absolutely smooth variation with frequency.On the contrary, the plot of intensity against frequency has an extremely spikyappearance in most regions of the spectrum, and it would be difficult to read an cu2dependence, for example, into the results, either within the low frequency band orprogressing through the spectrum as a whole. We have done these calculations withseveral different assignments of the effective charges, and also treating the non-bridging oxygen atoms on a separate basis. It is clear from the results that thecomputed intensities do not depend critically upon the effective charges.It seemsunlikely therefore that this type of calculation could be used to predict reliable valuesof the effective ionic charges in silica.Doz. Dr. Brigitte Eckstein (Technical University, Aachen, Germany) said : Theresults of the paper of Leadbetter et al. seem to indicate a fairly strong connectionof glassy structures with polycrystalline ones, in contrast to the assumptions of therandom network model. Quasicrystalline glassy structures, based on a defective-lattice model, have been proposed previously. Experimental results in favour ofquasicrystalline glass structures are compiled in ref. (2) and more recent resultsare given in ref. (3).One of the most simple models for non-periodic, quasicrystalline arrays is givenby the concept of the " paracrystal ".4 In a paracrystalline network, the next-' B.Eckstein, Mat. Res. Bull., 1968,3, 199.B. Eckstein, Phys. Non-Cryst. Solids, (Amsterdam, 1964), p. 77.J. Arndt and D. Stofller, Naturwiss., 1968, 45, 226.Publ. Co., Amsterdam, 1962).4R. Hosemann and S. N. Bagchi, Direct Analysis of Diffructiun by Matter, (North-HollanGENERAL DISCUSSION 111neighbour distances and thus the bonding angles vary withm a small range aboutthe mean value. The distribution functions for more distant neighbours are obtainedfrom the first neighbour ones by convolution products. Even for the most simplecase (network corresponding to a cubic lattice, distribution function N(r) of Gaussiantype), the array is described by a set of two independent parameters, the mean next-neighbour distance ro and the half width 6ro of the function N(r) for next neighbours.Variation of the value 8ro/r0 from zero to about unity gives the continuous transitionfrom crystalline to quasicrystalline to gas-like amorphous structures.In our presentdiscussions, no quantity connected with dro/ro (i.e., the actual range of the shortrange order) has been presented. Thus, an essential parameter is lacking. Appar-ently, we still lack an adequate basis for the description of vitreous structures.Prof. A. R. Cooper (Cleveland, Ohio) said: Has Springer any experience iwithcomparisons of the diffusion coefficient, D, measured by quasi-elastic neutron scatter-ing and by the homogenization of different isotopes?Prof.T. Springer (Germany) said: There is a number of measurements of self-diffusion coefficients by means of quasi-elastic scattering of neutrons (e.g., on argon,water, tellurium, hydrogen in metals, etc.) which agree quite well with the results ofmacroscopic measurements like isotope tracer methods or n.m.r. The order ofmagnitude of D accessible to the neutron scattering method is cm2/s. Withmodern instruments at high flux reactors one might attain cm2/s.Dr. A. J. Leadbetter (University of Bristol) said : Springer’s results on amorphousand polycrystalline selenium (fig. 2) show that there is a band of low frequency modes(E<7 meV) in the vitreous material which is not present in the crystal.Did heobserve any angle dependence of the scattering in this region, and has he any viewson the origin of these low-frequency modes and their relation to the structure of thevitreous phase ?Prof. T. Springer (Germany) said: Measurements have been performed only ina restricted range of scattering angles, so that I cannot answer Leadbetter’s firstquestion definitively. Concerning his second question I would like to point outthat the appearance of this intensity band in the spectrum of vitreous Se at smallenergies can have two reasons : either, there exist soft modes which do not exist inthe crystal; or there are phonons with relatively low energies in both cases whichcannot be observed in the crystalline sample as a consequence of quasi-momentumconservation K +q = G ; (G = reciprocal lattice vector ; q = phonon wave vector) ;this selection rule is, however, relaxed in the vitreous state because G becomes abroad and ill-defined quantity.Finally, I would mention that an anomaly in the specific heat at low temperaturesof vitreous Se has in fact been observed, and can be ascribed to a band of soft modesbetween about 1 and 6 meV.lDr.A. C. Wright (Reading University) said: An approximation to the X-rayor neutron scattering from a droplet-separated material, at high droplet concentra-tions, may be obtained using the reduced hard sphere structure factor Io(s), whencethe scattering for a collection of identical, spherical droplets may be writtenI ( S ) = F ~ ( s ) I ~ ( s )J.Lasjaunias Compt. rend., 1969,269, 763112 GENERAL DISCUSSIONper droplet, where F(s) is the droplet form factor and s the scattering vector, 4.n sin O/A.If the droplets are assumed of uniform excess scattering centre density Ap over thesurrounding medium thena(s) is a general scattering length, R the droplet radius and N the number of excessscattering centres per droplet (4nR3Ap/3). The use of the excess scattering centredensity, which may be positive or negative, has the result that the scattering from themedium in which the droplets are situated becomes a &function at s = 0 and canbe ignored.I&) may either be obtained from a random sphere packing, e.g., that of Scott,lmaking suitable corrections for the finite packing size,2 or alternatively from Percus-Yevick t h e ~ r y .~ The latter method will be used here as it is not troubled by statisticaleffects and gives an analytical form for I0(s). Thus, on integration of eqn (4) ofref. ( 3 ) and combination with (3),andr o w = 1lCl -#Wl) c o ~ 2 R s + ~ + % ] . Rs (Rs)Co to C5 are constants, being functions only of the packing density yCo = -37/(1 - v ) ~ ;C3 = 2-37+q3; C4 = (3v/2)(-2+4v+7v2); C5 = (3~/2)(2+q)~.C1 = 1 -6v+5v3; C2 = (37/2)(1 +271)2;The reduced scattering function I(Rs)/N2a2(Rs) is shown in fig. 1 for a range ofpacking densities from 0.0 (i.e., infinite dilution) to the maximum for a random spherepacking of 0.64. It should be noted that, at zero packing density I,(Rs) = 1 andI(Rs) reduces to F2(Rs).As q is increased, the limiting value of I(Rs) at Rr = 0,limR,+o~(Rs) = (1 - ~ ) ~ / ( l + 2 ~ ) ~decreases and the major scattering maximum occurs away from the origin, with ashape at high packing densities which is similar to that obtained from a spinodallydecomposed material (e.g., see ref. (4)). Thus, the shape of the X-ray or neutronscattering cannot be taken as a criterion for spinodal decomposition. In this casethe position of the above maximum, in terms of the corresponding spinodal wave-length & is given by s,,, = 27r/Am and it is interesting to observe that, at r = 0.4,the corresponding position for a droplet separated material is s, - n/R giving A,,, N 2R.A similar result is obtained from a comparison of the computer calculations ofCahn and Haller.6The introduction of a distribution of droplet sizes is approximately equivalentto a softening of the hard sphere potential, and this will result in the function I,(&)approaching its asymptotic value of 1 more rapidly with increasing Rs.F2(Rs)G. D. Scott, Nature, 1962,194,956.A. C. Wright, Ph.D. Thesis, (Bristol University, 1969).N. W. Ashcroft and J. Lekner, Phys. Reu., 1966,145, 83.K. B. Rundman and J. E. Hilliard, Acta Met., 1967, 15, 1025.J. W. Cahn, J. Chem. Phys., 1965, 42, 93.W. Haller, J. Chem. Phys., 1965, 42, 686GENERAL DISCUSSION 113RsFIG. l.--I(Rs)/N2a2(Rs) as a function of Rs for a range of packing densities.will assume some average value. Where the size distribution is not too large thequalitative shape of the scattering should be similar to that shown in fig.1 althoughsomewhat broadened.Dr. A. J. Leadbetter (University of Bristol) said: With reference to Wong andWhalley’s paper, there is a further possible contribution to the temperature-dependentabsorption below about 18 cm-l in addition to those mentioned (cf. Stolen l).Dielectric and mechanical loss measurements both show the presence of a relaxationprocess in vitreous silica and the reIation between the frequency and temperature ofthe maxima of the broad loss peaks is of approximately Arrhenius type.2* Extra-polating from the highest measured frequencies (30 GHz, loss peak at - 120 K)suggests a loss peak at about 10l2 Hz at room temperature. There will thus definitelybe a temperature-dependent contribution from this source to the infra-red absorptionbelow 18 cm-l although it is probably insufficient to account for all of the observedeffect.Dr.A. J. Leadbetter and Mr. M. J. Wood (University of Bristol) said: It nowseems to be clear that the continuous infra-red absorption (and Raman scattering)seen in glasses at low frequencies is a single quantum process arising from the atomicvibrations, all of which are rendered optically active by the disorder in the glass.R. H. Stolen,Phys. Chern. Glasses, 1970,11, 83.E. M. Amrhein, Glastech. Ber., 1969,42,52.A. S. Pine, Phys. Rev., 1969,185,1187114 GENERAL DISCUSSIONThe basic problem is the determination of the frequency distribution g(m) from theabsorption coefficient a@).The relation between the two may be writtenand Wong and Whalley suggest thatf,(m)ccm2, whereas if the absorption is due to arandom distribution of charges fI is independent of m.l Shuker and Gammon,,in interpreting their Raman spectra for silica, have also assumed that the Ramanequivalent of fI is frequency independent.We do not wish to discuss here the theoretical aspects of the problem, but to seewhat information experimental data can provide about fI(w). In order to do thissome information is required about g(w) and most is known about vitreous SiO,and GeO,. The most direct experimental information is provided by our inelasticneutron scattering experiments, which certainly show the main features of g(w),particularly at low frequencies, and at least for GeO, probably give a good approxi-mation to g(w) itself. There are also single crystal neutron scattering data on quartzwhich show that the acoustic branches, comprising about 11 % of all the modes arelargely located below about 120 cm-1 ; so as a very rough guide for both SO2 andGeO, we estimate that at least 10 % of the total modes are located below about100 cm-l.The low-temperature heat capacity also provides good informationabout the general shape of g(w) at low frequencies (say, co "< 100 cm-I). Finally,there are the results of the theoretical calculations of Bell and Dean which providean approximate description of the overall shape of g(m) for w r 100 cm-l.a(w> = m M 4 YI2 -I I 1.:/,/*' /- I,.--' /, //I00 2 00 300 4 0 0 5 0 0w/cm-la, measured on SiOz plates ; - - scaled nujol mull spectrum ; - - - a/02.FIG.1.Mra-red absorptivity of vitreous silica at T-295 K. -, absolute absorption coefficientInfra-red absorption data for vitreous Si02 and GeO, are shown in fig. 1 and2 where both the absorption coefficient ot(cr)) and a(w)/co2 are plotted. At lowfrequencies (in the acoustic region) use of a constant fI value predicts far too fewW. Bagdade and R. H. Stolen, J . Phys. Chem. Solids, 1968, 29,2001.A. J. Leadbetter, Phys. Chem. Glasses, 1968, 9, 1.A. J. Leadbetter and M. J. Wood, to be published.' R. Shuker and R. W. Gammon, Phys. Rev. Letters, 1970, 25,222GENERAL DISCUSSION 115modes (fig. 1 and 2). On the other hand, a/u2 appears to show the lowest frequencypeaks (at 40 cm-l for SiO, and 27 cm-1 for GeO,) which are observed in inelasticneutron scattering and heat capacity measurements, but gives too much weight tothe low-frequency modes, particularly for GeO,.Moreover, little detail is revealed-- 51210N 2.m0k i3 C5 -4 - . 8,';IIN2.__--I I100 2 0 0 3 0 0 400wfcrn-'efficient a, measured on GeOz plates ; - - scaled nujol mull spectrum ; - - - a/&.FIG. 2.-Infra-red absorptivity of vitreous germania at T--295 K. -, absolute absorption co -in the acoustic region apart from the single lowest frequency peaks. This might beexplained usingf, = Am2 with different A-values for each spectral branch. Further-more, we have observed differences in a(w) at low frequencies for different silicasamples, which are much bigger than can reasonably be explained by differences ing(m).This suggests that variable factors such as impurities (e.g., -OH) may playa significant role in determining absorptivity. To sum up, it appears that nosimple result like5 = Awn is generally valid over a wide frequency range.Dr. E. Whalley (N.R.C., Canada) said: To answer Leadbetter and Wood, it isperhaps worth stating the assumptions on which the theory is based since they are notgiven in the paper. They are as follows : (i) the glass is composed of units A , whichmight be atomic or molecular, arranged in an irregular glass-like manner ; (ii) thepotential energy Y can be written as the sum of square terms in the internal co-or d i n a t es s(Z, K ) ,2v = C4(tC>s2(W,1,Kwhere K identifies the kind of internal coordinate, such as bond stretchng, anglebending, etc., and I identifies the particular example of the rcth internal coordinate.The force constant 4 ( ~ ) is independent of I, but depends on K .It is important thatsuitable internal coordinates can be chosen so that the potential energy contains nocross terms in the internal coordinates; it is not, however, necessary that these co-ordinates be known. (iii) The dipole moment of the glass is linear in the internalcoordinates. The dipole moment derivatives areM W = f%Ulas(l,K)116 GENERAL DISCUSSIONwhere p is the dipole moment of the glass. This makes the theory inapplicable to,e.g., internal motions of the S102 group.It is assumed that values of M(2,1c)for neighbouring internal coordinates are uncorrelated with one another. It isalso assumed, rather reasonably for a glass, that the mean value of M(Z,K) averagedover all I is zero. (iv) The quantity M2(Z,rc) occurs in sums like1 3 %It is assumed that there is little correlation between the mechanical and the electricalproperties of the crystal, so that for any vibration this sum can be replaced by< 2 ( l p rc)) i { as([, .)/a Qk } 9K 1where ( means the average over all 1. It is further assumed that this average isindependent of the particular vibration. This should be true in the acoustic branches,but will not in general be true for the vibrations associated with defects in the glassstructure such as broken bonds. It is further assumed that the ratio (M2(Z7rc))z/q!(rc)is independent of K if more than one kind of internal coordinate is significant.Itfollows straightforwardly on the basis of these assumptions that the integratedabsorptivity of a normal vibration is proportional to the frequency squared, andconsequently the absorptivity of the glass is given byK(co) = const. co2g(co)where u) is the frequency and g(o) the density of states.This result, that the absorptivity can be related to the density of states withoutknowing anything about the normal coordinates of the states is a simple one. Ithas been obtained only with the help of assumptions that seriously restrict its accuratevalidity. In t h s respect it is not unlike the theory of the inelastic neutron scatteringof polyatomic materials as described in Leadbetter and Litchinsky’s paper.Themain conclusion from the theory is that in the region where the approximations arenot totally incorrect, i.e., in the acoustic branches the absorptivity divided by thefrequency squared Kfw2 is probably more closely related to the density of statesthan is the absorptivity itself but is not necessarily proportional to it. Consequently,features in the density of states might be recognized in the spectrum of K/w2.Bagdade and Stolen have used Vinogradov’s model as treated by Schlomannto represent a glass. In this model, ions having disordered charges are arranged on asimple lattice, and the infra-red intensity is due directly to the motion of the ions.Thelow-frequency waves in vitreous silica are due more to the movement of neutralSO, units. The infra-red intensity is due more to induced moments, and it seemslikely that it will not be well described by a model in which the neutral SiOz units arereplaced by ions having disordered charges.treatment of the Raman spectrum assumes that thevolume of the region of coherence of the vibrations is independent of frequency.While this may be a reasonable first approximation for the “ internal ’’ vibrationsof the SOz units, it is probably a poor one in the acoustic branches because we knowthat at Iow enough frequencies the volume of coherence becomes very large. In fact,Shuker and Gammon’sW. Bagdade and R.Stolen, J. Phys. Chem. Solids, 1968, 29,2001.V. S. Vinogradov, Fiz. Tuerd. Tela 1960,2,2622, (English trans. Siwiet Phys. Solid State, 1960,2, 2338).E. Schlomann, Phys. Reu. A , 1964, 135,413.R. Shuker and R. W. Gammon, Phys. Rev. Letters, 1970 25.222GENERAL DISCUSSION 117as shown in my comments in the paper of Stolen, Krause, and Kurkjian the low-frequency infra-red and Raman spectra appear to be similar functions of frequencywhen allowance is made for the factor v-l in the Raman spectrum.Dr. R. I(. MacCrone (Division of Materials Engineering Rensselaer PolytechnicInst., Troy, N. Y.), said : The relation between the low-frequency vibrational modesin fused silica, the excess heat capacity and the ultrasonic attenuation, as describedby Pine,l and discussed in Gaskell's paper, deserves further comment.Recently,Scott and MacCrone have measured the anelastic relaxation (homologous toultrasonic attenuation) in fused silica at audio frequencies (- 1 kHz) and foundseveral distinct relaxation processes superimposed upon the large well-knownadsorption maximum. These latter relaxations were also interpreted as beingdue to the Akheiser coupling between the strain and the excess low frequencyvibrational modes. The magnitudes of the relaxation and the temperatures of theiroccurrence were in good agreement with the published specific heat data. In addition,dielectric measurements were also made and revealed the presence of associateddielectric relaxations. It was shown that an applied electric field F results in a changeof frequency of the low-frequency mode in an analogous way to the (Griineisen)frequency change resulting from an applied strain. In both cases, anelastic anddielectric, the approach to equilibrium is governed by a relaxation time, whosetheoretical deviation is prohibitively difficult, but which experimentally is found tofollow a simple Arrhenius expression, z = T~ exp (Q/kT).Anelastic and dielectricrelaxations due to this mechanism are not confined to fused silica, but apparentlyoccur in, e.g., vitreous germania and in certain polymer^.^Dr. J. L. Finney (Dept. of Crystallography, University of Pittsburgh)" said:Many of the papers, and certain discussion contributions, have drawn attention toour inability to describe mathematically the disordered nature of the vitreous state.We have evidence-though much if it indirect-of the irregularity at the molecularlevel (subject to the relevant bonding restrictions) of many non-crystalline solids(see, e.g., structural work on Ni-P alloy^,^ vitreous silica,6 and amorphous ger-manium).' Up to a few years ago, it seemed possible that successful descriptions(in particular of the liquid state) of the " restricted irregularity " of dense, disorderedassemblies, might come from generalizing descriptions of the dilute gas phase.Recently, the need for new approaches to this problem has been more widelyrecognized.This is evident in the last few papers, where attempts were made todescribe specific aspects of the disorder of a system with respect to the particularmeasurements under discussion.The complexity of the normal modes obtainedfrom Bell and Dean's random network model of vitreous silica means they canpresently be described only by a statistical parametrization. Before a generaldiscussion of the vibrational spectrum of hsordered materials can be given, ananalytical description of the basic static structure must be substituted for the laboratorymodels. Similarly, Leadbetter's inelastic neutron scattering data on germania* on leave from Birkbeck College, LondonA. S. Pine, Phys. Reu., 1969,185,1187.W. W. Scott and R. K. MacCrone, Phys. Rev. B, 1970,1,3515.M. Zobel and A. Leadbetter, private communication (1970).S. Miller, M. Tomozawa and R. K. MacCrone, Some Polymers at Low Temperatures, ThePhysics of Non-Crystalline Solids III (Sheffield, Sept., 1970).G. S.CargilI 111, J. Appl. Phys., 1970,41, 2248,R. L. Mozzi and B. E. Warren, J. Appl. Cryst., 1969, 2, 164. ' R. Grigorivici and R. Manaila, J . Non-cryst. Solids, 1969, 1, 371118 GENERAL DISCUSSIONcannot be adequately interpreted because the unperturbed atomic arrangementscannot be described in a useful way. Gaskell has examined the effects on the vibra-tional spectrum of incorporating the known vitreous silica bond-angle distributioninto an essentially ordered arrangement, with some interesting results ; however,like Pople’s bent-bond water model, the resulting model is topologically inconsistent,and therefore can give-as Gaskell stresses-only limited information which mustbe interpreted with great care.Whalley too has stressed the intractability of thevibrational problem when crystalline simplifications are invalid ; in order to interprethis absorptivity measurements, he invokes as a model an orientationally disorderedcrystal, which may or may not bear sufficient similarities to the real gas to be useful.All these papers imply a need for new techniques to describe structural disorder;within such a framework, we could then proceed to discuss the electronic and vibra-tional processes with which we are concerned.During the past few years, there have been several ingenious approaches madepartially to solving this problem. Again, they have been direct attempts to explainparticular properties--e.g., Anderson’s use of randomly varying potentials,2 andZiman’s use of percolation theory and four-particle correIation functions allrelated to the electron localization problem (though generalizations might have beenpossible).Another apparently promising development via multiple scattering waspresented in the paper by Klima and McGill, who have abandoned the lattice conceptand concentrated upon the local molecular arrangements. It remains to be seenwhether or not consideration of first neighbour clusters is sufficient; in any case, acomplete solution will involve a mathematical description of local regions, a descrip-tion which will not ignore the structural differences between different volume elements.We need new methods of describing irregular arrays, which are not directlydependent upon particle positions ; upon the successful formulation of these methodsdepends a full understanding of the vitreous (and liquid) state.On a simple level,the Birkbeck studies on random packings of spheres have provided precise data onone kind of disordered system, and have led to statistical and geometrical quantitieswhich are invariant between arrays of the same den~ity.~ In the Dept. of Physicsand Crystallography at Birkbeck, we are attempting to take the first steps towards amathematical description which will fit the data obtained from the simple models.In contrast to the situation a few years ago, we are now optimistic about the chancesof success.Similar methods could be applied to the directionally-bonded systems relevant to thevitreous state-networks which are disordered subject to a set of angular dependentconstraints very different from those governing the simple sphere arrangements.Data on those systems are being collected from model studies, and attempts arebeing made to elucidate the invariant characteristics.The problem is difficult, butof great importance : it justifies much wider attention than it is currently receiving.Dr. A. J. Leadbetter (University of Bristol) said: Inelastic neutron scatteringexperiments have been performed on pile-irradiated vitreous silica (dose N 1020fast neutrons/cm2, for an idealized fission spectrum centred around N 1 MeV). TheJ. A. Pople, Proc. Roy. Soc. A , 1951,205, 163.P. W. Anderson, Phys.Rev., 1958,109, 1492.J. M. Ziman, J. Phys. C, 1968,1,1532.J. M. Ziman, J. Phys. C, 1969,2,1704.J. L. Finney, Proc. Roy. Soc. A, 1970,319,479.A. J. Leadbetter and D. Litchinsky, unpublished workGENERAL DISCUSSION 119data have rather poor statistical accuracy but show (fig. 1) a decrease in the scatteredintensity at w r 4 0 cm-1 and an increase at higher frequencies 40 < w < 80 cm-l,in agreement with the changes in infra-red absorption reported by Stolen et al.5 0 0 2 00 100 5 0 40 3 0o/cm-lFIG. 1 .-Inelastic neutron scattering spectra for unirradiated (solid line) and pile irradiated (points)vitreous silica at 90" scattering angle.The modes at a240 cm-l in unirradiated vitreous silica have been interpretedas being the broadened and damped transverse acoustic phonons observed at thisfrequency in cristobalite.The changes observed on irradiation are most simplyinterpreted as a general raising (by up to about 20 cm-l) of the frequency of this lowenergy TA branch and probably also enhanced disorder-broadening of the phonons.This is compatible with the observed reduction in the heat capacity on irradiation.Dr. R. H. Stolen (Bell Telephone Lab., N. Y.) said: In reply to Cooper, the idea wasto see whether the ultrasonic absorption could be related to the lowest frequencyoptical modes. Presumably the lower the frequency of such a mode the greater itseffect on the ultrasonic absorption. This is because in SiOz both the heat capacityaround 50 K and the Griineisen parameter are larger for very low-frequency modes.As such, the ultrasonic absorption would be expected to decrease whether the totalnumber of modes decreased or whether the modes just shifted to higher frequencies.With regard to the frequency shifts of the scattering peaks, infra-red active bandsat 1120 and 470 cm-l shift down in frequency after irradiation.Dr.E. Whalley (N.R.C., Canada) said: I have two comments on the paper ofStolen, Krause and Kurkjian. The first comment is on the best way to comparethe low-frequency infi-a-red and Raman spectra of glasses. As H a shave emphasized, the maximum in the low-frequency Raman spectrum observedat high temperature arises from the Boltzmann factor [exp (- hcv/kT) - 1]-2 (forthe Stokes lines) multiplied by a limiting low-temperature spectrum that falls rapidlywith decreasing frequency. Stolen and Stolen, Krause, and Kurkjian advocatecomparing the high-temperature Raman spectrum with the infi-a-red absorptivityK divided by the frequency squared.A serious disadvantage of this procedure isand StolenA. J. Leadbetter, J . Chem. Phys., 1969,51,779.M. Has, J . Phys. Chem. Solids, 1970,31,415.R. H. Stolen, Phys. Chem. Glasses, 1970,11, 83120 GENERAL DISCUSSIONthat while both have a maximum, the maxima have quite different origins, that in theroom-temperature Raman spectrum being due to the Boltzmann factor and that inthe infra-red spectrum being due to the maximum in the density of states.The integrated Stokes Raman scattering intensity 1, at low temperature of asingle oscillator k of wave number vk is &/I0 = const.(aa/aQk)2 l / v K , where 1, isthe intensity of the incident light, a is the polarizability,l and Q the normal coordinate.The integrated infra-red absorptivity Ak of vibration k is given by= const.(ap/aQ)k2, where p is the dipole moment.The scattering and absorption intensity of a glass is then the sum of the intensitiesof all the vibrations at frequency v,where s(V-vk) is the Dirac delta function. It follows thatIt would appear therefore that a comparison of the frequency dependence of vl(v)with that of K(v) gives directly the relative variation of the sums on the right side.Hass has commented that in the range 30-100 cm-l the low-temperature Ramanscattering appears to be roughly linear in the frequency, whereas according to Bagdadeand Stolen’s (and our) measurements, the infra-red absorption is more nearlyproportional to the frequency squared.Clearly, vl(v) and K(v) are behaving similarly.This is also clear in Stolen’s Raman spectrum of vitreous silica. At 10 cm-lintervals between 30 and 80 cm-l the ratio K(v)/vI(v), in which I(v) is defined onlyto a constant factor, using values of K from our paper are 6.0, 5.2, 5.1, 4.4, 4.4, 4.4,suggesting that the ratio on the right side of eqn (1) does not vary much with frequencyin this range.The second comment is that, at first glance, the results of Stolen, Krause, andKwkjian on the spectra of unirradiated and irradiated vitreous silica can be ration-alized qualitatively in a simple way. The unirradiated material can be considered asderived from crystobalite or tridymite, which are based on the cubic or hexagonaldiamond lattice and in which the lowest maximum of the density of states occurs atabout 40 C M - - ~ . ~ ~ On the basis of a simple Si-Si bond-stretching and Si-Si-Siangle-bending force field (ignoring the oxygen atoms), because of the local tetrahedralsymmetry, this maximum is almost entirely determined by the angle-bending forceconstant. In crystals with the diamond structure, for example, the zone-boundaryfrequencies in the transverse acoustic branch in the [loo] and [lll] directions arel Strictly, a = as+aa where as and au are the isotopic and anisotropic invariants of the polariz-bility tensor.M. Hass, J. Phys. Chem. Solids, 1970, 31, 415.W. Bagdade and R. Stolen, J. Phys. Chem. Solids, 1968,29, 2001.4R. H. Stolen, Phys. Chem. Glasses, 1970, 11, 83.A. J. Leadbetter, J. Chem. Phys., 1969,51,779.P. T . T. Wong and E. Whalley this discussion. ’ J. E. Bertie and E. Whalley, J. Chem. Phys., 1967, 46, 1271GENERAL DISCUSSION 121(12ke/m)* and (6k0/m)*, where ke is the angle-bending force constant and m themass of an atom. When vitreous silica is irradiated, it becomes compacted. Thelocal tetrahedral symmetry is removed by distortion of the Si-Si-Si bond angles,and the bond-stretching force constant contributes to the acoustic vibrations. Theirfrequencies therefore increase. In crystal quartz, the transverse acoustic maximumis at about 70 ~ m - l , ~ presumably because bond stretching contributes to the restoringforce

ISSN:0366-9033    

DOI:10.1039/DF9705000108

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Phase-separated SystemsBY JERZY ZARZYCKI"Centre de Recherches, Cie de Saint Gobain, 52 Boulevard de la Villette, Paris 19,FranceReceived 24th September, 1970The problem of immiscibility in glasses is reviewed. After defining the sub-liquidus immiscibilityand considering the stability regions within a miscibility gap, the dynamics of initial clustering arediscussed. The distinction between nucleation and growth, spinodal decomposition and coarseningprocesses is emphasized. Experimental studies of phase separation corresponding to the abovemechanisms are presented and, in particular, the possibility of verifying Cahn's theory is discussed.New results from the supercritical region enable the initial condition after quench to be assessed.Finally, the influence of demixing on crystallization is examined and the possibility of thermo-dynamical blocking of a stable phase indicated.It is the purpose of this paper to introduce briefly the subject of immiscibility inglasses.Interest in this phenomenon has rapidly increased since the discovery ofthe glass-ceramic process (controlled crystallization of glasses) in which immiscibilityis thought to play an important role. Phase-transformation concepts, familiar tometallurgists, were then transposed to oxide systems in the glassy state. But glasseshave several characteristics of their own which make them particularly attractivehere : they are isotropic, which simplifies many problems in the solid state, and theirviscosity increases considerably as the temperature is lowered, which permits at will,all transport processes under study to be slowed down.Glasses constitute a con-tinuous link between liquids and solids : thus, the study of immiscibility in glassesmay throw light on the kinetics of demixing in liquids which is in general too rapidfor present experimental techniques.SUB-LIQUIDUS IMMISCIBILITYDemixing of a glass may be considered as a particular case of phase separationof a liquid which has failed to crystallize and has thus become metastable withrespect to a crystalline phase. If we consider a simple binary system with a miscibiltygap and the associated free energy curves (fig. l), in the event of a crystalline phaseoccurring, the two-liquid field is bounded by the horizontal part of the liquidus.The evolution of the free energy curves shows then that, for a supercooled liquid theimmiscibility field extends below the liquidus, the two descending branches of thecoexistence curve simply being extrapolated by continuity towards lower temperatures.If, for kinetic reasons, this crystalline phase does not appear readily, the metastablebranches of the coexistence curve may be crossed reversibly and demixing studiedindependently of crystallization.Several silicate and borate systems belong to thiscase, a typical example being the BpOg+PbO system. It may happen that theimmiscibility field is entirely sub-liquidus (fig. 2). In this case, since the free energycurve of the liquid flattens considerably, the liquidus displays a nearly horizontal* Professor of Materials Science, Faculty of Science, University of Montpellier, France.12Tli):X YcompositionFIG.1. Left : Schematic phase diagram showing a two-liquid immiscibility region (cross-hatched)with a sub-liquidus extension (hatched). Right: related free energy curves of the liquid (L) andsolid (S) phases.1 X Y YcompositionFIG. 2 . T - f t : schematic phase diagram showing an entirely sub-liquidus two-liquid immiscibilityregion. Right: related free energy curves of the liquid (L) and solid (S) phasesT = TI% a b Gi/In recent years, numerous determinations of sub-liquidus immiscibility fields havebeen made.1-12 Experimentally, temperatures of the onset of turbidity or clearingtemperatures are determined by optical means (nephelometry). Quenched glasssamples can also be heat treated in a gradient temperature furnace and examinedby optical or electron microscopy.Differential thermal analysis and e.m.f. measure-ments l3 have also been used and effects of pressure investigated.14In ternary systems, the location of conodal lines poses a difficult problem due tothe fineness of precipitates. Composition determinations are often possible after acoarsening treatment either by selective dissolution and chemical analysis or by electronmicro-probe ana1ysis.l An interesting method is based on the determination ofglass-transition temperatures, these being invariant along the conodal lines. Whilesecondary phase separations connected with the asymmetry of the coexistence curve82i El8JERZY ZARZYCKI 125were observed,l’ the occurrence of three-liquid immiscibility has not yet beendefinitely proved.8*Prediction of demixing was originally attempted from crystal-chemical considera-tions. The extent of a miscibility gap was correlated with bond strengths, ionicpotentials e t ~ . ~ ~ - ~ ~ It is rather strange that, in these treatments of the thermo-dynamic problem of phase separation, the temperature factor is absent! Recentlya careful thermodynamic analysis of many silicate systems was performed andcoexistence surfaces calculated in good agreement with experimental e ~ i d e n c e . ~ ~ - ~ ~STABILITY REGIONS WITHIN A MISCIBILITY GAPAs was shown by Gibbs, a phase is always metastable with respect to the formationof an infinitesimal inclusion with a finite composition difference if this leads to aninterface with positive interface free energy.However, a phase may be unstablewith respect to the formation of large regions that are only infinitesimally altered incomposition. The limit between these two kinds of stability in a phase diagramis the spinodal, defined as the locus of points for whichf” = a2F2/aC2 = 0, corre-sponding to the inflection points i, j of the free (energy, composition) curve (fig. 3).In the regions (I), shown on this figure, (f”>O), the system is stable towardsinfinitesimal composition fluctuations which all raise its free energy, and metastabletowards occurrence of a phase widely differing in composition. The separation canonly occur by a nucleation andgrowth process which is controlled by a thermodynamicbarrier.In the region (11), ( f ” < O ) , the system is unstable towards all, even thesmallest, composition fluctuations ; the decomposition is controlled by dzflusion onZy.DYNAMICS OF INITIAL CLUSTERINGIn a single phase there is an equilibrium size distribution of fluctuations dependingon composition and temperature but independent of time. These fluctuations areparticularly marked in the critical region (111) (fig. 3) where they give rise to criticalopalescence. In a two-phase system, on the contrary, there is no equilibrium sizedistribution-the particles increase indefinitely in size by growth, then coalescence(until, theoretically, only one particle is left) in order to reduce the interface.Let us consider a system with two species of atoms28 ; if the preference for likemembers is small, the clusters have no permanence, they form and disappear atrandom, and any set-up concentration gradient vanishes-the system forms onephase. When the preference for like species is so great that the flux of individualsis up the concentration gradient, this leads to a spontaneous separation into twophases-the system is within the spinodal.The atoms sense the depleted zone andmove away from it, building a new cluster a short distance away-this leads to arapid and simultaneous formation of extremely small clusters approximately periodi-cally arranged in space. The concentration of the two c‘ phases ” varies continuouslywith time, a diffuse interface being present in the initial stages, which sharpensprogressively as the separation proceeds (fig.4a). To this mechanism Cahn hasgiven the name “ spinodal decomposition ”. Between these two extremes-whenthe attraction for like species is insufficient to cause a flux up the concentration gradientbut strong enough for a large supercritical cluster to retain the atom-lies the classicalcase of nucleation and growth. In the vicinity of a growing particle the matrixconcentration is reduced and feeding of the particle occurs by normal diffusiondown the concentration gradient (fig. 4b). This leads to particles randomly distri-buted in space and which appear in succession, the interface being sharp from thebeginning126 PHASE-SEPARATED SYSTEMSFrom the foregoing discussion, the essential characteristic of the spinodal decom-position is the dzflusion up the concentration gradient (uphill diffusion, or " active "diffusion of biologists).The spinodal thus marks the limit not only between stabilityregions but between modes of initial clustering. By comparison, the nucleation andgrowth mechanism is much more complicated since it involves the difficult conceptof the nature of the nucleus. This can either be a composition fluctuation or someheterogeneity to which enough individuals of one species are attracted. Suchheterogeneities play a much smaller role in the spinodal mechanism.EARLY LAT E R FINAL- distanceFIG. 4.-Schematic evolution of concentration profiles to illustrate the differences between (a) spinodaldecomposition and (b) the nucleation and growth mechanism.After Cahn.**One could ask why, in the spinodal region, the interfacial free energy does notstabilize the solution. This is due to the fact that the free energy may be decreasedby infinitesimal variations in composition and large regions can continuously changetheir composition. The system then tends to adopt a scale coarse enough to minimizethe contribution of the (diffuse) interface. The spinodal decomposition is, however,not the only mechanism possible in region (11) ; even though the average compositionlies inside the spinodal, there could be local regions present due to fluctuations whichcould be in other regimes and thus decompose by a nucleation and growth mechanism.The spinodal concept implies that the two phases separate by a continuousprocess; for solids, there must be a continuous free energy function from one phaseto another, i.e., the coexisting phases must have either the same structure or theremust exist a metastable phase of the same structure as one of the equilibrium phases.For crystals, the two phases forming by a continuous process start to be coherentand it is necessary to take into account the elastic energy of matching the lattices.29* 30This increases the free energy and has the effect of depressing the coherent spinodal(relative to the spontaneous separation) with respect to the incoherent (chemicalJERZY ZARZYCKI 127spinodal (fig.5). It is not yet clear to what extent these concepts apply to glasses,where strains should be fast relieved above the transformation range, but for lowertemperatures there could be a continuous transition between coherent and chemicalspinodal .FIG. 5.-Bottom : Schematic representation ofthe coherent (a) and incoherent (b) miscibilitygaps and spinodals.Top : related free energycurves for T = To : the temperature depressionAT corresponds to the increase A F of freeenergy for matching the lattices.1 I X YX YcompositionCOARSENING PROCESSESAfter the initial stage, which may be either nucleation and growth or spinodaldecomposition, the system tends to reduce its interface by a coarsening process.This may be coalescence by brownian movement, viscous flow, etc., or, for soliddistanceFIG.6.-Schematic evolution of concentration profile to illustrate the Ostwald ripening mechanism.Particles of a radius smaller than the critical radius r* dissolve and feed those with a radius greaterthan r*128 PHASE-SEPARATED SYSTEMSglasses, the Ostwald ripening process 39 based on difference of solubility of particlesaccording to their size. Small particles below a certain critical size tend to dissolveand feed, via diffusion, larger particles down the concentration gradient (fig. 6).THEORIES OF DECOMPOSITIONWe shall not discuss here the nucleation theories for which excellent reviewsexist.33 The basic weakness of these theories is the assumption that the embryohas the properties of a macroscopic droplet.Improvements allow for internalcomposition variations and diffuse interface. It is only for slight supersaturationsthat the critical nucleus may resemble the classical picture.The mathematical theory of spinodal decomposition originated by the work ofHillert 34 and developed by Cahn and Hilliard,35-37 is based on a general diffusionequation containing terms relative to the gradient of concentration. Cahn hassolved this equation in a linearized form by means of a spatial Fourier transformof the concentration, each spatial component of wavenumber fi being of the form :C(r,t) - Co = exp [R(P)t] cos Prwhere R(P) is an amplification factor depending on the physical characteristics of thesystem : atomic mobility, energy gradient terms, and which may contain contributionsdue to external fields.R(P) is positive only in the spinodal region, for wavenumbersp lower than a critical number pc and presents a fairly sharp peak for Pm (fig. 7).wavenumberFIG. 7.-Schematic representation of the amplification factor R(P) of Cahn’s linearized theory.Cahn’s treatment is restricted by the fact that a linearized equation is consideredand so the solution is limited to small perturbations. Only the wavenumber Pmwas considered important and the influence of other P neglected. This treatmentwas also extended to later stages involving coarsening 38 and to the problem ofdeveloping structures during the continuous cooling. 39 Solution of the diffusionequation for discrete atomic lattices 40 does not bring a significant improvement onCahn’s continuum theory.General salutions of the non-linearized equation havebeen obtained using step-by-step calculations by electronic computer^.^JERZY ZARZYCKI 1 29EXPERIMENTAL STUDYDISPERSED PRECIPITATESDue to the fineness of the separating phases, electron microscopy, small-anglescattering of X-rays, and light scattering 42 can be used. All these methods workwell when the precipitated particles are widely separated, i.e., form a minor phase,as is so in region (I). Although the average rate of growth of particles could befollowed in several cases,12* 43-45 a clear distinction between the nucleation andgrowth stages was not made until the pioneering work of Hammel on the SO,+CaO + Na20 system. Steady-state nucleation rates were measured ; their interpreta-tion, however, was strongly dependent on the solution model used to fit the freeenergy curve.Recently, nucleation and growth stages were clearly separated in2,5-Si02,Li02 glass.46A detailed study of the coarsening stage for the B203 +PbO system * has shownan Ostwald ripening process. From small-angle X-ray diffraction spectra the com-plete evolution of particle size distribution curves was directly obtained, and theapplicability of the Lifschitz-Slyozov-Wagner theory 3 2 9 47 demonstrated.DENSE SYSTEMSExperimental difficulties arise when the separating phases form a dense system,i.e., when neither of the two phases is minor. This is the case for region (11) wherethe interesting problem of distinguishing between spinodal decomposition andnucleation is encountered but not yet solved.STUDIES IN REAL SPACE-CONNECTIVITY CRITERIONTs it possible to distinguish between these two mechanisms from the morphologyalone ? The computed concentrations for cross-sections of a three-dimensionalregion during spinodal decomposition indicated a texture with a high degree ofconnectivity 37 (fig.Sa). As such a morphology is shown by many glasses under-going phase separation, connectivity was considered for a time as a criterion forspinodal decomp~sition.~~ However, it has since been shown 49 by a purely geo-metrical argument that a random array of spheres, such as results from a nucleationmechanism, also displays high connectivity (fig. 8b). Moreover when phases departfrom a 50150 volume proportion, a computed spinodal pattern loses connectivityand can be confused with a random array of particles.Numerous kinetic argumentswere raised to prove or disprove Haller’s scheme.50* 51 The problem will be furtherdiscussed at this symposium.STUDIES IN RECIPROCAL SPACESince Cahn’s theory is established in the reciprocal space, its most direct meansof verification is by a diffraction experiment. The diffraction spectra give directlythe distribution of squares of amplitudes of spatial components. This is a realprogress as the analysis of this type of dense structures is not straightforward. Anintensity maximum in a small-angle X-ray diffraction spectrum can, in principle,always be matched by various models of dense particle arrays.(The same difficultieswere experienced by metallurgists working on Guinier-Preston zones in Al-Cualloys. 53) For these textures with a high degree of connectivity, electron microscopyshows that the concept of geometrical particles is meaningless. It is thereforepreferable not to postulate a model in real space but to operate directly in reciprocal130 PHASE-SEPARATED SYSTEMSspace. Even the electron microscopic data should be evaluated by means of anoptical transform method.54 Due to the fineness of the structure (of the order of100 A), small-angle scattering of X-rays is indispensable for studying the initialstage^,^ light scattering being restricted to coarsening stages (textures of the orderof lOOOA and more).0.I 0.2 0.3 0.4 0.5 0.6 0-7 0.8FIG. &---(a) Cross-section of a three-dimensional spinodal structure simulated 011 the computeradding 100 random sine waves of wavelength A. The points define regions where the concentrationis greater than average ; after Cahn.37 (b) Cross-section through a three-dimensional matrix ofequal spheres. Positions randomized by computer ; after Haller.49Small-angle X-ray diffraction studies of the B203 + PbO + A1203 andSiOz +Na,O 5 6 systems have been carried out for region (11) ; they show that Cahn’slinearized theory is not strictly obeyed. In particular, cross-overs in the spectra(corresponding to the critical wavenumber PC) occur, not in the beginning, but in thelater stages of decomposition when Cahn’s treatment is no longer valid; the[R@)/P2,P2J plots are hyperbolic rather than linear as would be required by the simpletheory.Moreover? precise continuous recording of intensity components using asmall-angle X-ray goniometer equipped with a heating cell shows a sudden intensityincrease at the very beginning of the heat treatn~ent.~~CORRELATION WITH THE SUPERCRITICAL REGIONIf we discard the possibility of a nucleation and growth mechanism, thesediscrepancies may be ascribed either to poor initial conditions-appreciable decom-position having already occurred during quench-or to inadequacies in the linearizeJERZY ZARZYCKI 131theory to describe the phase-separation in these systems. To decide between thesehypotheses, we have attempted to determine directly the equilibrium spectrum ofsupercritical composition fluctuations in the liquid and to compare it to the initialspectrum of the glass obtained by quench. This was done using a specially designedsmall-angle X-ray goniometer equipped for the study of liquids at hightemperatures.57 *900rW t '/o A1203I I0 t 2 3 4 5 6 7 8 9 1025 20 15 10W t % PbO1 1 ~ ~ ' " " " ' ' ' ' ~compositionFIG. 9.-Section of the immiscibility surface (C.S.) of the system B203+PbO+A1203 showing thesupercritical region. Lines of equal Debye correlation length I of supercritical fluctuations obtainedfrom small-angle X-ray scattering are indicated ; after Zarzycki and N a ~ d i n . ~ ~The system chosen was B,O, + PbO + A1203, which has an iinniiscibility surfacewith a saddle point.Fig. 9 represents a section of this surface close to the con-solution crest. In order to show the extent of the supercritical fluctuations inregion (111), lines of equal Debye correlation length, obtained from small-angleX-ray scattering experiments, are indicated on this figure. Strong fluctuations716 "C50 100(wavenumber)2 x lo4FIG. 10.4rnstein-Zernike plots of small-angle X-ray scattering spectra for the liquid 77 % B203?18 % PbO, 5 % A1203 in the supercritical region and the quenched glass (TI)? (T2), after Zarzyckiand Naudin.'132 PHASE-SEPARATED SYSTEMSpersist in a temperature interval of up to 100°C over the saddle point of the consolu-tion surface. During quench, the liquid has to traverse this region. Fig.10 shows,superimposed on the same diagram, the Ornstein-Zernike plots (1 /I,p2) of the spectraof the liquid in the supercritical region and of the quenched glass. For high /I,both sets of O.Z. plots are parallel and the amplitude of the quenched fluctuationsdoes not exceed critical distribution of the liquid. This shows that our quenchconditions were satisfactory ; in the high B range, the spectrum of the glass correspondsto that of the liquid in the supercritical region.It would seem, therefore, that the observed discrepancies should rather be ascribedto the inadequacy of the linearized treatment. Solutions of the complete equationhave already been tested, but on systems in which initial quench conditions werefar from satisfact~ry.~~ Computer simulation of the evolution of unstable binarymixtures shows, anyhow, that the stage corresponding to the linearized case is onlya short transient, the system evolving continuously towards coarsening.6o In thelater stages, coarsening laws similar to those found in region (1) seem to apply, thecurvature of the interface replacing the radius of discrete particles.*OTHER STUDIES-INFLUENCE OF DEMIXING ON CRYSTALLIZATIONStudies of physico-chemical properties-viscosity, internal friction, conductivity,etc.,-might contribute to a better understanding of immiscibility in glasses. Numerousexperimental results have been described in a recent Russian symposium 61 entirelydevoted to the question of immiscibility phenomena in glassy systems ; others will beconsidered in these discussions.Finally, we may ask what exactly is the part played by demixing in crystallization.Composition changes brought about by phase-separation, and/or the extensivecompositionFIG.1 1 .-Thermodynamic blocking of a stable phase. From a metastable phase L of compositionC, the more stable phase S of composition C, cannot form, as this would correspond to an increaseof free energy (path AIB) until the phase L has undergone demixion. The precipitation then corre-sponds to a decrease of free energy (path A2B) ; after Cahn.6JERZY ZARZYCKI 133interface thus created, might promote crystal nucleation. 62 It is possible, however,as has been suggested by Cahn,63 that demixing is simply a precursor reaction;crystallization may be blocked, for thermodynamic reasons, until phase separationtakes place (fig.11). The stable crystalline phase S of composition Cs cannot pre-cipitate from the metastable phase L of composition co as this would require anincrease of the free energy (path AIB) until the liquid demixes (path A2B). Thisfeature, characteristic of multicomponent systems, offers an alternative explanationto the usual kinetic argument based on the differences in the speed of reactions.The study of immiscibility in glasses is therefore important not only for the develop-ment of the glass-ceramic process, but also for understanding the basic mechanismsof phase-separation in liquids.N. S. Andreev, D. A. Goganov, E. A. Porai-Koshits and Yu. G. Sokolov, Structure of Glass,(Consultants Bureau, N.Y., 1964), vol.3, p. 47.V. I. Averjanov and E. A. Porai-Koshits, Structure of Glass, (Consultants Bureau, N.Y., 1965),vol. 5., p. 63.V. I. Averjanov, N. S. Andreev, and E. A. Porai-Koshits, Physics of Non Crystalline Solids,J. A. Prins ed., (N. Holland, Amsterdam, 1965), p. 580.E. A. Porai-Koshits, D. A. Goganov and V. I. Averjanov, Physics of Non Crystalline Solids,J. A. Prins, ed. (N. Holland, Amsterdam, 1965), p. 177.J. J. Hammell, 7th Int. Congr. Glass (Brussels, 1964), (Institut National du Verre, Charleroi,Belgium, 1966), no. 36, vol. 1.S. M. Ohlberg, J. J. Hammel and H. R. Golob, J. Amer. Ceram. SOC., 1965,48,178.J. J. Hammel, J. Chem. Phys., 1967, 46,2234.J. Zarzycki and F. Naudin, Phys. Chem. Glasses, 1967, 8, 11.Y.Moriya, D. H. Warrington and R. W. Douglas, Phys. Chem. Glasses, 1967,8,19.G. W. Anderson and F. U. Luehrs, J. Appl. Phys., 1968,39,1634.J. F. MacDowell and G. H. Beall, J. Amer. Cerarn. Soc., 1969,52,17.l 3 Y. Abe, A. Naruse and K. Suzuki, J. Ceram. Assoc. Japan, 1968,76,86.l4 T. Sakasi, A. Makishima and T. Sakaino, J. Ceram. Assoc. Japan., 1969,77,31.l5 S . Scholes and F. C. Wilkinson, this Discussion.l 6 0. V. Mazurin, M. V. Streltsina and A. S. Totesh, Phys. Chern. Glasses, 1969, 10,63.l 8 W. Haller, F. E. Wagstaff and R. J. Charles, to be published in J. Amer. Ceram. SOC.2o B. E. Warren and A. G. Pincus, J. Amer. Ceram. Soc., 1940,23,301.21 A. Dietzel, 2. Elektrochem., 1942,48,9.22 E. M. Levin and S. Block, J. Amer. Ceram.SOC., 1957,40,95,113 ; 1958,41,49.23 F. P. Glasser, I. Warshaw and R. Roy, Phys. Chem. Glasses, 1960,1,39.24 R. J. Charles, J. Amer. Cerarn. Soc., 1967, 50, 631.25 R. J. Charles, Phys. Chem. Glasses, 1967,8, 185.26 R. J. Charles and F. E. Wagstaff, J. Amer. Cerarn. Soc., 1968, 51, 16.27 R. J. Charles, Phys. Chem. Glasses, 1969,10, 169.28 J. W. Cahn, Trans. Metal SOC. A.I.M.E., 1968,242,166.29 J. W. Cahn, Acta Met., 1962,10,907.30 J. E. Hilliard, Ind. Eng. Chem., 1966, 58, 19.31 J. Zarzycki and F. Naudin, J. Non-Cryst. Solids, 1969,1,215.32 C. Wagner, 2. Elektrochem., 1961,65,581.33 D. Turnbull in Solid State Physics (Academic Press N.Y., 1956), vol. 3, p. 226.34 M. Hillert, Acta Met., 1961,9, 525 ; 1962,10, 179.35 J. W. Cahn and J. E.Hilliard, J. Chern. Phys., 1958,29, 258.36 J. W. Cahn, Acta Met., 1961,9, 795.37 J. W. Cahn, J. Chem. Phys., 1965,42,93.38 J. W. Cahn, Acta Met., 1966, 14, 1685.39 E. L. Huston, J. W. Cahn and J. E. Hilliard, Acta Met., 1966, 14, 1053.40 H. E. Cook, D. de Fontaine and J. E. Hilliard, Acta Met., 1969,17,765.41 K. B. Rundman, D de Fontaine and J. E. Hilliard, to be published.42 M. Goldstein, J. Appl. Phys., 1963, 34, 1928.* O T. P. Seward 111, D. R. Uhlmann and D. Turnbull, J. Amer. Ceram. SOC., 1968,51,278.E. A. Porai-Koshits and V. I. Averjanov, 3. Non-Cryst. Solids, 1968, 1, 29.R. J. Charles and A. M. Turkalo, to be published in J. Amer. Ceram. Soc134 P H A S E- SEP A RAT ED SYSTEMS43 J. A. Williams and B. Phillips, Advances in X-ray Analysis, (Plenum Press, 1965), vol. 8, p. 59.44 S. M. Ohlberg, H. R. Golob, J. J. Hammel and R. R. Lewchuk, J. Amer. Ceram. SOC., 1965,45 D. J. Liedberg, R. J. Smid and C. G. Bergeron, J. Amer. Ceram. Soc., 1966,49,80.46 K. Nakagawa and T. Izumitani, Phys. Chem. Glasses, 1969,10, 179.47 I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids, 1961, 19, 35.48 J. W. Cahn and R. J. Charles, Phys. Chem. Glasses, 1965,6, 181.49 W. Haller, J. Chem. Phys., 1965,42, 686.50 T. P. Seward 111, D. R. Uhlmann and D. Turnbull, J. Amer. Ceram. SOC., 1968,52,634.51 W. Haller and P. B. Macedo, Phys. Chem. Glasses, 1968, 9, 153.52 R. W. Hopper and D. R. Uhlmann, this Discussion.53 V. Gerold, Phys. Status Solidi, 1961,1, 37.54 J. Zarzycki and F. Naudin, Cumpt. rend. B, 1968,266, 145.5 5 J. Zarzycki and F. Naudin, Compt. rend. B, 1967, 265, 1456.5 6 G. F. Neilson, Phys. Chem. Glasses, 1969, 10, 54.57 J. Zarzycki and F. Naudin, Cumpt. rend., 1968,266, 1005.5 8 J. Zarzycki and F. Naudin, to be published.5 9 M. Tomozawa, Thesis, (Univ. Pennsylvania, 1968), no. 69-15, p. 134.6o L. H. Shendalman and J. T. O’Toole, J. Colloid Interface Sci., 1968,27,145.48, 331.Immiscibility Phenomena in Glasses (First All-Union Cod. Leningrad, 1968), (Nauka, Lenin-grad, 1969), (in Russ.).62 M. Tashiro, 8th Int. Congr. Glass, (SOC. of Glass Technol., Sheffield, 1969), p. 113.63 J. W. Cahn, J. Amer. Ceram. Soc., 1969, 52, 1 1 8

ISSN:0366-9033    

DOI:10.1039/DF9705000122

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Small-Angle X-Ray Scattering and Scattering of Visible Lightby Phase-Separated GlassesBY N. S. ANDREEV AND E. A. PORAI-KOSHITSInstitute of Silicate Chemistry, Academy of Sciences of the U.S.S.R, LeningradReceived 26th May, 1970Modern theory and the experimental method of small angle X-ray scattering (SAXS) permit oneto find, without any a priori suppositions and with sufficient accuracy, a series of integral structuralcharacteristics of the sub-micro-inhomogeneous (supra-molecular) structure of glass. The tempera-ture dependence of the principal characteristics gives the possibility of describing the change of aglass structure during its heat treatment with great reliability. For example, the area of thegeometrical surface separating two phases has been determined, and the change of the " bidispersivestructure " with the heat treatment has been studied ; it was thereby proved that the high dispersivestructure is a result of the terminal velocity of cooling from high temperatures and that it also has aphase-separation nature (the secondary phase separation).Two methods-SAXS and scattering of visible light-have been used as experimental checks ofthe theory of spinodal decomposition, taking as an example sodium silicate glass, containing 12.5 %sodium oxide.Disagreement of the experimental results with the theoretical predictions has beenfound. This fact has been explained by the dependence of the diffusion coefficient and some otherparameters upon the concentration which have been disregarded in the approximate variant of thetheory.The main SAXS characteristics of the inhomogeneous glass structure have been defined,viz., the mean-square electron density fluctuation, the effective size and area of inhomogeneousregions and the distance between their centres.The failure of all attempts to obtain SAXS (small angle X-ray scattering) byglasses for almost thirty years has been considered as a proof of the structural homo-geneity of all g1asses.l In 1958 we obtained reproducible and distinct SAXS usingsodium-boron-silicate glasses and since that time the chemically inhomogeneousglass structure is now an experimentally proven fact. During the last ten yearsSAXS have been obtained for many glasses.Some authors have assumed the existence of phase separation (liquid immiscibilityor-in Russian-liquation) in glasse~.~ The strict quantitative proof of the phasenature of the supra-molecular glass structure using SAXS was given in 1966.4 Thisproof was facilitated by the development of modern experimental techniques and ofthe theory of SAXS methods by physicists of different c~untries.~~We can now use the following integral structure characteristics, obtained by meansof SAXS of the sub-micro-inhomogeneous (supra-molecular) structure of glasses :13136 X-RAY SCATTERING BY PHASE-SEPARATED GLASSESHere (Ap)” is the mean-square fluctuation of the electron density in a glassseparated into two phases (droplets and matrix) ; w1 and w2 are the volume fractionsof these sub-microphases (w, + w2 = I), which have average electron densitiesPI and p2, and V is the scattering volume; IsAxs is the SAXS intensity; (9 is thescattering angle ; s = (4n sin (+/2)}/a _N 2n$/A; A is the wavelength; usAXs, &Axsand ZsAxs are the effective volume, area and dimension of the scattering inhomogeneousregions (values for regions of a spherical form are given in square brackets).Allof these integral structure characteristics can be obtained without any a priorisuppositions about the structure of the scattering substance.The first (1) of the characteristics is “ substantial ” (it depends upon the composi-tions of the regions), the rest are geometrical. The interpretation of the geometricalcharacteristics ((2)-(4)) for the glasses is complicated (if the inhomogeneous regionshave no spherical form), because a mathematical description of the irregular structureis very difficult.But all of the characteristics are very sensitive to the slightestalterations of the supra-molecular structure of glasses when their composition orheat treatment is changed.In addition, we can determine other structural characteristics using different partsof the SAXS curve, e.g., the size of the radius of gyration R of the single region, whichcan be found from part of the smallest angle X-ray scattering curve by means of theGuinier equation. Then, by using S A X S at large enough values of the angle 4, wecan determine the area of the generalized interface 9 separating these regions (“ sub-microphases ”) by means of the equation,where S is the true (geometrical) interface.The exponent 4 refers to spherical regions.Finally, it is possible to calculate one more factor,03= s zSAXS/8vl w2 21 ISAXS(s)s4/ ISAXS(s)s ds/[m]290which determines the deviation of the inhomogeneous regions from the sphericalform and the degree of their interconnection (at sufficiently dense packing, the tworegions near each other can develop a region having an oval form), Therefore thesimultaneous use of all the characteristics (under certain conditions) makes it possibleto describe a sub-microscopical glass structure in detail.The mean-square fluctuation of electron density (1) was used to prove the phasenature of supra-molecular structure of glass.4 The values of (Ap)& were deter-mined in absolute electron units from experimental curves IsAm for a series of phase-separated glasses which had been heat treated at different temperature without anyinformation on the composition and volume of each phase.The absolute valuesof the mean-square difference between the electron densities of phases enriched insodium oxide and silica, (Ap)&lc., can be also calculated from the binodal curvefor all glasses under the cupola of immiscibility.A comparison of plots of (Ap)&,,. and (Ap)Zalc. against the composition of theglass at different temperatures for sodium silicate glasses was made, and the agreementwas very good.’ The phase nature of the supra-molecular glass structure wasproved; such agreement also proved the correctness of the SAXS data.Moreover,it is possible to use the binodal curve for determining the compositions of the separatedphases and their alteration during heat treatment or as a result of changing theglass composition. And we can conclude that both the supra-molecular structureand the properties of the phase-separated glasses which depend on this structurehave become controllable.~N. S. ANDREEV AND E. A. PORAI-KOSHITS 137However, we have only considered the simplest case. It is not always possibIeto delineate the complete cupola of immiscibility even for two-component systems(e.g., if for no other reason, because of the possibility of crystallization), and it ismore difficult to obtain results in three-component and more complicated systems.In the latter cases a more difficult problem of deriving conodes, which help us todetermine the composition of co-existing phases, arises. Moreover, such an approachgives no information about the sizes, forms, space distributions of the inhomogeneousregions, and the interface of the co-existing phases etc.Therefore is is importantto use all the information which the methods of SAXS can provide without anya priori suppositions.An example of such use of the SAXS method was given by us * for sodium silicateglass with 14 mol % Na,O. We recall that the generalized interface of phaseswas determined from the curves of the dependence of log ZsAxs on log 4. The lineardependence on the slope (- 4) denotes a spherical form of the inhomogeneous regions(see eqn (5)).A bi-dispersive structure was discovered at a certain temperature (fig.1). Wecan use two straight lines to divide the original SAXS curve into branches whichcorrespond to large and small regions and then estimate their sizes : they were equalto about 500 and 60A. These sizes correspond well with those determined by usingthe tangent method and from electron microscope data.1-00 2-001% 9FIG. 1.-The dependence of log I s m s upon log s demonstrating a bidispersive structure.Above and below the immiscibility temperature a fine inhomogeneous structure isformed (small droplets) with sharply different values of (AP)~ and s. The nature-of this " fine " structure was discussed earlier.' The explanation for the appearanceof the " fine " structure as a secondary phase separation as a consequence of over-saturation of a primary separated phase (matrix or droplets) by silica or by alkalihas been given in our paper on the electron microscope study of the phenomena ofsecondary (tertiary, etc.) decomposition in silicate gla~ses.138 X-RAY SCATTERING BY PHASE-SEPARATED GLASSESThese investigations prove the sensitivity of all SAXS structure characteristicsto small changes of structure and, in particular, to small changes of the differencesbetween the electron densities of regions and those of the matrix, the latter beingimportant for studying the initial stages of phase decomposition.Tn this respect theSAXS method has an advantage over the electron microscope method which givesmore detailed information on the geometry of regions but is of low sensitivity totheir changes of composition.Some years ago many researchers have consideredthe possibility of experimental verification of Cahn’s spinodal decomposition theoryby means of the SAXS and light scattering methods. According to current ideas 9 9 lothe process of phase separation in a spinodal region in immiscible solutions includingglasses is given by the general diffusion equation and its solution can give, in principle,comprehensive information about all the stages of structure evolution of glass duringits phase separation. At present, this equation has not yet been strictly and generallysolved. In the special case of the initial stage of decomposition the dependence of thediffusion coefficient D = M(a2f/dc2) cf is the free energy of the system, M = uc,where u is the atomic mobility) and M upon the concentration c (r,t) at time t at apoint of space defined by the position vector r can be apparently neglected, i.e., thethird and fourth derivatives of the free energy with respect to composition arenegligible.g-llwhere all the values correspond to the initial average concentrations of c,, the diffusioncoefficient Do being negative.Under such conditions the general Cahn’s equation has the form :aclat = D , v ~ C - ~ ~ M , , (6)Eqn (6) can be solved to givec(r,t) = co + ( l / 2 ~ ) ~ A(fl,t) exp (iflr) dp, (7) s where A(fl,t) is the Fourier amplitude of the wave vector component at the time t.I B I = 27t/A, where A is the spinodal wavelength.The amplitudeA(fl,t) = [ [c(r,t) - c,] exp (- ipr) dris related to the concentration c in the same way as the structure amplitude F(s,t) isrelated to the electron density,VF(s,t) = [p(r,t)-pPo] exp (- isr) dr,/ Vwhere V is the scattering volume and po is the average electron density.wheref, andf, are the atom factors of the two phases andif s = p, i.e., if (4n sin ( 4 / 2 ) ] / A = 2n/A, or A = 2A sin (4/2). This is Bragg’s law.This means that if solution (7) is correct, the scattering intensity of electromagneticradiation by macroscopic isotropic bodies can be related to the interdiffusioncoefficient and to the mobility through the amplification factor P(s) in the followingway:wherecan now be controlled.Actually,p(r7t)--p0 = dfi -f2)[c(r7t)-ccol,F ( s 9 t ) = (fl - f 2 > A ( p > t ) ; (8)I(s,t) = I(s,O) exp {2P(s)t 1, In I(s,t) = In I(s, 0) - 2P(s)t, (9)(10) P(s) = - D0s2 -2yM,s4N .S . ANDREEV AND E . A . PORAI-KOSHITS 139From (10) it follows that at s = 0, P(s) = 0, and at s = sm = (l/2)[(-d2j/ac2)/2ylf,the amplification factor P(s) reaches its maximum value Pm, and at s = s, = smJ2it again returns to zero.The formula (9) allows an experimental check on the theory of spinodal decom-position in its initial stage. The experiments confirmed that, in agreement with (lo),the dependence of P(s)/s2 upon s4 is expressed by a straight line cutting the ordinateaxis at a point numerically equal to the diffusion coefficient Do.12 The study ofphase separation can be carried out more completely by using two complementarymethods-visible light scattering (small s) and small-angle X-ray scattering (larger s).The results of such investigations of a sodium silicate glass with 12.5 mol % Na,Oare reported below.The initial glass samples had been obtained by sharp quenchingof the melt in a steel mould from 1600°C. The chosen temperature of heat treatment,530"C, was 300°C lower than the temperature of immiscibility on a binodal curve.13The experimental curves of visible-light scattering intensity are listed in fig. 2.Fig. 3 shows the dependence of the SAXS intensity upon s. The latter also showsthe corresponding visible light scattering curves drawn to scale which permit asmooth change to curves obtained in an X-ray region.500 88 4 0 - .t: 1Llc302 01008o 70 t----6 0 I-L/ l4 // l 320 40 60 80 100 120 1404, deg.FIG.2.-Intensity curves of visible light scattering, temp. 530°C. 1 , 1 h ; 2, 5 h ; 3, 7 h ; 4, 10 h ;5, 14 h; 6, 16 h; 7,20 h ; 8, 23 h ; 9, 25 h; 10,28 h ; 11, 30 h ; 12, 33 h ; 13, 38 h; 14,42.5 h.Examples are found in fig. 4, where the dependency of ln I upon time for several sis shown ; in the region td20 h this dependency is rectilinear. Consequently, underour conditions, the check of the applicability of expressions (9), (10) must be limitedto 20 h. Accordingly, we have calculated the dip angle tangents for the rectilinearparts, and half-values are labelled P. In fig. 5, curve 1 depicts the experimentaldependency P(s), whereas curve 2 is a result of the calculation of P(s) according toeqn (lo), where values Do, Mo are taken so that the maximum values of P on th140 X-RAY SCATTERING BY PHASE-SEPARATED GLASSES84, 0.01 002 0.03 0.04 0.05 0.06 0.07s, A-'FIG.3.4ntensity curves of X-ray and visible light scattering plotted as q function of s. Visiblelight scattering occupies region s<sl = 0.004 A-l. The dottled line show interpolation regions.'Temp. 53OoC, 1 , 1 h ; 2,lO h ; 3,12 h ; 4,16 h ; 5,20 h ; 6,25 h ; 7,30 h ; 8,42.5 h.. 10 20 4 0time, hFIG. 4.-The dependence of In I upon time f ; 1, s = 0.001 65 8, ; 2, s = 0.004 A-' ; 3, s = 0.01 A-' ;4, s = 0.025 A-1 ; 5, s = 0.037 A-', (1,2, visible light, 3, 4, 5, X-rays)N.S . ANDREEV AND E. A . PORAI-KOSHITS 141experimental and calculated curves coincide. From a comparison of curves 1 and 2in fig. 5 it follows that the general nature of the behaviour of the P(s) experimentalcurve can hardly be represented by formula (10). This is definitely confirmed byfig. 6a (visible light) and fig. 6b (X-rays), from which is it evident that in contradictionwith (10) the experimentally obtained dependency P(s)/s2 as a function of s2 is notrectilinear.0'902 0.03Is, A-1FIIG. 5.-The dependence of the amplification factor P upon s.scattering data of visible light ( ~ < 0 . 0 0 4 ~ - ~ ) and X-rays.under the conditions : Do = -6.2 x lo-'' cm2 s-l, 2M0y = 6.2 x1, experimental curve plotted from2, curve plotted according to formula (5)~ m - ~ s-l.Thus, the process of phase separation in these glasses does not obey theregularities which can be inferred from eqn (6).Before giving an explanation ofthis fact it is expedient to analyze the experimental intensity curves from the standpointof the general theory of small-angle X-ray scattering, which provides information onthe inhomogeneous glass structure irrespective of its formation mechanism. Thisanalysis also includes light scattering data, since intensity is considered as a functionof s.The results of the calculations of (Ap)Zxpt., Isaxs/2 and qSAm (see eqn (I), (3) and(4)), and the mean distance between the centres of the inhomogeneous regionsLsAXS = 27z/s&, where SA determines the position of the maximum directly on theintensity curves (fig.3), are shown in fig. 7. The dotted line of fig. 7 illustrates thevalue (Ap)&lc. = 58.6 x (eZ/A3)2 which corresponds to the complete separationof the glass into two phases at 530". The values have been obtained by using therule of lever in the immiscibility curve and from the density of developing phases.The ratio $ = (Ap)Zxptt./(Ap)&lc. characterizes the degree of immiscibility that hasbeen achieved at a given stage of the process. For the initial glass,--(Ap)Zxpt. = (2342) x (eZ/A3)' and # = 0.391 42 X-RAY SCATTERING BY PHASE-SEPARATED GLASSESThe question is at what stage of decomposition does this value correspond?The calculations based on the theory of regular solutions l4 have shown that the value(Ap)", which corresponds to the attainment of spinodal concentrations of thedeveloping phases at 530", is equal to 20 x ( e Z / ~ f ~ ) ~ which is close to the experi-mental value for the initial glass. The use of other formulae from ref (14) gives stilllower values.0 5 10 15 0 200 400 600 800 1000s2 x 1O-lo, c t r 2 s2 x cm-2FIG.6.-The dependence of P(s)/s2 upon sz from scattering data of visible light (a) and X-rays (b).From these facts it follows that the degree of glass immiscibility occurring duringits quenching from the melt (the main effect) and during 1 h exposure at 530" (theadditional effect) is such that on further heat treatment the composition of theinhomogeneous regions changes between the values determined by the spinodaland binodal positions.Once inhomogeneous regions have reached spinodalconcentrations, the second concentration derivative of the free energy of the systemreaches zero and on further processing becomes positive. Evidently, under suchconditions, D(c) # Do. Thus, on correlation of the experimental results with thetheory, conclusions will be possible only after solving the general Chan's equation,which takes into account the dependence of the diffusion coefficient and other para-meters upon the concentration.Though the value (AP)~ has not been determined by Zarzicki l5 there is a reasonto believe that the degree of phase separation of the initial glasses (reported in thepaper) is considerably lower than that in our investigation and is therefore a betterapproximation to the conditions under which the equation (6) holds true.The dataof ref. (15) must be considered as a confirmation of the theory of the initial stage ofspinodal decomposition only to a first approximation, because a specific critical values, was not observed by himN . S . ANDREEV AND E. A . PORAI-KOSHITS 143Of particular interest are the results of Neilson l6 who used glasses close incomposition to those in our investigation. First, the experimental results of glassesof group B (ref. (16)), for which the initial degree of phase separation I) at 600°C was0.35 also could not be interpreted on the basis of the theory for the initial stage ofspinodal decomposition. Although, in principle, the process may proceed by amechanism of nucleation and growth, we nevertheless would explain the negativeresult by the fact that separation of glasses at their quenching had been too fardeveloped.ttime, hFIG.7 . T h e dependence of parameters characterizing inhomogeneous glass structure upon time.From this point of view the convincing but still incomplete agreement of theexperimental results with the predictions of the theory for considerably more weaklyquenched glasses of group A is unexpected. For glasses of this group, II/ = 0.6and the diameters of inhomogeneous regions are in the range of 300-600 A. If seemsthat under such conditions we cannot neglect the dependence of the parameters in thegeneral Cahn’s equation upon the concentration. These discrepancies can beresolved only after obtaining the strict solution of the general equation, which describeall stages of decomposition.It is also essential to discover the initial conditions forsatisfying the general equation as it describes the change in the modulated inhomo-geneous structure with time, but gives no information about the mechanism of thedevelopment of this structure.Heused the solution of the general Cahn’s equation given by de Fontain l 8 in order tointerpret two unusual maxima in the amplification factor curves for sodium silicatcglasses. However, we think the results of this work need further discussion.Recently, Tomasava has made an interesting attempt in this directi0n.l144 X-RAY SCATTERING BY PHASE-SEPARATED GLASSESl see e.g., R.James, The Optical Principles of the Diffraction of X-rays, (London, 1950).E. A. Porai-Koshits and N. S. Andreev, Nature, 1958,182,336 ; Dokldy Akad. Nauk U.S.S.R.,1959,118,735; J . SOC. Glass. Techn., 1959, 43,235; N. S. Andreev, E. A. Porai-Koshits, andJu. G. Sokolov, Isvestia Akad. Nauk U.S.S.R., O.H.N., 1959, 4, 636.M. E. Nordberg, J. Amer. Ceram. Soc., 1944, 21, 299 ; W. Hintz and P. 0. Kunth, Glastechn.Ber., 1961, 34, 431 ; R. Roy, Symp. Nucl. Cryst. in Glasses and Melts; Amer. Ceram. Soc.,1962, 39 ; F. Ya. Galakhov, Isvestia Akad Nauk U.S.S.R., O.H.N., 1962,5,743.D. A. Gaganov and E. A. Porai-Koshits, Doklady Akad. Nauk U.S.S.R., 1966, 167, 1266 ;N. S. Andreev and T. I. Ershova, boklady Akad. Nauk. U.S.S.R., 1967,172,1299.0. Kratky and G. Porod, J. Colloid Sci., 1949, 41, 35 ; 0. Kratky, G. Porod and L. Kahovec,2. Elektrochem., 1951,55,53; G. Porod, KolloidZ., 1951,124,83; 1952,125,51.V. N. Filipovich, J. Techn. Phys., U.S.S.R., 1956, 26, 398; 1957, 27, 1029. ’ E. A. Porai-Koshits and V. I. Averianov, J. Non-Cryst. Solids, 1968, 1, 29.* D. A. Goganov and E. A. Porai-Koshits, Doklady Akad. Nauk U.S.S.R., 1965, 165, 1037.J . W. Cahn, Acfa. Met., 1961, 9, 795.V. N. Filipovich, 1. Neorganich. Mat. U.S.S.R., 19b7, 3, 993, 1192.I t J. W. Cahn, J. Chem. Phys., 1965,42,93.l 2 K . B. Rundman and J. E. Hilliard, Acta Met., 1965, 15, 1025.l 3 N. S. Andreev, G. G. Boiko and N. A. Bokov, J. Non-Cryst. Solids, 1970, 5, 4154.l4 H. E. Cook and J. E. Hilliard, Trans. A.I.M.E., 1965, 233, 142.l5 J. Zarzicki and F. Naudin, J. Non-Cryst. Solids, 1969, 1, 215.l6 G. E. Neilson, Phys. Chem. Glasses, 1969, 10,54.M. Tomasova, Thesis (University of Pennsylvania, 1969).D. de Fontain, PhB. Thesis., (Northwestern Univ., 1967)

ISSN:0366-9033    

DOI:10.1039/DF9705000135

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Phase Separation in Glass and Glass-Ceramic Systems *BY G. F. NEILSONOwens-Illinois, Inc. 1700 N. Westwood Ave. Toledo, Ohio 43607 U.S.A.Received 16th June, 1970The capabilities and limitations of the small-angle X-ray scattering technique which have beenfound for the establishment of the mechanisms of early stages of phase separation in glass systemsare examined. This includes the evaluation of the criteria for distinguishing between the mechanismof initial separation such as nucleation and spinodal decomposition. The application of this techniquefor directly following nucleation processes is given. The information obtained employing thesemethods concerning qualitative differences in the early stages of phase separation as functions ofcomposition and temperature of Zr02-containing phases in two MgO -A.1203 *3Si02 -X ZrOzglasses are presented.The X-ray scattering results show that precipitation of a ZrOz phase is nota necessary prelude to the crystallization of the glass.In our laboratory, a research programme is in progress to obtain informationon the nucleation and growth mechanisms that lead to phase separation in SiOzglasses. A primary goal has been to establish the capabilities and limitations ofsmall-angle X-ray scattering (SAXS) for such studies. In the course of this work,the kinetics of the early stages of phase separation has been studied by followingvariations in the totally accessible SAXS curves of quenched glasses after isothermalheating for increasing time. Variation in glass structure with heat treatment wasalso studied by electron microscopic examination, when possible.We have studiedglass systems in which glass-in-glass immiscibility is known or was suspected tooccur. Low temperature studies where spinodal decomposition may occur, studiesat intermediate temperatures where non-classical nucleation mechanisms may beoperative, as well as high temperature studies in the classical metastable region havebeen made.The most detailed SAXS study of the kinetics of the latter stages of phase separationwas reported by Zarzycki and Naudin.2 Several SAXS investigations have beenreported on the kinetics of phase separation at temperatures where spinodal decompo-sition may be occurring. 3-5In this paper some observations made during SAXS studies of the kinetics ofphase separation of three glasses are reported : (1) a Na,O-SiO, glass containing12.6 mol % Na,O, (2) and (3) gtasses of stoichiometry Mg0-AI2O3-3SiO2 withadditions of 10.3 and 7.8 wt % ZrO, respectively. Glass (1) is in the system whichhas been extensively studied by Porai-Koshits and associates.However, theseworkers have employed SAXS data mainly to determine the characteristics of fullyphase separated Na,O-SiO, glasses, and they have been less concerned withempIoying the technique for the study of the separation processes themselves.Glasses (2) and (3) form glass ceramics upon heat treatment, with the ZrO, requiredas a nucleating agent. There has been some controversy regarding the role of thenucleating agent in promoting homogeneous crystallization in this type of glass* Research partially sponsored by Air Force Office of Scientific Research, Directorate of PhysicalSciences, under Contract No.AF 49 (638)-1701.14146 PHASE SEPARATION I N GLASS SYSTEMSceramic system, which the present investigation may tend to clarify. Glasses (2)and (3) are particularly suitable for SAXS studies of the nucleation processes leadingto the formation of a Zr-rich phase. This technique is very sensitive to such changesbecause of the relatively high atomic number of Zr with respect to the other ionicspecies present in the glasses.EXPERIMENTALThe glasses employed were all prepared from reagent-grade materials. The NazOglass was poured as a slab and annealed at 450°C for 4.5 h.The 10.3 % ZrOz glass wasalso poured as a slab but annealed at 791°C for 15 min. The 7.8 % ZrOz glass was pulledas 5 in. rod and air-quenched. For scattering measurements, glass platelets of nominal0.14 mrn thickness for the Na,O glass and of 0.09 mrn thickness for the ZrOz glasses wereprepared after heating, by the method previously de~cribed.~The type of instrumentation used and its method of employment has also been de~cribed.~However, for this work, two Kratky cameras were employed in conjunction with a copper-target X-ray tube. One camera was of low resolution and was employed for obtainingscattering data in the intermediate and higher angular ranges (about 5 to 100mrad). Theother high-resolution camera was used to obtain scattering data to angles as low as 0.4 mad.Some of the SAXS curves obtained (and shown here) represent composite curves obtainedwith the two instruments under varying slit conditions as dictated by the magnitude andangular rate of change of the scattered intensity. Slit-height corrections were made onlyfor the angular range in the Guinier region to allow more accurate determination of the ( D2)and I(0) parameters.RESULTS12.6 Na,O-SO, GLASSScattering measurements on the course of phase separation have been made atseven temperatures within the spinodal region in the temperature interval from 500to 660°C and at two temperatures, 770 and 790°C, in the metastable region abovethe spinodal.A more detailed account of our study of the kinetics of nucleationand growth in the metastable region than can be given here will be published elsewhere.During the early stages of phase separation at all temperatures the theoreticalhigh-angle asymptotic behaviour given by h3F(h) = constant, where F(h) is theexperimental scattered intensity for infinite slit-height, and h = ( 4 4 4 sin 0, wasnot obeyed. Instead, scattering curves typical of the curve A shown in fig.1 wereobtained. However, at times sufficiently long such that the phases have probablyattained their equilibrium compositions, the limiting law is reasonably obeyed asshown by curve B in this figure. It appears doubtful that the determination ofinterfacial areas S from such data is very meaningful before the equilibrium conditionsare obtained.We also note that before the asymptotic scattering behaviour isexhibited, the product hF(h) does not converge toward zero with increasing h, andthus the invariant Q defined byQ = 1 co hF(h) dh,0cannot be accurately evaluated without making arbitrary assumptions concerningthe nature of the background scattering.Guinier plots of the scattering exhibited by the glass initially and after short-timeheat treatments at temperatures in the spinodal and metastable regions, respectively,are shown in fig. 2. Extended linearity is observed for the initial glass (curve A)suggesting the presence of particles ; and indeed, micrographs of this unheated glassdo show the presence of discrete, but densely packed, particles whose sizes are iG. F. NEILSONnC3rA u ._E a;;j 10-8UWc, x3 0 3 * -l *147L --- - -----I I 1 I I I I0 2 4 6 8 1 0 1 2 1 4128 (mrad)FIG.1 .-Log experimental intensity against log scattering angle curves for NazO glasses after heatingat 600°C for (A) 3 h and (B) 32 h. Lines drawn show slope of -3.100148 PHASE SEPARATION IN GLASS SYSTEMSreasonable agreement with that determined from the slop of this curve. However,linearity is exhibited for a larger angular range than would be expected by the Guinierapproximation for a monodjsperse system of spheres, as is discussed later. However,after short-time heat treatments in the spinodal region at all temperatures (exemplifiedby curve B), no lineatity is observed and the intensity maxima become more pro-nounced.Also, the angle of cross-over of the curves (the critical wavelength) isconstant as a functioa of time. In contrast, upon heating in the metastable region(exemplified by curve C), the intensity maximum rapidly disappears, with linearityin the Guinier plots being maintained to lower angles. Although the high-angleintensity components also rapidly disappear at these higher temperatures, the cross-over of the scattering curves continuously shifts to lower angles,From the electron micrographic study of the glasses heated in the metastableregion, the continuous growth of roughly-spherical particles is observed. Themeasured size distributions of the growing particles are in acceptable agreement withthe r.m.s. diameters (D2>* of the particles as determined from the limiting slopes ofthe Guinier plots of the scattering data.Also of interest is the variation with timeof I(O), the extrapolation to zero angle on a Guinier plot of the slit-height correctedintensity, i.e.whereI(h) = I(0) exp [ -(D2)h2/20], (1)Z(0) = KNAp2(V2>, (2)N being the concentration of particles in the glass, Ap the difference in electrondensity between particle and matrix, and ( Y2> the mean-square volume of a particle.We set <V2> = (,/t/6)2<02>,3 with <Da> determined by eqn (I), and have determinedAp at each of the two temperatures studied from the known coexistence curve. Fromthese data and the measured Z(0) as a function of time, the variation of N with timeand temperature was determined.For times of 1 h and longer, N varied inverselywith time at both temperatures, in excellent agreement with the micrographic results.For times of 3 h and less, N as calculated from the scattering data was about twicethat determined from the micrographs. It is in accord with other results that thediscrepancy at early times is due to a concentration of Na20 in the regions surroundingthe forming particles in excess of the equilibrium concentration. From these andsimilar results obtained on other glasses, it is believed that I(0) measurements canprovide useful independent information readily obtainable concerning details ofphase separation processes in glasses, so long as only particle formation and growthis occurring and the linear Guinier region of the scattering curves is accessible tomeasurement.For the glasses heated at 790°C, the occurrence of two apparently independentprocesses were followed by the SAXS data.This temperature is only about 10"below the immiscibility temperature for this particular composition, which is in thetemperature range where a classical nucleation mechanism should occur. In fig. 3is shown the type of variation of behaviour exhibited for two of the heat treatments.In a h3F(h) against h plot, a double maximum is usually indicative of bimodal sizedistributions. It is suggested that the increase with time of the smaller size fraction(the component at the high h values) may correspond to continuous formation ofnuclei, whereas the increase of the larger size fraction is due to particle growth.Itis possible that the development of the small size structure corresponds to the forma-tion of the secondary microstructure found by Porai-Koshits.6 However, we didnot observe the continuous formation of this small structure for the glasses heatedat 770°C. This would indicate that the process occurs at the temperature of heaG. P . NEILSON 149treatment and not during the subsequent quench through the spinodal region, aswas cited for the origin of the secondary microstructure.28 (mrsrd)FIG. 3.-Log h3F(h) against scattering angle curves, wbere F(h) is experimental intensity for Na20glasses after heating at 790" (A) for 5 min and (B) for 60 min.Mg0-A1203-Si02 + 10.3 % ZrO2 GLASSIn fig.4 are shown typical experimental SAXS curves for glasses heated at 791,850 and 903°C for intermediate times during the precipitation of a Zr02-rich phase,but prior to the crystallization of the glass itself. At 850" and above, the scatteringarises from the presence of particles in the glass of a crystalline ZrOz species, and adetailed account of the kinetics of growth of these crystallites at these higher tempera-tures will be published elsewhere. However, several of the results obtained from theinvestigations (SAXS, X-ray powder diffraction, and electron microscopy) at thesetemperatures are pertinent for the present discussion. The pronounced maximaobserved in the scattering curves after heating the glass at 850°C were found to arisefrom the presence of diffusion depletion regions around the particles similar to aWalker-Guinier type.' The crystallite size as determined from Guinier plots of thescattering data from the high-angle side of the maxima were in good agreement withthose measured from the corresponding micrographs over a wide range of sizes(90-400 A diam.). Also, analysis of the scattering data of glasses heated at highertemperatures indicates the independent formation and growth of both large andsmall size particle fractions.Thus for curve D of fig. 4, analysis indicates thepresence of particles 114A diam. plus another group of particles of narrow sizedistribution whose largest size is 7SO'A. The development of the smallerparticle fraction with time was interpreted as the development of subcriticalprecursor to critical size nuclei of the Zr02 phass which with increasing timeapproaches the steady-state concentration150 PHASE SEPARATION IN GLASS SYSTEMSThe combined investigations mentioned above indicate that at 79 1 "C the initialseparation through 64-h heating occurs through the formation of amorphous regionsenriched in ZrOz.In fig. 5 are shown Guinier plots for various heating times atthis temperature. Again, for longer heating times, maxima in the scattering curves20 (mad)FIG. 4.-Plots of experimental scattering curves against scattering angle of (A) initial 10.3 % ZrOzglass showing variation in relative intensity and angular behaviour after heating (B) at 791°C for64 h, (C) at 850°C for 8 h, and (D) at 903°C for 90 min.develop as shown in the insert.In this system also, the background scattering ishigh and fairly constant as the Zr02 phase separation occurs, but largely vanisheswhen the crystallization of the glass commences. Thus, again it is difficult to obtainreliable values of Q and S for the early stages of separation. For determination ofthe scattering characteristics of the developing phase the extrapolated backgroundscattering (dashed line in fig. 5) was subtracted from the measured intensity. Theresults of this procedure for the shorter heating times are given as Guinier plots infig. 6. Here the arrow indicates the value of 28 for which ho = 2/R, where the radiusR = 13 A was determined from the slope of the curves. Linearity is maintainedover a much larger angular range than that for which the Guinier approximationfor spheres is valid, as given by this ho value.For times of 0, + , 2 and 8 h the relativevalues of I(0) were determined as 2.0, 3.7, 4.5 and 5.4, respectively. These scatteringresults suggest that the separating regions have properties associated with discreteparticles, rather than those properties associated with separation by means of spinodaldecomposition. The development of pronounced scattering maxima at the latertimes when growth of amorphous particles occurs is then interpreted as arising fromthe formation of limited regions about each particle which become depleted in Zrion concentration. Thus, for the 64-h heat treatment when the calculated particlediameter is 68 A, it is estimated that the depletion region extends about 90 from theparticle surface.These results, together with the observation that the slit-correcteG . F. NEILSON910 Lx1510 5(20)2 (mad2 x10FIG. 5.-Guinier plots (experimental scattered intensity) of (A) initial 10.3 % ZrOL glass and afterheating at 791"Cifor (B)L8 h (C) 32 h and (D) 64 h. Insert shows intensity against scattering anglescurves (linear scales)-with expanded 20 scale to show behaviour exhibited at low angles.lo2 0 1) I(2€I)2 (mad2 xFIG. 6.-Guinier plots (corrected scattered intensity) of (A) initial 10.3 %Zr02 glass and afterheating at 791°C for (B) + h and (C) 8 h. Here intensity is obtained from the measured intensityminus the dashed line fig. 5152 PHASE SEPARATION IN GLASS SYSTEMSscattering curve extrapolates to the neighbourhood of zero intensity at zero scatteringangle, strongly implies that the growth process is diffusion controlled at this tem-perature. The result that no particle growth is observed during the first 8 h of heatingsuggests that the nucleation process may involve the continuous increase of Zr ionconcentration to regions which do not increase in extent.This conclusion appearscompatible with the Cahn-Hilliard theory of non-uniform nucleation.Mg0-A1203-3Si02 +7.8 % ZrO, GLASSScattering studies of the kinetics of phase separation were made at the sametemperatures for this glass as for the preceeding one. Some of the SAXS dataobtained at 903°C are shown in fig. 7. It is most significant that after 10 h of heatinga crystalline SiO, phase can be detected by X-ray powder diffraction, and the phaseis well developed after an additional 1 h of heating.However, through 10 h, theSAXS data show conclusively that no macroscopic phase separation involvingZrO, has occurred. Rather, the SAXS results indicate that during this period anordering process involving 21-0, on a scale of about 25 A occurs continuously from6-10 h heating. No measureable change in the glass structure takes place duringthe first 6 h. It is estimated that no more than 1 % of the available ZrO, is involvedin the structural rearrangement of the glass at the time crystallization commences.Apparently, during crystallization of the glass (after 11 h heating), an exsolution ofthe ZrO, occurs, resulting in the precipitation of a finely divided ZrO, phase.TheSAXS data indicate that little particle growth occurs during this process through15 h. The phase can be detected after 14 h by X-ray powder diffraction. Qualita-tively similar results to these were obtained at 850 and 977°C.I 1 I I I 10 10 20 3 020 (mad)FIG. 7.-Plots of experimental scattered intensity (log scale) against scattering angle of (A) initial7.8 % ZrOz glass and after heating glass at 903 C for (B) 8 h, (C) 10 h and (D) 11 h.DISCUSSIONThe results presented indicate that with suitable glass systems the nucleationprocesses involved in phase separation can be followed directly and concurrentlywith the growth processes by the SAXS technique.In discussing the early stages oG . F. NEILSON 153separation of amorphous or solid solution phases, it is convenient to differentiatebetween (1) the classical nucleation region, (2) the non-classical Cahn-Milliard region,and (3) the spinodal region. For the direct formation of crystal nuclei in an in-coherent matrix, probably only region (1) is meaningful.In region (l), which occurs in a limited temperature range below the equilibriumtemperature, the nuclei and growing particles have constant internal compositionand well-defined composition boundary with the matrix. The critical radii of thenuclei in this region vary inversely with the supercooling. After a steady stateconcentration of nuclei have formed, if the nucleation frequency is much less thanthe growth rate of the super-critical particles, a bimodal distribution of particlesizes will result.The results obtained on the 790" study of the Na,O glass appearin accord with this interpretation, as do the results obtained on the studies of the10.3 % ZrO, glass at temperatures of 903" and 977°C where crystalline nuclei probablyform.Region (2) is characterized by the formation of nonhomogeneous (but independent)nuclei with a diffuse and extended composition interface between particle and matrix.This region is of particular importance at temperatures in the vicinity of the spinodalcurve. It is speculatively suggested that the extended linearity (well beyond theextent of validity of the Guinier approximation for spheres) observed in Guinierplots of glass heated at temperatures which may fall within this region, may arisefrom the formation of such diffuse nuclei.If a spherically shaped forming nucleushas a radial composition profile which is Gaussian in shape, its scattering intensitycurve is given by I(h) = h-210 exp (-Kh2) for all angles, which at higher anglesclosely approximates the Guinier curve. The above anomaly is not observed tooccur during particle formation upon heating glasses at higher temperatures.Region (3) is characterized by the development of periodically modulated andcontinuous composition variation in the glass. Here the SAXS behaviour is dueprimarily to a cooperative diffraction effect, rather than to the scattering characteris ticsof individual regions.Accordingly, the scattering maximum which develops duringthis process can be identified with the periodicity wavelength. For this case anyattempt to analyze the shape of the scattering curve as one would for a system ofincoherent particles is meaningless. Unfortunately, we have noted that scatteringmaxima at low angles can, and do, arise also from glasses containing isolated particleswhen diffusion depletion regions of limited extent develop about them. However,in this case the position of the scattering maximum cannot be interpreted in terms ofa repeat distance, whereas the Guinier analysis of the scattering curve on the highangle side of the scattering curve is meaningful. It does not appear to be possibleto distinguish with certainty between these two extreme cases from the shape of ascattering curve alone.Thus, the type of analysis which is carried out on the scatter-ing data must depend, at least in part, upon other independently known characteristicsof the system.CONCLUSIONSFor studying the nucleation and early stages of particle growth in glasses bySAXS, the I(0) and (D2) parameters obtained from the scattering data may provideinformation most amenable to interpretation of the kinetic processes, so long asspinodal decomposition is not occurring. Under favourable conditions the nuclea-tion and growth processes may be followed concurrently and independently withSAXS measurements alone. Development of maxima in scattering curves cannotbe regarded as conclusive evidence of the occurrence of spinodal decomposition,since such maxima can also form during particle growth as diffusion depletion regionsabout the particles develop154 PHASE SEPARATION IN GLASS SYSTEMSWith respect to the glass-ceramic systems studied, the results indicate that depend-ing upon the heating temperature and concentration of Zr02 in the glass, eitheramorphous phase separation involving Zr02, formation and growth of crystallitescontaining ZrO,, or only local small-scale ordering processes involving the ZrO,may occur prior to crystallization of the glass. The SAXS data conclusively showthat the precipitation of a ZrOz phase need not be involved in the mechanism bywhich the crystallization of the glass occurs.J. W. Cahn and J. E. Hilliard, J. Chem. Phys., 1959, 31,688.J. Zarzycki and F. Naudin, Phys. Chem. Glasses, 1967,8, 11.M . Tomazava, H. Herman and R. K. MacCrone, Proc. Inst. of MetaZs Symp. Mechanism ofPhase Transformations in Crystalline Solids, (Univ. Manchester, 1968), p. 1 .G. F. Neilson, Phys. Chem. Glasses, 1969,10, 54.J. Zarzycki and F. Naudin, J. Non-Crystal. Solidr, 1969,1,215.E. A. Porai-Koshits and V. I. Averjanov, J. Non-Crystal. Solids, 1968, 1, 29.C. B. Walker and A. Guinier, Act. Met., 1953, 1, 568

ISSN:0366-9033    

DOI:10.1039/DF9705000145

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Investigation of Liquid-Liquid Phase Transitions in OxideMelts by Viscosity MeasurementsBY JOSEPH H. SIMMONS, PEDRO B. MACEDO,* ALBERT NAPOLITANO,AND WOLFGANG K. HALLERInstitute for Materials Research, National Bureau of Standards, Washington D.C.20234 U.S.A.Received 8th June, 1970Results from viscosity measurements conducted both above and below the liquid-liquid phasetransition of a series of molten oxide glasses are reported in order to analyze the effect of supercriticalcomposition fluctuations on viscous flow, and to investigate the mechanisms of phase separation.Measurements of four oxide mixtures with similar high temperature structures and widely differentcritical temperatures, revealed an anomalous increase in viscosity at temperatures above the criticalpoint.The anomalous increase occurs when large composition fluctuations characterizing thecritical point are present. The effect is explained in terms of an interaction between viscous flow andthe supercritical fluctuations through the structural relaxation process. An analysis of this inter-action is presented.Measurements conducted at temperatures slightly below the critical point of one of these glassesindicate that the microstructure resulting from the phase separation is highly sensitive to the precedingheat-treatment. Phase separation by the formation of isolated spheres of the silica-rich componentis identified a few degrees below the critical point. Further measurements of viscosity by a fibreelongation method, conducted far below the critical temperature, are reported in order to analyzethe growth mechanisms occurring in the separated phases.In this case, the rearrangement stage ofphase separation is characterized by a growing interconnected structure.The processes which control liquid-liquid phase transitions in molten oxideglasses are characterized by the interplay of thermodynamic and kinetic propertiesof the materials investigated. While the thermodynamic conditions determine theequilibrium states of the materials as a function of temperature, pressure and compo-sition, the kinetic properties control how the systems approach and reach theseequilibrium states. The mechanisms of phase transitions in molten oxides, inparticular, are highly influenced by the kinetic properties since such nlixtures have, ingeneral, a large viscosity and thus a low diffusion coefficient at the transitiontemperature.Phase separation occurs in a system if the free energy of the state having twoimmiscible constituents is lower than the homogeneous state.The surface of thephase diagram representing the locus of the maximum temperatures at which phaseseparation occurs is called the coexistence or consolute surface (or curve for binarysystems). This represents the temperatures and concentrations at which the chemicalpotentials of the two immiscible phases are equal. The critical composition of thesystem is represented by the concentration of the constituents which has the highesttransition temperature. The free energy and its fist derivatives are continuousacross this critical point, so that many of the theoretical analyses applicable to criticalsecond-order phase transitions are applicable here. Continuity of the free energy* present address : Catholic University, Washington, D.C., 20017, U.S.A.15156 VISCOSITY STUDY OF PHASE TRANSITIONSand its first derivatives across the critical point suggests that the unstable compositionfluctuations present below the transition temperature T, are also expected to occurabove T,.These fluctuations in the supercritical state have lifetimes determined bythe kinetic properties of the material and are precursor of the subcritical state. Forexample, if the diffusion coefficient is low, then the fluctuations will take time to growto the amplitude characteristic of the thermodynamic state.The subcritical state is far more complex. For simplicity, we consider phaseseparation between two liquids.This applies to most oxide systems includingternary systems. From thermodynamic equilibrium conditions, the mixture mustseparate into its constituent immiscible phases when the temperature is below thecorresponding transition temperature on the coexistence surface. The first complica-tion arises when one considers the process by which phase separation may occur.Two models have been proposed for the initial decomposition mechanism of thematerial at the critical composition of a system. The older model of nucleation andgrowth proposes that phase separation occurs by the formation and subsequentgrowth of small nuclei of one phase within a matrix of the other phase.In thisprocess, a small amount of nucleation energy is required before the process canprogress. This is called the nucleation barrier and represents the excess surfaceenergy of a small nucleus. The second mechanism is called spinodal decompositionand involves a spontaneous phase separation into large interconnected regions. Thisprocess is characterized by a decrease in the free energy of the system in the initialstages and thus has no so-called barrier to overcome. There have long been dis-cussions as to the occurrence of either mechanism in various systems. At the criticalcomposition, it is thermodynamically possible for either mechanism to occur belowthe transition temperature.There are, however, further complications introduced by the kinetic propertiesof the material.Spinodal decomposition only occurs following large-scale fluctua-tions in the composition of the material. The minimum size of these necessaryfluctuations decreases as the temperature goes further below the critical. Thus,near T , , large migrations of the various constituents are necessary to set up the requiredfluctuations. If the diffusion coefficient is low, as is generally the case with moltenoxides, such large-scale migrations take a long time and nucleation which requiresonly small displacements of the constituents is likely to prevail. As a result, whilethermodynamic equilibrium considerations lead to a greater likelihood for theoccurrence of spinodal decomposition at the critical composition, the kinetic propertiesof the material may favour nucleation when the temperature is close to T,.Fortemperatures far below T,, spinodal decomposition requires less mass transport andmay become as probable as nucleation, particularly if the material decomposes intophases with a large nucleation barrier.RESULTS FROM SUPERCRITICAL VISCOSITY MEASUREMENTSSeveral models have been proposed to explain the interaction between the super-critical composition fluctuations, and the shear viscous relaxation Ingeneral, the supercritical state is characterized by fluctuations in the local composition.These space (r) and time ( t ) dependent fluctuations, c(r,t), are small for temperaturesfar above the critical, T%Tc, and grow to macroscopic sizes near T,.A spatialFourier decomposition of the fluctuations gives insight into their lifetimes :c(r, t ) = c(k, t ) exp (ik r) dk. (1) SSIMMONS, MACEDO, NAPOLITANO A N D HALLER 157Using the classical Ornstein-Zernike theory,l we find that the probability for theoccurrence of each Fourier component is given by :kT<c(k)2> = a( TIT, - 1) + bk2 ’where k is the Boltzmann constant and a and b are constant parameters. Thisequation shows that the long wavelength components (k = 0) dominate the fluctua-tions for temperatures near the critical. Each Fourier component of the fluctuationsis also associated with a temperature dependent, most probable lifetime :‘loOC [a(T/T,- 1) + bk2]k2 ’ (3)where qo represents the expected static and macroscopic viscosity or inverse of themobility for a homogeneous, non-critical system. This equation implies that as thewavelengths of the fluctuations increase in size near Tc (k goes to zero), their lifetimesdiverge.The interaction between these fluctuations and the viscosity may now be con-sidered.Fixman and Kawa~aki,~ while using very different coupling mechanisms,conclude that as the critical point is approached, an excess viscosity appears as aresult of the occurrence of the fluctuations. This excess viscosity Aq is defined as thedifference between the measured viscosity and the non-critical viscosity functions,qm and qo, respectively :This excess viscosity is inversely proportional to the mobility and proportional tothe range of correlation A, which diverges as (T/T,- 1)-* as the critical temperatureis approached.The corresponding proposed divergence in viscosity results fromthese authors’ assumption that the shear viscous relaxation process interacts withall the Fourier components of the fluctuations, including those of macroscopic wave-lengths (k = 0). At Tc, the long wavelength components dominate the fluctuationsand have infinite relaxation times or lifetimes, leading to the divergence in viscosity.In order to investigate the anomaly, we have conducted viscosity measurementson a series of molten oxides with widely varying critical temperatures but with similarhigh temperature structures.6 The samples were the critical composition of thesodium-borosilicate system ’ : 70.5 % SO2, 27.7 % B203 and 6.8 % Na,O by mole,which we shall denote as sample 1, and a series of glasses, samples 2 to 4 made byadding small doping concentrations of selected oxides in ofder to vary T,.Sample1, the base glass, has a critical temperature of 754°C. Sample 2 made up of the sameconcentrations of NazO, B203 and SiOz as the base glass, plus 2.1 mol % CaO has acritical temperature of 830°C. Sample 3 made up of the base glass plus 1.8 mol %A1,03 has a critical temperature of 643°C. Finally, a sample 4 was made by adding1.05 mol % CaO and 0.9 rnol % A1203 to the base glass in order to separate theeffect of doping from the supercritical effects. Sample 4 has a critical temperatureof 741°C close to that of sample 1.Sample 3 provided a material with a criticaltemperature lower than the others by 106, 187 and 98°C respectively. Criticaleffects are not expected to appear in this material at temperatures above 741°C sothat its viscosity curve may be assumed to represent the non-critical viscosity behaviourin the supercritical temperature ranges of the other glasses.Theresults varied somewhat at high temperatures among the samples, reflecting the effectof doping (log viscosity of 2.41, 2.32, 2.71 and 2.46 at 1300°C for samples 1, 2, 3 andAv = ~ln-Vo- (4)The viscosity was measured by a rotation viscometer described earlier.*158 VISCOSITY STUDY OF PHASE TRANSITIONS4 respectively). When the viscosity curves were normalized by the values at 1300°c,the critical point effects could be clearly seen.This normalized viscosity :r*(T) = Ylm(WYlm (1300°C) ( 5 )was calculated for each sample, and is plotted against temperature in fig. 1. It isevident from the figure that each curve departs from the curve of sample 3 as therespective critical points are approached from above. Using the viscosity of sample3 to represent the non-critical viscosity function yo at temperatures above 741°Cwe may now calculate the excess viscosity defined in eqn (4).temp."CFIG. 1.-Fractional viscosity increase from the value at 1300°C for each glass melt exhibiting thesupercritical excess viscosity. The critical point of each glass is represented by an open circle on therespective viscosity curves.Comparison of the curves of samples 1, 2 and 4 shows that the excess viscosityAq is proportional to the non-critical viscosity qo, or inversely proportional to themobility as predicted.Thus, the normal viscosity function qo was divided into theexcess viscosity in order to separate the variations in the normal viscosities of thesamples from the excess viscosities :A@( T ) = as*(T)IVXT) = [V*(T) - VXT)lId(T)* (6)The normalized excess viscosity function A@(T) is plotted in fig. 2 for samples 1, 2and 4, as a function of reduced temperature (T/T, - l)%. All samples exhibit thesame behaviour and follow the same curve when the temperature is properlynormalized by the transition temperature. Fig. 2 shows that no divergence isindicated in the behaviour of the excess viscosity down to a reduced temperature,TIT,, of I .0009, despite the large excess viscosity, Ay*-2y: measured at 7''.This isin contrast to the prediction of Fixman and Kawasaki.In view of these viscosity measurements, and a series of ultrasonic relaxatioSIMMONS, MACEDO, NAPOLITANO AND HALLER 159measurements conducted on the same glasses,l0 a theory was proposed l 1 for shearviscous relaxation near a critical point, which predicts a finite non-divergent excessviscosity at T,. This model, which we shall only briefly outline here, limits theinteraction between the fluctuations and the shear viscous relaxation process byassigning a limiting range ro to the relaxation process. The result implies that whenthe fluctuations grow to sizes larger than yo (A % ro) near T,, their effect on the viscositysaturates as the relaxation process becomes insensitive to further increases in thesize of A.This limiting range ro is a property of the viscous relaxation process onlyand is not influenced by critical point phenomena. The existence of such a parameteris based on the concept that the molecular forces which affect the relaxation andtherefore the viscosity of a molecule are limited in extent. This model accountswell for the observations from the supercritical viscosity and ultrasonic relaxationmeasurements on the aforementioned critical oxide glasses. An extension of thislimited range concept has proved successful, even in the analysis of viscosity measure-ments on non-critical systems.reduced temp.(T/T,-- 1)10-4 I O - ~ ~ X I O - ~ lo-' 2.5 X lo-' 4 X lo-'1 1 I I I I 1 1c. 0 SAMPLE I\ A SAMPLE2SAMPLE4 - \\ A\ A-\A\\A\\o \ 00 . A O. 0I 1 I1 I I I I I t0.1 0.2 0-3 0 4 0.5 06 07reduced temperature, E = (T/T'.- 1)3FIG. 2.-Behaviour of the normalized excess viscosity as a function of reduced temperature forsamples 1, 2 and 4.In these past measurements, we have investigated supercritical compositionfluctuations in thermodynamic equilibrium. The viscosity of the sodium-borosilicateglasses has been low enough (7 x lo5 to lo7 poises) at the transition temperature toallow the fluctuations to reach their equilibrium states before the measurements couldbe made. Measurements of viscosity by fibre elongation conducted in a relatedinvestigation l2 demonstrate the effect of kinetic properties on the approach ofsupercritical fluctuations to thermodynamic equilibrium.The measurements wereconducted near the transition temperature of a barium crown and a chemical-resistantsodium-borosilicate glass at viscosities of 4.3 x 1014 and 6.1 1 x 10l1 poises respectively.While these glasses are not at the critical composition of their systems, supercriticalfluctuations are expected to occur near T, nevertheless. Fig. 3 shows results fro160 VISCOSITY STUDY OF PHASE TRANSITIONSthe fibre elongation measurements at T/Tc = 1.01 and 1.003 for the barium crownglass (glass A) and the sodium-borosilicate glass (glass B) respectively. The viscositycurves exhibit three regions.The first apparent viscosity increase, at short times,is due to delayed elastic effects and was observed each time the weight was removedand replaced. Glass A at a higher viscosity has a longer relaxation time than glass B.The second region, approximately from 10 to 100 min, repreSents the excess viscosityarising from supercritical composition fluctuations. The increase in viscosity withtime reflects the growth of these fluctuations according to eqn (3), since the fibreswere made by quenching from a high temperature where fio fluctuations were present.In this case, the diffusion coefficient is low enough to allow the measurement todetect the approach of the fluctuations to the thermodynamic equilibrium state.One is thus observing the slow increase in size of the fluctuations as the excess viscosityincreases to its equilibrium value.The third region demonstrates no further changein Viscosity, and gives the normal viscosity behaviour of the materials.I 10 I00 I ,000 10,000elapsed time (min)FiG. 3.-Viscosity of a barium crown (glass A) mixture, and a chemically durable borosilicate (glassB) mixture at temperatures above their transition temperatures.SUBCRITICAL VISCOSITY MEASUREMENTSMEASUREMENTS NEAR THE CRITICAL TEMPERATUREThe following measurements were carried out on sample 1 only ; it has a compo-sition of 70.5 % SO2, 22.7 % B203 and 6.8 % Ns20 by mole and a critical tempera-ture at 754°C. The viscosity of sample 1 at the critical point is approximately 6.8 x lo6poises.The viscosity measurements were conducted in the range 745-751 "C usinga rotating cup viscometer described elsewhere.8* Measurements made in thisrange can give insight into the decomposition mechanisms occurring near Tc.This glass phase-separates into a silica-rich phase and a sodium-borate-richphase with a resulting large variation in viscosity between the immiscible phases.lViscosity measurements at temperatures below T, will therefore be sensitive to thSIMMONS, MACEDO, NAPOLITANO AND HALLER 161microstructure of the material. If an interconnected network is formed, the viscositywill increase as a function of time as a silica-rich structure grows in the material.If nucleation prevails, and the silica phase forms isolated, freely moving spheres, thenit is expected that the viscosity will decrease as silica is precipitated from the matrix.I I I I I1 1 I 174 5 750 755 760 765temp., "CFIG.4.-Viscosity of the critical sodium-borosilicate glass below the phase transition is plotted for3 heat-treatment conditions. Solid circles represent stress-dependent behaviour.The results obtained indicate that nucleation of silica spheres is occurring inthis material just below the critical temperature instead of spontaneous spinodaldecomposition. The phase separation, however, appears to depend upon the super-critical heat-treatment history of the material. Some results from the measurementsare shown in fig. 4, for three different supercritical heat-treatments. The times andtemperatures measured during the heat-treatments are listed in table 1.Inheat-treatment A, the sample is cooled from a high temperature, 11OO"C, far abovethe critical point, to a temperature 749"C, a few degrees below T,. Then the tempera-ture is maintained at that level for a long time and the resulting viscosity is monitored.The viscosity decreases below the extrapolated curve at temperatures between 747and 749°C. The configuration present appears to remain stable for 3 155 min (52.6 h).A further measurement at 7080 min follows a stress-dependent behaviour as shownin fig. 5. The glass is cooled from1100 to 755"C, a temperature close to, but larger than T,. After an elapsed timeof 60 min, the temperature is lowered below T, to 752°C where the viscosity measuredis close to the extrapoIated supercritical curve.After a total elapsed time of 105 minHeat-treatment B presents a different result.1 62 VISCOSITY STUDY OF PHASE TRANSITIONSthe viscosity has increased only by a small amount. However, decreasing thetemperature further, to 750°C led to an increase in viscosity of 2.5 times. All measure-ments in schedule B at times below 1410 min were Newtonian, while the last threetaken after 1410 min were stress-dependent. The viscosity in this series of measure-ments did not drop below the extrapolated curve. Increasing the supercriticaltemperature at which the heat-treatment was started to 762"C, as shown in scheduleC, leads to a lowered viscosity value below T, as in schedule A.TABLE 1 .-SUBCRITICAL MEASUREMENTS CONDUCTED ON SAMPLE 1 WITH A CRITICAL POINTOF 754°Cschedule A schedule B schedule Ctemp."C time min temp. "C time min temp. "C time min1100°C748.7 60748.8 205749.3 365747.6 1455747.5 1730747.2 2865747.5 3155749.1 70801 I0o"C755752751.7750750.1751.7750.5750.61100°C60 767.5 100105 762.0 170270 751.4 220390 748.4 2701410 750.8 340206029203100Thus it appears that the critical composition of the sodium-borosilicate glasssystem phase separates by nucleation and growth at temperatures just below thecritical. This is evidenced by the decrease in viscosity below T, as in schedule A,attributed to the formation of a low-density distribution of freely moving, isolatedsilica spheres which must grow to fairly large sizes in order to occupy the totalvolume of the second phase.However, no large degree of interconnectivity developssince the viscosity remains below the extrapolated curve for measurements within3 155 min. This result indicates that if collisions occur as described in Haller's paperon rearrangement kinetics below T,,l3 then the surface energy is high enough toapplied stress (mV)FIG. 5.4tress against strain rate plot of the non-Newtonian behaviour of sodium-borosilicateglass showing a non-zero interceptSIMMONS, MACEDO, NAPOLITANO, A N D HALLER 163reform spheres and prevent an interconnected structure from developing. The largeincrease in viscosity observed after 7080 min is linked to stress-dependent behaviouras shown in fig.5. This stress-dependent behaviour appears to be due to damageinflicted upon some structure in the glass by the viscosity measuring apparatus, andthus is probably due to crystallization in the melt.A glass heat-treated just above the critical point, before being lowered to thesubcritical region showed no decrease in viscosity as shown in schedule B. Spinodaldecomposition is discounted in this case as no large immediate increase in viscosity,characteristic of the spontaneous formation of an interconnected network, wasobserved. This surprising difference between the results of schedules A and Bis due to the heat-treatment at 755°C ( T / T C ~ 1.001) in schedule B. A 1 h heat-treat-ment at this temperature allowed the stabilization of the large supercritical compositionfluctuations characteristic of temperatures near T,.These fluctuations are thenlikely to provide a high density of nucleation sites for the silica phase. The increasein viscosity occurring before 1410 min indicates that some degree of interconnectivityis achieved in this glass. The growth, however, is slow and probably occurringfollowing Haller's model for the development of an interconnected phase from dis-persed spheres by collisions only. The measurements for times longer than 1410 minwere again stress-dependent as represented by the solid circles and are thereforeindicative of crystallization. Heat-treatment C shows that if the supercriticalholding temperature is too far above T, (i.e., 762"C), then the viscosity drops as inschedule A representing a stable configuration of isolated silica-rich spheres in asodium-borate-rich matrix.MEASUREMENTS FAR BELOW THE CRITICAL TEMPERATUREA series of measurements of viscosity were conducted on the same glass at 600and 560°C by a fibre elongation method described elsewhere.* In this case, allsamples are fibres quenched from high temperatures where no fluctuations werepresent.After an initial period of relaxation of delayed elastic stresses, the viscosityincreased by several orders of magnitude, finally appearing to reach a saturationvalue after 20 000 min. Fig. 6 shows two measurements conducted at 600°C. Thesolid circles represent data obtained with a 513 g weight, while the open circlesrepresent data obtained with a 214g weight.For the fibre measurement describedby the solid circles, we removed the weight after 160 min and replaced it after 1 000min. The fibre was left in the furnace during the entire heat-treatment time. Thisprocedure was again repeated between 1 600 and 5 600 min. It is seen from fig. 6that, in both cases, the viscosity had continued to drift without the weight and themeasurements coincided with the open circles representing data from a fibrecontinuously loaded. The solid lines at 1000 and 5600 min represent loadingtransients due to replacing the weight.The results, showing an identical viscosity increase for two fibres under differentloads, indicate that the viscous flow below T, is stress-independent. In this case,the loads are small and the measurement only probes the state of the material withoutdamaging the structure.The large viscosity increase is due to the rearrangementstages of phase separation. The initial stages occur during the relaxation of thedelayed elastic effects and cannot be analyzed. The viscosity continued to increase,for the data represented by solid circles in fig. 6, despite the removal of the measuringload. This indicates that the observed viscosity drift is not stress-induced, but rathera consequence of the mechanisms of phase separation only.Fig. 7, showing data from measurements at 600 and 560"C, demonstrates that th164 VISCOSITY STUDY OF PHASE TRANSITIONSelapsed time (min)FIG. 6.-Viscosity data taken on the critical sodium-borosilicate glass at 600°C.The solid circlesrepresent measurements with a load of 513 g and successive loading and unloading conditions, whilethe open circles correspond to a load of 214 g applied continuously.-1 I I I I10 100 1,000 10,000 l00,000 Ielapsed time (min)FIG. 7.-Viscosity data at cioo"C, open circles; and at 560"C, solid circlesSIMMONS, MACEDO, NAPOLITANO, AND HALLER 165same effect is occurring at both temperatures. Between 2.5 and 100 min, both fibresappear to have the same viscosity despite the difference in temperatures. Thisunexpected result implies that since the original state of the fibres was identical, theearly decomposition process is not strongly temperature dependent. Both measure-ments then show a large time-dependent increase in viscosity.These increases by4 and 5 orders of magnitude at 600 and 560°C appear to saturate as the viscosityapproaches a plateau after a sufficiently long time period. As expected from normalviscosity behaviour, the plateau is reached faster at 60O0C, and the viscosity of thisplateau is lower than at the lower temperature.Since the viscosity increases, it can be concluded that the two phases are inter-connected after an initial period whose behaviour is hidden by the relaxation ofdelayed elastic stresses. The process observed is a rearrangement process wherebycoarsening dominates as the phases attempt to reduce their interfacial energy. Theincrease in viscosity of the fibre is due to the addition of pure silica to the inter-connected structure as phase separation progresses.The appearance of a plateauafter the long heat-treatments indicates that the phase separation process is finallyreaching an equilibrium state.CONCLUSIONSViscosity studies of systems whose immiscible phases have widely differentmobilities can yield much information about the mechanisms of phase transitions.We have presented an analysis of the interaction of viscosity with supercriticalcomposition fluctuations in some sodium-borosilicate glasses. We have observedthe appearance of a large excess viscosity when the critical temperature was approachedfrom above. This excess viscosity, however, appeared to remain finite at the criticalpoint, in contrast wj th previously published theories. Supercri tical measurementson fibres at high viscosities demonstrated the effect of low diffusion coefficients onthe time required to approach thermodynamic equilibrium.Subcritical measurements on the critical composition of the sodium-borosilicatesystem have shown that phase separation occurs just below the critical temperatureby nucleation and growth of freely moving isolated spheres of the silica-rich phasein the sodium-borate-rich phase.This structure appears to remain stable for longtime periods. Far below the critical temperature, the rearrangement stage of thephase separation proceeds by the growth of an interconnected structure of the twophases, which appears to begin immediately after the initial phase separation.L. S. Ornstein and F. Zernike, 2. Phys., 1926, 27, 761. L. D. Landau and E. M. Lifshitz,StatisticalPhysics (Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958), chap. 12, p. 368.M. E. Fisher, J. Math. Phys., 1964,S, 944.M. Volmer and E. Weber, 2. phys. Chem., 1926,119,277.J. W. Cahn, Acta Met., 1961,9, 795.M. Fixman, J. Chem. Phys., 1962,36,310.K. Kawasaki, Phys. Rev., 1966,150, 291.J. H. Simmons, A. Napolitano and P. B. Macedo, J. Chem. Phys., 1970,53,1165.W. Haller, D. H. Blackburn, F. E. Wagstaff and R. J. Charles, J. Amer. Ceram. Soc., 1970,53, 34.A. Napolitano, P. B. Macedo and E. G. Hawkins, J . Res. Nat. Bur. Stand. A , 1965, 69,449.lo J. H. Simmons and P. B. Macedo, J. Chem. Phys., 1970, 53, 2914.l 1 J. H. Simmons and P. B. Macedo, J. Chem. Phys., 1971, 54, 1325.l 2 W. Haller, J. H. Simmons and A. Napolitano, Viscosity Drqt Technique for the Determinationof the Critical Liquid-Liquid Immiscibility Temperature of Glasses (Amer. Ceram. SOC. Meeting,Philadelphia, Pennsylvania U.S.A., May 1970), J. Amer. Ceram. soc.,1971, 54, 299.l 3 W. Haller, J. Chem. Phys., 1965, 42, 686.* A. Napolitano and E. G. Hawkins, J. Res. Nat. Bur. Stand. A , 1964, 68,439

ISSN:0366-9033    

DOI:10.1039/DF9705000155

出版商:RSC    

年代:1970

数据来源: RSC