REVISTA MEXICANA DE FislCA 44 SUPLE~lENTO 3. 8G-84

Boroxol rings and the stochastic matrix method

D1C1EMHRE 1998

G.G. Naumis and R.A. BarrioInstituto de Fisica, Universidad Nacional Autónoma de México, Aparrado postal 20-364, 01000 México, D.F, Mexico

R. Kerner and ~1. MicoulautLaboralOire CeR, Tour 22-12, .J.eme érage. Baile /42 Ullil'ersiré Pierre el Marie Curie, 4. Place lussieu, 75005 Paris, Frailee

Recibido el 10 de enero 1998; aceptaJo el 6 de marzo 1998

A statistical model bascd on the stochastic matrix mcthod is dcvclopcd and uscd lo find Ihe fraetion of boron atoms bclonging to boroxoJrings in a horoo oxide (B203) g:lass. Thesc results are compared wilh recent experimcnlal data from inelastic neutron scattering estimates.Thc melhod aho allows lOcvaJuate the energies rclated (O {he formation of a single 8.0-8 unir in an oxygen bridge or in a boroxol ringoThequalítative behavior of (he heat capacily Cp(T) during the glass transition is reproduced. with the inflcxion point at the tempcrature given bythe expcriment. The model gives a reasonable qualitative prediction for the growth of micro-clusters.

Keywords: Glass transition: glasses; boron oxide

Se desarrolla un modelo estadístico basado en el método de la matriz estocástica y se usa para encontrar la fracción de átomos de boropertenecientes a anillos boroxol en un vidrio de óxido de horo (820)). Estos resultados se comparan con datos experimentales recientes deestimaciones de dispersión inelástica de neutrones. El método permite también evaluar las energias relacionadas a la formación una unidadsimple 8.0-8 en un puente de oxígeno o en un anillo boroxol. El comportamiento cualitativo del calor específico Cp(T) durante la transiciónvítrea se reproduce. con el punto de inflección a la temperatura dada por el experimento. El modelo da una predicción cualitativa razonablepara el crecimiento de micro-cúmulos.

Descriptores: Transición vítrea: vidrios: óxido de boro

PAes: 6470.Pf; 61.43.Fs: 61.43.Bn

I. Introduction

Boroxol rings are particular structures that ha\.c heen sug-gcsted lOexist in grcat numhcrs in vitrcous 820j" Thcy rep-resent the hest example of interl1lcdiate range onJcr, as de-lined by Galeener [1] and are six-mcl1lbcred planar and iCg-ular rings B3-03 in which lhere should be a subslanlial rc-duction of lhe oxygen-bridge angle, since lhe average O-B-Oangle in the network is 1300

• The abundance ol' boroxol ringsin the network should be substantial to cxplain (he cxperi.I11CtltS,particularly the extremely sharp pcaks l'ound in vibra-¡ional spectroscopic studies, as Inl'rared and Raman scaltcr-ing [2,3). However. the existence al' boroxol rings in the cor.rect concemra{ions has beeo questioncd. due 10 the fact thatrecent molecular dynamics ca1culalions have found il difficulIto produce boroxol rings, c1aiming that they are not neededto explain mOSlol'the experimental data" One should mentionthat there is not yet a model ol' a structurc containing man)'boroxol rings able lo predicllhe density of lhe malerial.

There are various kinds of thcoretical calculations. rang-íng from clleclivc I1lcdium ll10dels l4]; to 1110redelailcdOrlCSl5], which are able to explain all the experimental fea.lures only if one assumes large quantilies of boroxol ringsin the disorder network. A more definite argument in favour01'lhe existence of boroxol rings comes from reccnt neutronscattering experimcnts [6), \\'hich can be explained coher-cnll)' only if "'-'80'7< of [he Boron atoll1s belong lo rings.

This paper has lhe purpose of seltling lhis question aboulthe existence and abundance of boroxol rings in lhe amor-phous nelwork of boron oxide. Evidcntly, the [ormarion oí"rings should be a consequence of the way [he solid is grown,sincc Ihey are nOI present in any of the cryslalline forms ofB20;I, and some kinetic models are necded to explain lheirformation. Molcrular dynamics is unlikely lO give definiteanswers, sincc the maio rcason for the formation of ringsshould be a peculiar lhree body force lhal allows lo modifythe hridge angle. Therefore, we have devcloped a theoreticalmodel for the solid growth, hased on the original ideas givenin Refs. 7-10, and applying lhe slochaslic malrix melhod forsolid growlh. "rSI pUl forward by Kerner [11]. The cumplelOand delailed description of lhe melhod can be found in a for-Oler paper [12].

2. Stochastic matrix method description of thegrowth of a solid

Let us start hy assuming that v.'hile cooling down the Illclt.SOIllCclustcrs of alOms are fonncd through an agglomera-lion proccss. Ouring the solid growth through agglomeration.Illany complicated and competing processcs take place, butgradually, biggcr c1usters and parts of the network appear e\"-erywhere. \Vhatever their shape, they can be divided in twoparts: lhe rilll, (or Ihe borda), composed of alllhe enliliesthal offcr a potcnlial possibility for a new entity to stick and

HOROXOL RINGS AI'O THE STOCHASTIC MATRIX METHOD ~l

agglorncratc. and lhe bu/k, (m lhe interior), Ihal is, all lheunits tl1at have satisflcd all their honds already.

The elementary cntitics composing lhe rim are found in afinile (usually quite small) numbcr ol' local gcornctrical situ.aliaos otlering alle, two or more possibilitics for anothcr cn~tity 10 stick too \Vc shall cal! IhcI11.'lites, amI \Ve shall assulllc1hat lhe prohability of a simullancous agglolllcr;.lIion 01' twoor more atoms al a single site is ncgligihlc.

Whilc lhe IClllpcraturc 510\\lly dccrcascs. (he average si/.cof clusters grows due 10 lhe progrcssivc agglolllcration ofncwat0l115Ihal stick lo lhe rimo Aftcr a characlcrislic lime a !lewlaycr ol' atollls is crealeJ. thus transforming Ihe prohabiliticsof observing various silcs on the rimo

This process of growth al the rim can he describcJ hya linear Iransl"onnalion, rcprcscnlcd hy a malrix aL:ting onJ vector, whose componenls are the prohahililies of finJinga given site on the rim uf a cluster. Thc matrix Iransfonnsthis vector inlo a ncw onc. hccause the rim is changed af-ter adding one alom lo Ihe c1usler. The lransformalion 01"Iherim depends 011 the sile 00 which Ihe ne\\' alom slicks. Now.each sticking proccss has a certain probahilily to occur. thus,lhe matrix elements cont;.lin lhe probabililies 01'transformingcach kinJ of sile ¡nto olhers. Thc probahilil)' factors shouldinduJe two conlributions: (1) the statistical wcight ror eachprocess. thal is. Ihe numhcr of ways Icading lo the sume finalrcsult, anJ (2) lhe Bollzmann factor laking into account Iheenergy barrier of forming a certain kind of hond.

Lel us apply these ideas to lhe B203 continllous oel\',:ork.Thc elementary unit. dictated hy the hond chcmistry is a lri-angle I3(O~ lo [9i), \Vhich \Ve shall call a "singlet". T\Vo sin-glcts can he connccled only lIsing one hond to form a "dou.hiel". since olher \\'a)'s 01' connecting two units would pro-duce a two-fold ring which is nol seen in the real ne(work.Lct liS assumc that the encrgy cosl to fOl"mIhis honding isEl. Afler a douhlcl is produced, two situalions can occur ifa ncw singlct is added: the newly arriving singlet forms alonger chain (a "triplet") or il can close a ringo with a diffcr~enl energetic cost (El). since one has to defonn lhe bridgeangles in a ringo The agglolllcration process nccurs at a givelllemperature T. al which the individual honds reach cquilih-rium. Therefore, we shall use Ihe notation:

p-' ::: (,/-:1//.:1'.

anJ

where k is the Bollzmann constanl.The possihle conflgur¡¡liolls. or siles are shown in I'ig. l.

\Vc shall denote lhese configuralions by .1', ,/)' :. f. ll. and /l'.

In principIe one could have also con side red longcr chains 01'singlets. bUl Ihis \\'ould Iead lo the Illultiplkation 01'silcs amitransilions not expecled in a real glass, sincc the creation 01'longer chains in covalent glasses is ncgligihle. oecause oneshould avoid the formation 01' local roids.

w

y

~/Rim border

x

FIGURE l. A typical cluster in vitreous 8203 showing the six typesof silcs considcrcd on lhe ri rn.

The transition probability for each elementary step can begiven in the following \Vay:

.1' =} y 1'(.1'. y) 3e-f,

y=} : I'(y.:) 6 -,e ,

y I'(:.y) 2 x 3e-(.z =} {m 1 P(z,t) 6e-~,

01' IV P(:.ll') 12e-J].

y+:: PI(/,y) and I'(t.:) ......3e-(.

{"';t' + 2,// I'I(/'X) ano P,(t, y) -..,3e-',

t=}01' .IJ + 1I P,,(t. y) and P(t, u) -.., 12e-'I .01' J' + 1L' P,(/'x) and P(/' /V) .....•121"-'1.

ti =} .l' 1'(11.1') 3 -,r .

w => y+u I'(w,y) ano P(w,u) 6 -,e .

Here \Ve took into account the purely statistical factorsaccording to the numher oC free bonds and the multiplicityavailable for each transition, which can be quite easily foundoul from purely geollletrical considcrations, and Ihe l1ollz-mann factors according lo lhe new bands formed.

The factors 1'(1'. X), P(x, y) •... etc., that define slalis-tical weighls al' transilians resulling in the correspondingtransformarions ofsites can be displayed as a 6 x 6 matrix:

o O O P(/,x) O OP("',y) O I'(z, y) P(t,y) P(u.y) P(ll', y)

O P(y.: ) O P(/, z) O OO O 1'(:,/) O O OO O O 1'(1, n) O P(ll', u)O O P(:.lL') P(t. lL') O O

Re". ,l/ex. Fú. +l S3 (199X) XlJ-84

X2 G.G. NAUMIS. RA HARRIO. R. KERNER. AND M. MICOULAUT

Note rhat the lCroS correspond lo the abscncc 01' manytransitions, e.g.. r => .l", Y => r. ctc. In this matrix cach cnlrysymholizcs the sum of a1lpartial prohahilitics \\.'hcncvcr (hefeis more Ihan one pathway leading lo (he givcn contlguration,e.g. P(U.) = PI (t, l.) + Pz(t, l'), ele.

Thc cxplicit form of the matrix is

o O O 3e-t + 12e-1/ O O:~(,-{ O G -, ge-{ + Ge-r¡ 3e-{ G('-(eO G('-{ O 3e-f O O

(I)O O 6e-f O O OO O O 6e-'1 O Ge-'() O 1') -r¡ 12e-'1 O O_f

el, lhen, in the limit al' large j. vj converges lo lhis eigcn-vector, independcnlly of lhe inilial conditions.

As a consequence.lhe evolulion ol'the rim attains a slablestalistical regime after successive sleps of growth; this regimeis govcrned solely by the statistics represented by the eigcn-vector wilh eigenvaluc onc. This eigenvector determines lhedistribulion (P:r.Py,P:,PhPll,Pw)1XJ to which Ihe averagedslatistics tenJs asymplotically. This is also the stalistics oí"lhe bulk if the clUSlers are really large. For c1usters 01' inter-mediate si7.e. one should ralher average over the sum 01'manylayers.

3. The statistical analysis of horoxol rings

The aboye matrix is supposed lo act.ol1 a column vec.tor rcprcscnting (he probabilities Px. Py. Po:, PI. PIl' Pw. whichadd up lo one. After applying lhe malrix we wanl lo ohlainanorhcr distribution ol' probabilitics, satisfying the samc 110r-

malization condition. Thereforc the sum 01'the cntrics in cachc.:olulTln af the abo ve matrix must be set to one. Afler nor-malization. we get a stochastic matrix (Al) thal transformslhe prohahilities 01'f1nding one conflguralion on the rim of acluster, (P:r. [Jy' P:. Pt, Pu, Pw) into él new set 01'prohahilitics(jJ~, p~.J)~. ]J~. P:l' ]J~,) aftcr an cntire ncw layer of atoms hashccn grown. wilh Olle new atom at cach available site:

In our Slalislical model for Ihe B203 glass, lhe only free pa,rameter is ~. or lhe excess free energy when closing a ring(£2 - E¡) = kTln (O = F. To fix lhis parameler. we shallsludy lhe hehavior of lhe inlernal energy U, and Ihe specilieheal ep' near the glass lransition temperature. and lhen tit lhercsult with available experimental dala.

The internal energy involved in a process 01'growing inlhe i-Ih layer is,

3~T _ E2Pj,(T) - £¡(I - Pá(T)],

Using lhe ahove matrix. lhe growth 01'c1USlers is mod-cllcd by a successive application 01'the matrix on an arbitrary¡nitial vector va. Thus. lhe evolution 01' lhe prohabilitics onlhe rim after j steps is given by

O () O1 + 4~5 + 12~

I 3 + 2~O

2 + 2~ 5 + 12~I

O O5 + 12~,\1 = I

O O O2 + 2~

O O O2~

5 + 12~O O

2~ 4~2 + 2~ 5 + 12~

O

(J

O

O

O

O

I2

O

()

I

2O

where Ihe firsl lerm is lhe kinelie eonlribulion, and Pá (T)is lhe probahilily 01' forming a ring from the i-th rim to thei+ 1, sill1ply ohlained hy eounling lhe proportion of rings Ihalwerc fonncd during this proccss. This inforrnation is encodedin the matrix as the prohability al' the processes: Z'i -4 tL'j+l,

and ti -4 "i+ l. 1Oi+l. Then.

Now, Ihe internal encrgy is an extcnsive parametcr. the lo-tal energy aftcr N SICPSof growing is the sum al' the encrgiesin each layer 01'lhe growth:

'(T) 3NkT £ \' (£ £) ~ P' (TÚ = -2- + l' + 2 - 1 L ¡¡ ).

i=1

The specific hcal is ohtaincd by taking lhe derivative ofU with respect to thc Icmpcrature.

3Nk dP"'(T)ep(T) = :2+ (£2 - £¡)N ~T' (4)

and if we calculale ",,(T) for a hig numher of sleps of grow-ing, Ihen Pj, (T) can he replaeed by ilS limil value on lhestalionary rcgimc.

In glasses. il is gencrally observed lhat Ihefe is an int1cx-ion point ami a precipilouS dccrcase in the hcal capacily [13]at lhe glass lransition temperalurc (Tg). Therefore. we de-

(3)T) - 3Nk (£. _ £ );.-. dPá(T)cl'( - 2 + .! 1 L- dT .

í=l

(2)

The final configuration depends only on lhe eigenveelors01'the stochastic matrix. It is easy to prove that a maIrix withalJIhe columns normalized to one has aIleast une eigenvaluclXluallo one, while the olhcrs can be real. complex or imagi-nary, dcpcnJing un lhe valucs ofthe paramctcrs involvcd. ThecompleXocigcnvalues indicate the presence al' an oscilfat01)'

regime 01'growth. usually damped by lhe nonTI of lhe eigen-value. if it is less than 1. Due to this exponenlial damping.only eigcnveclors with norm one remain after many succes-si\'e applicalions of lhe stochaslic matrix. lf we supposc that.H has only one such eigenvalue (,\. == 1). with eigenvector

Rev. Mex. Fis. 44 53 (1998) 80-84

83BOROXOL RINGS AND THE STOCHASTIC MATRIX METHOD

Il.X

ll.h

Ftjl

nA

11.2

" "1"

20j

311

(6)

Nh = "; + lU; + 2z; (Al.,) + 2tJ(M"t + ,\/.,).

In Fig. 2 we show the cvolution of F as a function of thenumber of elcmcntary steps of growth (j), with lhe encrgy

and lhe fraclion F is oblained by dividing lhis number by lhetotal numbcr of alOms on the rim, thut is

Once the energy differencc (E, - E¡) is fixed. we c.ncalculalc the fraction 01'boroo atoms in the glass that are inboroxol rings (F == .V Il / NT)' Al this point, wc musl remcm-bcr tha1. in princip\c. lhe cvolution orthe vector v j gives on\yinfonnation on lhe cvo\ution of lhe probabilitics 00 lhe rim.while Ihc f,"clÍon F is • propcrlY 01' lhe bulk. Howcvcr. thisbulk is formed by the addilÍon 01' conscculÍve layers. Thcre-rore. by adding lhe number of atotnS lha1 are trappcd iOlOhoroxol rings in cach S1CP01' gro\\'ing we can flnd NB. An-olher problcm relllains 10 he sol ved: Thc crcalion of a ringnccds three slCPS 01'agglomeration, thus. 10 be able lo decideir an a10111is ¡nsitie a boroxol ringo we muSl consider mofethan une step 01"agglomeralion.

This counling can he aehieved in thc fo\lowing way: Thenumbcr of boron alOms in boroxol rings cxpcetcd in thc j.th

stcp is

(5)

E2 _ Et = 0.242 cV = 5.572 Kcal/mol.

Thcse va\ues are vcry clase to othcr estimutes found in thc lit-eraturc. Walrafcn and co-workers 114\ have found 6.4:r 0.04Keal/mo\, oblained from experimental data. 5nyder 1151es-timated 6.0 Kcal/mol. obtaincd from (lb illitio quantum mc-

chanical ca\culations.

2) E2 _ Et = 0.068 eV = [,566 Keal/mol,Howevcr. only the first one corresponds to the real glass.

sincc only this cncrgy differencc \cads lO the concet behaviorof cp(T). Another litling is possible when Tg = 530K, which

gives.

Thc I.tler eondition fixes the parameter ~ .t Tg, and lhisqu.ntilY is well known from v.rious experimenls lO vary fromT

g= 470 K lo 530 K (see ReL \4). For Tg = 470 K, we have

founo. that condition (5) is salisficd for twO valucs:1) E2 _ El = 0.214 cV = .\.927 Kcal/mol,

and

iJ2fj2f [Cp(T)]T=T, = O,

.nd by using (4) we get.

,1'd'T {p¡¡O(T)lT=T, = O.

FIGURE 2. Example of the evolulÍonof F(j) a>a function of the .gglomer.tion ,ter j.

ReI'. Mex. Fis . .\4 53 (1998) 80--84

84 GG NAUMIS. R.A HARRIO, R K.ER,'\'ER.At"D ,1\1~lICOULAUT

Inserting the appropriate cnergy valucs for Ty :: ...(70 Kinto Ihe lasl cxprcssion \Ve get that p= = 81.3%.

This rcsult is in very good agreclllent \vith Olher Iheorel-ical and experimental results Iike lhe 83% proposed hy Jclli-son el al. [16J, 80% by Hannon el al. [17J and 84',í( hy Mi-coulau! el al. [9J.

difference lixed al: £, - El = 0.21,1 eY for a completelyarbitrarily chosen ¡n¡tial conditions. As discusscd prcviously.(he dampcd oscillations are c1early visible al the ¡n¡tial slagcs,hut lhey rapioly disappear as lhe systcm H~achcs lhe sIal ion.ary regime. \Vc can ¡nfer 1hm (he final configuration 01' Iheglass is delcrmincd only by fe\\' local conligurations. NoticcIhat Ihe oscillations cxhibit a pcriod nI' roughly thrcc sters.UUClo lhe faet lhal for complcting olle horoxol ring. al leas!thrce steps of growth are ncccssary.

lhe stationary value is near 80% of boron atoms Irappcdin boroxol rings. This numbcr can he obtained more cxactlyir we use lhe eigcn\'cctor el, which is the onl)' one Iha! re-mains after many steps of agglomeralion. Thcn the \'aluc 01'F is gi\'en by the asymptolic expression F ~ FXJ, wherepx is obtaincd by substiluting the eOlllponenls 01' Ihe cigell-veelor wirh eigenvalue one intó Eq. (6). In Ihis model. theeigcnveclor is,

el1

84C + 107~ + 25

1 + .t~2le + 34( + 92.te + :J4~+ 10

12( + 5

3((4(+3)

2~(12~ + 7)

4. Conclusions

In Lhis papel' \VC Ilavc shO\\'n that a simple model 01" slatislicalagglomeralion can gi\'c "cry valuable and sensitive informa-tion ahOlu (he o\'crall structurc 01' a glass, or a solid in gen-eral. Thi ...rael has far rew.:hing cOllsequcllees. since one canexplain ho\\" the existerll.:c ot' onl1' ver1' shon correlations canproduce i111ermcdiate rangc order in a glass. This gives valid~il)' 10 calcu]alÍons dOlle in small c]ustcrs and 0pposcs 10 thewidely sprcad idea of a glass hcing a trapped metastahlc slatein a phase-spacc landscapc.

\Ve han:, abo shown tha! with Ihis model one can con-nect lo experimental dala through lhe cakulalion 01 thermo-dynamically :lvcragcd quantiLies. a.\ the interna] cnen!y, orthe speeifle heat. that can he reaJil1' ohlaincd. Thc m<.;¡;¡ rc-sult of this papel' is Ihat ollr model forces us to cOllclllJC Ihatin Ihc sil ucture oí" vitrcous Ih03 l"OlIghly S(j1X 01"!he horonatoms ar(' forming boroxol rings, a 4uestion un"ol\"l;'d for along lime. There remains so me importanl fCa!lln.~...to be ex-plained, in particular, one shoulJ cakulalc Ihe dcnsil)' of ¡heglass, sinec Ihere is no a slruelural model that is ahle to prc-dict the Illeasured valuc. Many similar prohlcms in oll1er Illa-tcrials can also be invcstigalcd with Ihis simple lechnique. asquasicryswls, covalent glasscs and nanoslruelured material".\Ve shall address these prohlellls in the futurc.

Acknowlcdl(ments

This work was supported hy lhe project IN] 04296 fromUNt\M-DGt\f't\, and by CONACyT granls 0088P-E,2677P-E and 25237-E. GGN is grateful for the hospilalit)'reccived at Ihe Université Picrre et I\1arie Curic uuring a post-doctoral slay thCfl~.

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R,,'. Mex. Fú. ,¡.¡ S3 1199R) RO-X4