Solid StateCommunications, Vol. 9, pp. 1087—1090,1971. PergamonPress. Printedin GreatBritain

STRESSRELAXATION FUNCTION OF GLASS

C.K. Majumdar

Tata Instituteof FundamentalResearch,Bombay5, India

(Receivedin revisedform 24 April 1971 by A.R. Verma)

It has beenfound by Douglasandothersthat the stressrelaxationfunction of glass hasthe form S = S0 exp[ — (t/~7)°9.Simplephenom-enologicalconsiderations,utilizing the theory of Brownian motion,canexplain this law and yield propervaluesof a and~

IT HAS BEEN shown’ that the stressrelaxation boundaryconditions. Equations(2), (3) and(4)function in stabilizedglassis of the form are usedby Zener

2in his theory of anelasticityS = S~,exp[—(t/TfI, (1) andplasticity of metals.We do not adopthis

assumptionsin this problem,but merely regardwhereS~is the initial appliedstressandS the (2) as a phenomenologicalpostulate.Itsstressat time t at constantstrain. For t <<f, a dynamicalbasisis the fact that stressrelaxationhas valuesabout ~-; for t ~Ta is about ~ and in glassoccurby microbrownianmotion of itswhen 1 >> ~ a has the value 1. Thereis also molecularconstituents.It is thennatural tosomeinconclusiveevidenceof a ~ 0.2before invoke the Einstein relation3for the diffusionthe regiona = ~. We shall show that a law of coefficient.D is thereforeinversely proportionalthe form (1) follows from simplephenomenological to viscosity ~. Note that eachrelaxationtimearguments. is directly proportionalto viscosity.

We first note the fact that glass is a linear As (3) providesa completeset of functions,solid.’ We assumethe the stressfield S (x,t) we writeat any point in glass can be decomposedinto S(x, t) = ~ akcsk(x)exp(—Dk2t). (5)elementarystressrelaxationmodes,which are

Icsolutionsof the linear diffusion equation

Vile shall haveto determinehow much eachmodet9U — DV2 U. (2) contributesto stressrelaxation.-a;-

Glass is usuallydescribed4as a super-Eachmode decaysexponentially,with arelaxationtimer, U(x, t) = ~cx) exp[—(t/T)] cooled liquid madeof a randomly connected

network of tetrahedra.It is known from X-raythat work that glasshas well-definedshortrange

D 2 ~ + = 0. (3) order such that the local environmentof aparticular atom is moreor lessthe sameas in

Solving (3), we find the elementarymodes~k(x) a regularsolid. The randomnessarisesfromvariationof the relative orientationof adjacentand

tetrahedra.The rangeof order A is estimatedI = Dk2, (4) from X-ray dataor the model structuresof Dean‘7-

andBell,5 and canbe taken to be 5 x 108cm.where k is the usual wavevectordeterminedby

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1088 STRESSRELAXATION FUNCTION OF GLASS Vol.9, No.13

On a shorttime scale(not so short that — 47Tk~

macroscopicdescriptionscannotbe used),glass — iV~ ~Ik1~(xX’7~/3DO exp[_(t/’r1)I, (8)

will respondlike a rigid disorderedsolid. Considerwith

a relaxationmode of wavelengthA. The lineardimensionof materialsustainingsucha mode 4 A

2mustbe of theorder of A. If A is much larger than = 0.15A2/D. (9)the short rangeorder~, the modeinvolves motion

of partsof glass only incoherentlylinked and Note that the exponentialfactor, exp (— A/A), isshould be suppressed.It is a well known result characteristicof equilibrium situationsasof thetheory of disorderedsolids that correlations typified by theIsing model theories.6Henceitfall off exponentially(seefor instancetheIsing canhold for timesshort comparedwith f whichmodel theoriesof Glauberand Wu).6 The probability providesthetime scalefor motions.of excitationof wavelength A is then takentobe exp (— A/A). Hence At longer times the natureof relaxation

changesas theparts of glass in which theci IkHttAmin relaxationis more or less completeadjust to

~ f c~k(~)e~~2te~”~d~k oneanother.We considertwo extremesituations.

S(x,t) = \k~=1/Amax First, considerthat mismatchesbetweenthekI~i/Am~ small regionswhich haverelaxedare adjustede~~’Xd3k by slipping motion along surfaces,thus making

= 1 /Amax additional contribution to the stressrelaxation.(6) Let abe the activation energyper unit areafor

ci is the volume of the specimen.The smallest flow alongthesesurfaces.Then the total energywavelength A ~ is of the orderof the dimensions involved is 7TA2a, where,on accountof theof the tetrahedra.A max is of theorderof the isotropy of glass, we takethe relaxing region to

samplesize, or determinedby the sizeof the be a sphereof radius ~ A. The exactshapeof theregion of perfect glass, free of mechanicaldefects, region is, however, not important. The probabilityetc. The denominatoris certainly less than that this modeis available for relaxationat

(IZ/8’rT3). (4n/3A~~).Becauseof the two expo- temperatureT is expjI—(7TA2a/k~T)1.a can benential factors the major contribution comesfrom estimatedby this fact that if the modeis muchmodeswith k, determinedby longer than the short rangeorder A, it is

/3 suppressed.HencekBT/TTa A2 this gives(7) a 10-2eV per atom. This is of the sameorder2Dk, I — = 0, or, k 1 as the energy barrier for bond rotation in organic

The other factors dependweakly on k. Hence molecules,polymersetc. With this weighting

factor, we get4~k~ ifAminS (x, z) = Ic (x) ‘~kO~)e~2t e_A2~2d3k

ifArnin S(x,I) = tfAmax

j exp _Dk2I_.~ dk. ~f Xmjn e_A2~~~~2d3k

i/Amax i/Amax

W~denotesthe integral of the denominator,and

~k,~(x) is the sum over all functions with the = W2 k2((x)(~/4Dt)

2 exp[_(t/~)I~2I,samemagnitudek

1 of the wavevectork. We can (10)approximatethe integral, in view of (7), as with

4nk2 / 1 ~

S(x,t) = ~‘ k~(X) exp[—(t/~)”1 (p~2) T

2 = X2/4D. (11)Glass, however, is certainly not madeof crystal-J ~ )2 dk line aggregates;and the surfacesarerather thick.

- DC

Vol.9, No.13 STRESSRELAXATION FUNCTION OF GLASS 1089

To usea ferromagneticanalogy,the domain In view of the crudenessof the calculation,thewalls areas thick as thedomainsthemselves, agreementwith the experimentalvalue 4 x 10~secUnderthesecircumstances,onecan go to the is fair (estimatedfrom Fig. 10 of referencela,otherextremeand considervolume relaxation; seealso reference8).this will bein line with the critical volumetheoriesof Cohen andTumbull.7 One has Since the temperaturedependenceof ? is

i ~ almostentirely dominatedby the exponential~k(X) e_T)~~2te”3”~3 d3k changeof viscosity with temperature,this

max providesa straightforwardexplanationof theS(x,t) = i/Amin

j e’~~3”‘~ d 3k well-known time-temperaturecorrespondence:’

4~k~i/Amax —Te______ ~

= ~ ~i’3’ (x)(~/5Dt)~”2exp[—(t/T~)3”~1, texp =

(12)= exp[(A/k

8) (1/Te~,,)~(lIT8) 10].with ( 3 \1/5 3 2~’~X2

k3 = 2Dt~3) ‘ ~ = ~‘(~)‘~. It is perhapsworth while to point out the

following interestingfact. If we regardthe lawThe difference between~ and ~ is not much; the (1) as the resultof superpositionof manysimplecorrectanswerprobably lies somewherein exponentialdecays,we get a remarkabledistri-between.Onecan check that the transition from bution function for relaxationtimes. Write~ to ~ occurswhen I ~ T2_ ‘7-3•

As t .~ co, glass becomesmore and more exp[—(t/?)~i = 5 f(r) e_t~~TdT

relaxed,and finally only the longest modecontributes; Do

= 5 f(s) eSt ds (17)S(x,t) = f~ktx e~

2t .~‘k— d3k e_t’T4, 0 S2~

(13)with Hencethe distribution function f(r) is obtained

‘T4 = A~ax/D• (14) by an inverseLaplace transform. For large

relaxation times, which areof interest, this canNow we have a single relaxationtime. be evaluatedby a simple saddlepoint calculation.

The result, valid for a < 1, isFrom (9), (11), (12), ~ T2~‘T~ differ by

minor numerical factors and, within experimental

accuracy,may be consideredequal. To estimate f(T)

weuse Einstein’s relation3 iCrgeT r2127Ta(1— a)]

kBTD=—. (15)

6 7777a

Although data are rare, CohenandTurnbull think exp [(1 — a) 1 ] (18)that it is probablyvalid in glass;7 in any case

(15) should not bp used for F~in (14). a may betakento be 2 x 10_B cm, somewhatgreaterthan Acknowledgements— I would like to thankthe Si—O distance1.6 x 108cm. Hencefor ProfessorS.F. Edwardsof the Manchester

77 = iO’~poise, T = 800°K University for suggestingthe problemandProfessorR.W. Douglasfor an informative

1 67T77aA2= ~x kT - ‘-~ 10~sec. (16) discussionanda critical readingof the manu-script.

1090 STRESSRELAXATION FUNCTION OF GLASS Vol.9, No.13

REFERENCES

1. DOUGLAS R.W., (a) Proc. 4th mt. Cong.on Rheology,Providence,R.I., 1963, part 1, (editorsLEE E.H. and COPLEY A.L.) Wiley, New York (1965)pp.3—27; and (b) Br. J. appl. Phys. 17,435 (1966).

2. ZENER C., Elasticity and Anelasticityof Metals, p.76 et seq.The University of ChicagoPress,

Chicago, (1948).

3. EINSTEIN A., Ann. Phys. 17, 549 (1905).

4. ZACHARIASEN W., J. Amer. Chem.Soc. 54, 3841 (1932).

5. BELL R.J., and DEAN P., Nature, (Lond.) 212, 1354 (1966).

6. GLAUBER R., J. Math. Phys.4, 294 (1963); WU T.T., Phys. Rev. 149, 380 (1961).

7. COHEN M., andTURNBULL D., J. Chem.Phys. 34, 120 (1961).

8. DOUGLAS R.W., DUKE P.J., andMAZURIN O.V., Phys.Chem. Glasses,9, 169 (1968), [fig. 16].

Nach Douglasu.a.hat die Spannungsrelaxationfunktionvom Glasdie GestaltS = S

0 exp[—(t/?)~].Dies Gesetzmit richtigen Wertenvon a und ~ folgt aus einfachenphänomenologischenBetrachtungendie die Theorie derBrownschenBewegunggebrauchen.