VOL. 81, NO. 17 JOURNAL OF GEOPHYSICAL RESEARCH JUNE 10, 1976

ExperimentalDetermination of the PressureDependence of the Thermal Diffusivity of Teflon, Sodium Chloride, Quartz, and Silica

SUSAN WERNER KIEFFER

Departmentof Geology,University of California,Los Angeles,California 90024

IVAN C. GETTING AND GEORGE C. KENNEDY

Instituteof Geophysics,University of California,Los Angeles,California 90024

The thermaldiffusivity of Teflon,sodium chloride, quartz, and silicaglass was measured at 40øCto pressuresof 35, 18,30, and 36 kbar, respectively.A transientline sourcemethod was modified for usein a piston-cylinderhigh-pressure cell. Pressure gradients were determined by experimentswith bismuthfoils. The pressuredependence of thethermal diffusivity at 40øCfor thesubstances studied may be represented as follows(K in squarecentimeters per second,P in kilobars):for the low-pressurephases of Teflon, Teflon I-II, P < 5.5 kbar, K = 0.0012+ 3.6 X 10-sP;for the high-pressurephase, Teflon III, 5.5 kbar < P < 35 kbar, • = 0.0012 + 8.0 X 10-5 P; for polycrystallinehalite, P < 18 kbar, • = 0.0031 + 9.5 X 10-4 P; for quartz,perpendicular to the c axis,P < 30 kbar, • -- 0.031+ 5.3 X 10-4 P; for silicaglass, P < 36 kbar, • = 0.0068- 6.7 X 10-6 P. The diffusivityof silicaglass decreases with pressure,in contrastto the diffusivityof its crystallinecounterpart, quartz, which increaseswith pressure.In addition to the diffusivitythe thermalconductivity of Teflonwas determined by measuringthe powerapplied to the heaterwire. The thermalconductivity of a TeflonI-II mixtureis approximately constant at 0.0075-0.0078 cal/cms øK to 5.5 kbar.Above 5.5 kbarthe conductivityof TeflonIII isgiven by K: 0.0062+ 4.0 X 10-sP.The specificheat of Teflondecreases with pressureand decreases discontinuously by 15%across the TeflonlI-lII phasechange, in goodagreement with the decreasepredicted from thermalexpansion and compressibilitydata.

INTRODUCTION determined,either allowing the specificheat to be determined There are only a few reported data in the literature on the through c = K/pK or providing a consistencycheck if the variation of the thermal conductivityor diffusivityof minerals specificheat is known; (2) the early part of the temperature- with pressure[Bridgman, 1924; Fujisawa et al., 1968;Schloessin time history may be used, thus minimizing violations of and Dt;orak, 1972].We report herenew thermaldiffusivity (K) boundaryconditions which are in generaldifficult to maintain data for Teflon, sodiumchloride, quartz, and silicaglass and in the pressureapparatus; (3) thermal propertiesare measured thermalconductivity (K) and specificheat (c) data for Teflon. in a singleplane, perpendicularto the line source;(4) sample Thesesubstances were chosenfor measurementsfor a variety designis amenableto the inherent pressurevessel geometry; of reasons:(1) Teflon, as a material that is commonly used (5) sample size maximized so that experimental errors due to in pressure cells, as a material for which values had been the finite size of the heater wire and the thermocoupleare published[,4ndersson and Bg'ckstr•m,1972], and as a material minimized; (6) sample preparation, while it is not trivial, is with phase transitions in the range of pressuresattainable relatively simple.The main disadvantageof the method is that [Pistorius,1964]; (2) sodium chloride (polycrystalline),as a data reduction is complex. crystalline substance whose atomic structure allows theo- An inherent difficulty with the application of most methods reticalcalculations of the thermal properties[e.g., Mooney and of simultaneouslymeasuring the thermal diffusivityand con- Steg, 1969]; (3) quartz, as a framework silicaterepresentative ductivity to high pressureis in providing an accuratedescrip- of minerals in the earth's crust; and (4) silica glass, as an tion o[ the boundaryconditions on the sample.The original amorphousmaterial for comparisonwith its crystallinecoun- line source method of Jaeger and Sass [1964] applies to a terpart, quartz. sample which is perfectly insulated or has only small heat losses.It is not possibleto insulatea mineral sample(which is THEORY: LINE SOURCE METHOD itself a good insulator) in a high-pressurevessel. We per- We modifiedthe transientline sourcemethod first proposed formed severalexperiments which showedthat the assump- by daeger[1959]and later developedby daegerand Sass[1964] tions of perfectinsulation or small heat lossesfrom the sample for application to a piston-cylinderhigh-pressure cell. In this are not even approximated at high pressure. We therefore method a thin linear heater serves as the heat source; the decidedto make the cylindricalsurface of the sampleas nearly temperature rise (t;) above ambient is recordedas a function of isothermalas possibleby placing the samplein contact with time (t) at a known distance from the heater. The thermal good thermal conductors.The following theory, then, is a diffusivityand conductivitymay be determinedfrom this tem- modification of the Jaeger and Sass [1964] theory to this perature-time(t;-t) curve and a knowledgeof the power output boundary condition. of the heater. Many advantagesof the method as used for An infinitely long circular cylinder 0 < t, < a (or equiva- work at 1-atm pressurewere listedby daegerand Sass[1964]. lently, one which is perfectly insulatedat the endsso that the In addition,the methodhas the followingadvantages for high- axial flux is zero, dv/dz = 0) is assumedto have a thermal pressurework: (1) the diffusivity and conductivityare both conductivityK, densityp, specificheat c, and thermal diffusiv- ity • = K/pc. K, •, p, and c are assumedto be independentof Copyright¸ 1976by the AmericanGeophysical Union. temperature. The initial differential temperature between the

3018 KIEFFERET AL.' EXPERIMENTALDETERMINATION OF THERMAL DIFFUSIVITY 3019

cylinder and its surroundings(v) is assumedto be zero (v = +z vo(r) = 0) at time zero (t = 0). The outer surface(r = a) is Ht.'• T.C. assumedto be an isothermalsurface (v = v(a) = 0 for all t > 0). For all time t > 0, there is a line sourceof infinitelysmall diameter parallel to the axis through the point whose polar coordinates are (r', 0). This sourceemits heat at the constant •r rate Q per unit length per unit time. The basicequation of heatconduction in cylindricalcoordi- tainless nates is --Pyro. 7 - = o + --Brass The ditTerentialtemperature v (henceforthreferred to simply Teflon as the temperature) at the point (r, 0) at time t (t • 0) is obtained by applying, successively,a Laplace transform, the dv =0 --Lead addition theorem, the inversiontheorem, and the theory of dz residues,or by direct integrationof the Green's function [Car- --Silver slaw and Jaeger, 1959, p. 385]. For a line source and the thermocoupleat equal distances,r' = r, the solution is ChromeI-Alumel v(t)=0-- hermocouple •Kv(r)_o 21 {--ln (•)+ In -- Z• qnCOS H0Z• •xp[-- K.n,,• 2t/•2] Jn2[a .... (r/a)] •Heater n=o m=o (•n,m•/a•)Jnt• (•n,m) ' (2) d._•v=0 where O is the heat emissionper unit length,K is the con- dz ductivity, • is the diffusivity,r is the radiusto the heaterand foil thermocouple(assumed equal), a is the cylinderradius, •, = 1 + Moly-coat for all n = 0, •, = 2 for n 2 1, andan,m are the positiveroots of J,'(a) = 0. This solution consistsof two terms: the first, a steady state term; the second, a transient term which is a flon complicatedfunction of the samplegeometry.

EXPERIMENTALPROCEDURE AND DATA REDUCTION The experimentswere conducted in an end-loadedpiston- cylinder apparatusto a maximum pressureof 35 kbar. The initial temperature(To) wasroom temperature (about 27øC). 2.54 cm (1 in) The high-pressurecell whichcontained the samplewas placed Fig. 1. High-pressurecell for measurementof K, K, and pc by the in a largepressure vessel (3.18 cm (1{ inches)in bore diameter line source method. by 15.24 cm (6 inches)) in length. The outside of the vessel was cooled by continuouslycirculating water. The high-pressurecell is shown in detail in Figure 1. The immediatelyin contact with the samplebecause of its high sample(5.08 cm (2 inches)in length, 1.27cm (• inch) in diam- thermal conductivity.To assurereasonable stress distribution, eter) contained two diametrically opposing slots, passing however, a weaker bushing of lead (a slightly poorer con- through the points (r, 0) and (r, •); r was generallychosen ductor) was placed outside the sample over a substantial to be 0.7 of the sample radius. The heater wire and thermo- length of the cell. An insulator,Teflon, was placedat the ends couple (Chromel-Alumel) were placedi n opposedslots. For of the sampleto givea reasonableapproximation to the Teflon samples the wires were packed into the slots with boundary condition that the axial heat flux be zero (dv/dz = Teflon; glass was melted into the sIOtsof quartz and glass 0). samples;and sodiumchloride was packedinto the slotsof Ideally, the sampleenvironment should be hydrostatic.This the sodium chloride sample. The four wires from the heater conditionis impossibleto attain in a solid mediumpressure and thermocouplewere brought out from the high-pressure apparatus becauseof the finite shear strengthsof the various environment through a four-hole ceramic tube at the top of cell components.Weak materials(lead and Teflon) were used the assembly. wherever possiblein order to minimize shear stresses. The cell was designedto achievetwo conditions:(1) the Stress distribution in the cell was studied in a separate desiredthermal boundary conditionson the sidesand endsof experimentof geometry similar to the thermal measurement the cylinder,and (2) a conditionof uniform stressdistribution. assembly.Bismuth foils were placed at the endsof a Teflon The samplewas surroundedin the radial directionby good sample,as is shownin Figure 2. As the pressurewas increased conductorswhich coupledit thermallyto the pressurevessel and decreasedthrough two cycles,the Bi I-II and II-I transi- and cooling system.Such thermal couplinggave a reasonable tions in each foil were observedby monitoringelectrical resis- approximation to the boundary condition of the cylinder sur- tance.Although thesefoils are expectedto respondmost sensi- facethat the differentialtemperature remain zero (v(a) = 0 for tivelyto theaxial component of stress,we have taken them to all t • 0). Silver was chosenas the material to be placed be reasonableindicators of pressure.On compression,the Bi 3O2O KIEFFER ET AL.: EXPERIMENTAL DETERMINATION OF THERMAL DIFFUSIVITY

Pressure Gradient Model effectsto be reasonablysymmetrical in the bismuthexperi- Cell Geometry ments,we chose the midpoint of hysteresisloops as the pres- sure at the center of the sample. In the central region of the loop (•7 to •27 kbar) the correction is 2«-3 kbar (Figure 3b). The ends of individual hysteresisloops are complicatedby pressuregradient reversals.At the lower ends they are also •ISTON BIf•)ll 1 Bifoil 2 complicatedby phasechanges in the Teflon. Data from these 4O regionsare not consideredin determiningfriction corrections. On the 35-kbar excursion,complete pressure gradient reversal upon decompressionappears to have occurred by 26 k bar (Figure 3a). This value is in reasonableagreement with the value required to reversethe gradient in the vicinity of the sample center (from the linear model of Figure 2). Bit The temperature-time(v-t) history of the thermocouplewas recorded(generally for about2 rain)on a stripchart recorder. 2O Data reduction was accomplishedby using the method of Jaeger[ 1959]using only the earlypart of the temperature-time curve, typicallythe first 30 s to 1 min. The theoreticalsolution (2) of the temperature-timecurve is of the form

IOi 2 4 6 8 I0 12 14 v(t) = A(K, •)f(•t/a •) (3) PISTON Distance (in undeformed cell, cm) where A is a constantdepending on K and •. From this, it Fig. 2. Pressuregradients in the high-pressurecell. Data pointsA, follows that B, C, and D indicate applied pressureat which the bismuth l-ll transition (A and B) and the ll-l transition (C and D) were observed v(nt) l(nt•t/a•) at the two foils. Closed triangles representtransitions in foil 1; open (4) triangles, foil 2. In this plot, linear pressure distributions are con- v(t) - l(gt/a•') structedto passthrough thesepoints and the correspondinglocations (a, b, c, and d) at which the transitionsactually occurred. The top line where n is a positivenumber, generallytaken to be an integer. showsthe assumedpressure distribution at an appliedpressure of 35.2 With the aid of a theoretical curve or table of the function kbar; reductionof this applied pressureto 30.0 kbar is necessaryto just reversethe pressuregradient at the center of the cell. Complete f(n•t/a•)/f(•t/a •) a valueof •t/a • can be foundfrom an experi- reversalof the gradientsis not attained until the pressureis reducedto mental value of v(nt)/v(t) [Jaeger, 1959]. 23.0 kbar, coincidentwith point C. In practice, convenientunits of time t = toand n are chosen, e.g., to = 2 s, n = 2. Values of v(mto) are read from the strip chart at timesmto, rn = 1, 2, ß... The ratios v(nmto)/v(mto)are 1-11 transition pressurewas taken as 25.5 kbar [Heydemann, 1967];the Bi II-I transition on decompressionwas assumedto calculated and compared to tables of the function be about 0.5 kbar lower [Davidsonand Lee, 1964]. Upon compressionthe foil nearer the piston underwentthe l-If transitionat a pressureof 27.3 kbar appliedto the piston; Diffustvityvs•ominal Pressurg: TEFLON• I ß Up1 the more remote foil, at 28.8 kbar. On decompressionthe foil nearer the piston underwent the II-I transition at 23.0 kbar; the more distant foil, at 21.1 kbar. Thesevalues are symmetri- u Down2 •/,. / /,,j 4 _^Up3 / / / • cally located about the true transition pressuresto within 0.2 v Down3 / /' -/./" / kbar for the foil nearer the piston and 0.6 kbar for the more v Down4 / •/% / distant. The departuresfrom symmetrywere of oppositesigns X Second / • / / / at the two ends of the sample. The applied pressuresat which the transitionstook placeimplied an axial pressuregradient of 0.4 kbar/cm (1.2 kbar/inch) over the 5.08-cm (2.0 inches) initial length of the sample. The thermocoupleused in the thermal measurementswas locatedat the midpoint of the sample'slength. Thermal scaling • v 4t of the experiment shows that during a typical run of 100-s duration the thermocouplehistory is most influencedby mate- rial within a distanceslightly lessthan one cylinderdiameter, that is, by material within 1.27 cm (,• inch) of the thermo- (• couple. Thus we estimate that the pressurevariation within I0 20 30 40 material which influencesthe experimental results is about 1 kbar. Since the pressuregradient is probably uniform in the central region, it is reasonableto assumethat the pressure differencesare largelyself-compensating and that the thermal I I propertiesare averagedover +• kbar about the mean pressure. In order to estimatethe mean pressureat the center of the Fig. 3. (a) Diffusivityof Teflon versuspressure. Four hystereses loops are shown. The center (heavy) line representsthe midpoint sample, diffusivity measurementswere made on compression values. The light line shows the relative values of Anderssonand and decompressionover severalpressure ranges. The data for Backstrbm[1972] normalizedto our l-bar value. (b) Hysteresisloop Teflon are shownin Figure 3a. Having observedthe frictional width versuspressure. KIEFFERET AL.: EXPERIMENTALDETERMINATION OF THERMALDIFFUSIVITY 3021

f(nKt/a2)/f(Kt/a2) generatedfrom (2). From these tables a z o value of K is determinedfor eachratio. In theory,each ratio z v(nrnto)/v(rnto)gives the samevalue of K,or valueswhich agree _o I-- z within experimentalerror. In practice,we observedsystematic z drift of the valuesin all but the earliestparts of the v-t curves, TEFLON TEFLON an indication that the theory did not perfectly representthe •T I-- z 1T o. 40 z _ experimentalsituation. In these caseswe extrapolatedthe E • 3o- drifting valuesback to the nominal value at t = 0. When K has been determined,the value of K can be found from (3) (if the IO- heaterpower, Q, has been measured)by comparingthe ob- servedvalues of v(t) with the theoreticalvalue A(K, K)f(Kt/a2). Time(increasing-.-,-) (~15sec) From K and K the specificheat can be determined from c - Pressure(decreasing K/OK. Fig. 5. Heat of reactionof the Teflon III-II phasechange recorded The observeddiffusivity values vary somewhatwith varia- as the pressurewas decreased through the phasechange region. Values of applied pressureduring the decompression(not correctedfor fric- tions in the power appliedto the heater wire. Higher heater tion) are recorded beside the curve. power causesa higher averagetemperature in the sample.We thereforeexpect the observedvalues to reflectthe temperature dependenceof the diffusivity. This is indeed the case. For rocks, minerals, and polymers, we chose Teflon as the sub- Teflon, higher valuesof diffusivitywere measuredat larger stancefor usefor our initial samplesbecause (1) diffusivityand power outputs(e.g., a factor of 2 increasein power appliedto conductivityvalues for Teflon with small experimental uncer- the heatercaused an increaseof averagediffusivity of about tainties had been published by Anderssonand Biickstrbm 10%).The publishedvalues of diffusivityof Teflon do indeed [1972]; (2) Teflon is commonly used in laboratory pressure increasewith temperaturein the range 30øC-60øC [Shelley cells, and knowledgeof its thermal conductivitywould aid in and Huber, 1968]. For quartz the diffusivitygenerally de- designingexperiments; and (3) Teflon is easilymachinable and creasedslightly with appliedpower, as would be expectedfrom therefore easy to use for developmentalexperiments. the temperature dependenceof the thermal conductivity [Clark, 1966, p. 466], but the dependencewas slight. The RESULTS diffusivityof glassdid not vary with changesof a factor of 2 in Teflon. Teflon shows several kinds of phase transitions applied power. No data were obtained on the power depen- (Figure4) [Pistorius,1964; Shelley and Huber, 1968]. At 1-bar denceof diffusivityof NaCl becauseof experimentalproblems. pressurethere is a strongfirst-order thermodynamic transition In orderto minimizethis temperature effect we used the min- at 20øC (Teflon II-la in Figure 4; [Rigby and Bunn, 1949]). imum power required to give a resolvablev-t curve for data This transition is believed to be a screw dislocation, order- reduction.The measuredvalues of diffusivityare appropriate disordertransformation, in which the helical form of the poly- to a temperature averaged over the sample volume for the mer chainchanges [Shelley and Huber, 1968].At 30øC a transi- duration of the experiment.The value of temperaturewhich tion (Teflon Ia-I in Figure 4) occurs as the polymer chain we specifyfor our measurementsis the highesttemperature unwinds. The high-pressuredata of Pistorius [1964] show a attainedat the thermocoupleduring the durationof the experi- high-pressurephase, Teflon III, stable above about 5 kbar. ments. Actual temperaturesattained in the material between These data show the Teflon I-II-III phaseboundaries but do the heaterwire and the thermocoupleat the end of the experi- not resolvethe I-Ia transition at high pressure. mentwould have beenslightly higher, but the valuespecified is Data in the literature regardingthe behavior of the thermal probably a reasonableapproximation to the mean temper- diffusivity and conductivityacross the phase boundariesare ature of the sampleintegrated over the duration of the experi- inconsistent.The l-atto data of Kirichenkoet al. [1964]and ment. Ozawa and Kanari [1967] do not show discontinuitiesat the Becauseof the difficulty of the experimentswe wishedto II-I and l-la transitions;the data of Shelleyand Huber [1968] determine the diffusivity and conductivityof a sample for clearly resolve the transitions and show large variations in whichthe values were previously known. After consideration diffusivity through the transitions.The pressuredata on diffu- of previouslypublished data on diffusivityand conductivityof sivity of Anderssonand Bi•ckstrbm [1972] do not resolve the II-IIl transition. Our experimental diffusivity data for Teflon are shown in Figure 3. Below 5.5 kbar the diffusivityis givenby K(cm• s-•) = 0.0012 + 3.6 X 10-* P(kbar) at average temperature 40øC. The nominalvalue of the diffusivityextrapolated to I bar is 0.0012 cm• s-•. Although pressureuncertainties in a piston- cylinderapparatus are relativelylarge at pressuresof a few kilobars, so that we could not resolvea phasechange in this range of pressure,it is probable that the low-pressuredata apply to a mixture of Teflon I and II, in which the proportion of Teflon II presentincreases with increasingpressure. Phase diagram of The diffusivity is discontinuousacross the Teflon II-III TEFLON phase boundary. The transition occurred gradually in the (after Pistorius, 1964) high-pressurecell at pressuresbetween about 5 and 8 kbar. I I Ia 5o •00 [5o zoo Although we did not seeany evidenceof the I-II transitionat Temperature (øC) low pressures,the heat of reaction of the II-III phasetransi- Fig. 4. Phasediagram of Teflon (polytetrafluoroethylene). tion was clearly recordedon the strip chart recorderby varia- 3022 KIEFFERET AL.: EXPERIMENTALDETERMINATION OF THERMALDIFFUSIVITY

io 2O

And• Bi•C-• / --<--And & BSc.

•O12 TEFLON

:10 I01 dO Conduchv•tyvs Nominal Pressure Pressure (kb) TEFLON Fig. 8. Specificheat and volumeof Teflon versuspressure. 2

o pressurizedlength of copper leads, the final heater resistance Pressure (kb) was observedto vary from 0.63 •2 initially to 0.50 •2 at 35 kbar. Fig. 6. Thermal conductivityof' Teflon versuspressure. Same nota- The thermal conductivity data obtained for Teflon are tion as for Figure 3. shownin Figure6. Below 5.5 kbar the conductivityis approxi- mately constant at K = 0.00075 cal cm-• s-• øC-'; the con- tions in thermocoupletemperature as the pressurewas in- ductivity doesincrease slightly with pressurein this range,to a creasedor decreasedacross the transition (Figure 5). The value of K = 0.00078 cal cm -z s -z øC-' at 5.5 kbar. These pressureat which the heat of reactionwas recorded agrees well values are attributed to a Teflon I-II mixture or to Teflon II. with the pressureat which the phasetransition was reported The conductivity is discontinuousacross the Teflon II-III (Figure 3) and with the measureddiscontinuity in the thermal phase boundary. Extrapolated to 5.5 kbar, it has the value diffusivity.The diffusivityhas the value 0.0020 cm• s-• at 10 0.00088 cal cm-• s-' øC-' and by 30 kbar increasesto 0.00187 kbar and increaseslinearly to.0.0045 cm• s-' at 40 kbar. In this cal cm-• s-' øC-'. This increasein conductivityfor Teflon III range the increaseis 8 X 10-5 cm• s-' kbar -•. The values of is 5.5% kbar -z, somewhatlower than the (7.6 + 0.8)% kbar-• thermal diffusivityare somewhatlower than thosereported by reported by Anderssonand BiickstriSm[1972]. Anderssonand Bi•ckstrbm [1972], who did not resolve the The experimental values for K and K are summarized in II-1II phase transition. It is possible that variations in the Figure 7. From thesevalues and Bridgman's[1948] compres- sampleproperties (degree of crystallinity,method of manufac- sion data the specificheat was determined from c = K/t•K ture) accountfor the lack of agreementbetween the two setsof (Figure 8). The specificheat decreasesnonlinearly by 17% results,although the densityof the samplesused was nearly from 0 to 5 kbar and decreasesby 15%across the Teflon II-III identical(2.15 gcm -• [Anderssonand Bfickstrb•, 1972], 2.144 phasechange, in good agreementwith the decreaseacross the gcm -• (this paper)). phasechange predicted by Weir [1954] from thermalexpan- In order to determinethe conductivityK from the diffusivity sion and compressibilitydata. Kit is necessaryto measurethe power dissipated per unit length Quartz. The thermal diffusivity of single-crystalquartz in the line source.Limited by the numberof wiresexiting from wasmeasured perpendicular to the c axis(Figure 9). Although the cell,we werenot ableto measurethis power in situ.Instead, two cyclesof compressionand decompressionwere measured, the initial resistanceof both the heaterand its copperleads was the data shown in Figure 9 are selectedfrom only two tra- measuredbefore assembly into the pressurevessel. During the verses.Data from a smaller,interior hysteresisloop werebadly run the heater current and total resistanceof the leads plus scatteredand are not shown. The hysteresisloop width ob- heaterwere measured. The copperleads inside the cellchanged tained from the first compressionand last decompressionis in resistanceowing to the effectof pressure.Bridgman [1938] observeda 5% decreasein the resistanceof copperfrom room pressureto 30 kbar. After this correctionwas appliedto the

4O -2O

15T b

_

• 2o lOb 30 _ •5i/vv.?? _ TEFLON 26 _L.tO C-•xis ,'o ,'o ½o 24 I I I I I I I 280i 5 I0 15 20 25 Quartz•0 •5 40 Pressure (kb) Pressure (kb) Fig. 7. Summary of thermal diffusivity and conductivitydata for Fig. 9. Thermal diffusivityof quartz, perpendicularto the c axis, Teflon. versusapplied pressure. KIEFFERET AL.' EXPERIMENTALDETERMINATION OF THERMALDIFFUSIVITY 3023

i cm-x s-x kbar-1, in excellentagreement with the value 1.3 X Si02 Glass 10-4 cal cm-x s-x kbar-x reported by Schloessinand Beck [1970]. v Down Sodium chloride. Sodium chloride was ground into a fine F1 Up2 powder and pressed, slightly damp, to 14 kbar to form a u Down 72 sampleof 97% theoretical density. It was not possibleto form a single sample of 5.08 cm (2.0 inches) in length, so the final ß.-. 7O sample was composedof two shorter segments.The thermo- u u u couple broke during this run at an applied pressureof 20 kbar E during the first increasein pressure,so the four data points i 0 6.6 which were obtained were corrected to a final pressureby Au subtracting3 kbar from the pressureapplied to the piston.This .>_ 64 value of the half-hysteresiswidth was observedin all of our

._ experiments (except those on quartz) in which one or more ø62 hysteresisloops were obtained. The four data points give a diffusivity of 0.031 cm•' s-1 at I bar (40øC) and a rate of

Pressure (kb) increaseof diffusivity with pressureof 3.2% kbar -x to 18 kbar. Fig. 10. Thermal diffusivity of silica glass versuspressure. Two The l-bar value of the diffusivity is closeto the value of 0.029 hysteresisloops are represented.Solid line representsleast squares cm•' s-1, which can be calculated from published values of linear fit to data. Dashed line indicatesprobable behavior discussedin thermal conductivity data [Clark, 1966, p. 465]. text. Silica glass. The diffusivity was measuredon samplesof homogeneoussilica glass to 35 kbar (Figure 10). Upon exam- anomalouslylarge, between6 and 8 kbar. The large value of ination after the experimentsthe samplestypically showed3-5 hysteresisloop width indicatesthat the pressuregradients in major fracturessimilar to thoseobserved in quartz. If a linear the cell containingquartz are larger than thosemeasured in the fit is made to the data to 35 kbar, the diffusivity is given by Teflon cell used for calibration. We believe that this effect is •(cm •' s-1) = 0.0068 - 6.7 X 10-•P(kbar). The zero-pressure causedby the larg•estrength of the quartz. We thereforeig- value agreeswith that which can be calculated from the ther- nored data from the interior hysteresisloop becauseit was mal conductivitydata of Ratcliffe [1959]. Note that the diffu- obtained over a pressurerange which may have been sub- sivity of silica glassdecreases with pressure,in contrastto the stantially affectedby incompletepressure gradient reversals. diffusivity of its crystalline counterpart, quartz, which in- Upon examination at the end of the experiments, quartz creaseswith pressure.The data show a systematicdeparture samplestypically showed 3-5 major fracturesroughly parallel from the straight line rifted to the data, being consistentlylow to the cylinder axis. These fracturesgenerally initiated at the in the •15- to •25-kbar region and then rising with further slots which contained the heater and thermocouple wires. In increasein pressure. the experiments on glass and quartz, loud cracking noises Comparisonwith theory. An approximate expressionfor which may have been related to the formation of these frac- the pressure dependenceof the thermal diffusivity has been tures were heard upon compressionat pressuresnear 10 k bar. given by Fujisawaet al. [1968]: The fractures probably remained closed at pressuresabove a few kilobars and did not appreciably affect the diffusivity measurements.The two low data pointsat the lowestpressures Ko (Figure 9) suggestthat the fracturesmay have openedbelow a few kilobars pressure. VmO• dP/oJ From the data in Figure 9 the nominal 1-bar value of the diffusivityfor quartz is 0.031 cm2 s-l, and the rate of increase where B(kbar) is the bulk modulus, v•(km/s) is the phonon with pressureis 1.7% kbar -1. The 1-bar value is within the velocity, and the subscript 0 refers to the values of these range of values of diffusivity which can be calculatedfrom quantitiesat l-bar pressure.Values of the parametersrequired thermalconductivity data givenby Clark [1966,p. 466]. These in this equation are listed in Table 1. In the case of NaC1, for values range from K = 0.029 to K = 0.033 cm•' s-x (with the which(dv•/dP)o is kfibwn,the lastterm of thisequation reported values of Clark interpolated to 40øC--K = 0.0145 contributes lessthan 10% to the calculated Changeof diffusiv- cal cm- xs- • øC- xand 0.0157 cal cm- • s- 1 øC- l, respectively-- ity. The values of this derivative are not knoTM for the other and with the assumptionsthat p - 2.65 g/cm 8 and c - 10.9 substances;however, they are likely to be of the same magni- cal/mol øK). The rate of increaseof diffusivity with pressure tude as for NaCI, so the lack of knowledgeof this term does is equivalent to an increaseof conductivity of 1.2 X 10-4 cal not seriouslyaffect the approximation,

TABLE 1. Parameters for Equation (5)

Predicted Observed dv,ddP, Change in Change in Substance Bo,kbar dB/dP Vr•o,km s -• km s-• kbar-• K,% kbar -• K,% kbar -•

NaC1 245 5.3 2.9 1.5 X 10 -•' +2.5 +3.2 Quartz 374 6.4 4.5 ... + 1.7 + 1.7 Silica 370 -7.3 4.1 .... 4.0 -0.1

All data are from Clark [1966] except dB/dP and dvr•/dP, which are from Roberts and Ruppin [1971] and Fujisawa et al. [1968], respectively. 3024 KIEFFERET AL.: EXPERIMENTALDETERMINATION OF THERMALDIFFUSIVITY

The valuesof Kpredicted by (5) are in goodagreement with Birch, F., Compressibility:Elastic constants, Geol. Soc. Amer. Mem. the measuredvalues for quartzand sodium chloride (Table 1 ). 97, 97, 1966. Equation5 alsocorrectly predicts the decreaseof diffusivity Bridgman,P. W., The thermalconductivity and compressibilityof severalrocks under high pressures,Arner. d. Sci., 7, 81, 1924. for fusedsilica but severelyoverestimates its magnitude.The Bridgman,P. W., The resistanceof nineteenmetals to 30,000kg/cm 2, decreaseof diffusivitywith pressureis associatedwith the Proc. Arner. Acad. Art s Sci., 72, 157, 1938...... I"us bulk -,,,,4,,• us ...... ,4,•..:.... :...... •, ,•,Uich is' •,,•Uaracteristic -• n,;,•..... t• W r•....gU ..... },.... ions of 177 ;•LIUbLCtlIL,...... C;b tO •• fusedsilica and Pyrex glass but notof mostother glasses [e.g., kg/cm•, Proc. Amer. Acad. Arts Sci., 76, 1948. Birch, 1966].We thereforeexpect that the negativepressure Carslaw,H. S., and J. C. Jaeger,Conduction of Heat in Solids,2nd ed., 510 pp., Oxford University Press,New York, 1959. dependenceof diffusivityis characteristicof silicaglass and Clark, S. P., Thermal conductivity,Geol. Soc. Amer. Mem. 97, 459, will not be found in glasseswhich do not haveanomalous bulk 1966. modulusderivatives. From (5), however,we seethat this de- Davidson,T. E., and A. P. Lee, lhe studyof the structuraland creasewill persistonly in that range of pressureswhere the transformationcharacteristics of the pressure-inducedpolymorphs in bismuth,Trans. Met. Soc. AIME, 230, 1035, 1964. bulk modulusis behavinganomalously. At highpressures the Fujisawa,H., N. Fujii, H. Mizutani, H. Kanamori,and S. Akimoto, glass'stiffens,' and the diffusivityshould increase with pres- Thermaldiffusivity of Mg•SiO4,Fe•SiO4, and NaCI at highpres- sure.Our data suggestthat this reversalin gradientoccurs at suresand temperatures,J. Geophys.Res., 73, 4727, 1968. •25 kbar. It is worthnoting that Bridgman's[1924] measure- Heydemann,P. L. M., lhe Bi I-II transitionpressure measured with a mentsof thermal conductivityfor Pyrexglass, which has a dead-weightpiston gauge, J. Appl.Phys., 38, 2640, 1967.(Correc- tion, J. Appl. Phys.,38, 3424, 1967.) negative, but smaller, bulk modulus derivative than fused si- Jaeger,J. C., The usesof completetemperature-time curves for deter- lica, showthat the conductivityincreases slightly with pres- minationof thermalconductivity with particularreference to rocks, sure. SinceK = Kpc,this observationsuggests that if the Aust. J. Phys., 12, 203, 1959. predictedsmall diffusivity decrease occurs, it is offsetby the Jaeger,J. C., andJ. H. Sass,A line-sourcemethod for measuringthe density increase. thermalconductivity and diffusivity of cylindricalspecimens of rock and otherpoor conductors,Brit. J. Appl. Phys.,15, 1187, 1964. CONCLUSIONS; SUMMARY OF DATA Kirichenko,Yu. A., B. N. Oleinik, and T. Z. Chadovich,Thermal propertiesof polymers,J. Eng. Phys.,7(5), 70, 1964. The pressuredependence of thethermal diffusivity at 40øC Mooney, D. L., and R. G. Steg,Pressure dependence of the thermal conductivityand ultrasonic attenuation of non-metallicsolids, High for thesubstances studied may be represented as follows(• in Temp. High Pressure,1, 237, 1969. squarecentimeters per second,P in kilobars)' Ozawa, T., and K. Kanari, Thermal conductivityof poly- Teflon I-II, P < 5.5 kbar K = 0.0012 + 3.6 X 10-sP tetrafluoroethylene,J. Polym.ScL, Part B, 5, 771, 1967. Teflon III, 5.5 kbar < P < 35 kbar K = 0.0012 + 8.0 X 10-sP Pistorius, C. W. F. T., Transitions and melting of poly- Quartz, perpendicularto c axis, tetrafluorethylene(lefton) underpressure, Polymer, 5, 315, 1964. P_<30kbar g= 0.031+5.3X 10-4p Ratcliffe,E. H., Thermal conductivitiesof fusedand crystalline NaCI (polyxl), P _< 18 kbar • = 0.031 + 9.5 X 10-4P quartz, Brit. J. Appl. Phys., 10, 22, 1959. Silica glass,P _<36 kbar g = 0.0068 - 6.7 X 10-6P Rigby, H. A., and C. W. Bunn,A room-temperaturetransition in polytetrafluorethylene,Nature, 164, 583, 1949. The thermalconductivity of TeflonI-II is approximatelycon- Roberts,R. W., andR. Ruppin,Volume dependence of the Grfineisen stant to 5.5 kbar, in the range 0.0075-0.0078 cal/cm sø C. parameterof alkali halides,Phys. Rev., Ser. B., 4, 2041, 1971. Above5.5 kbar the conductivityof TeflonIII is givenby' Schloessin,H. H., andA. E. Beck,On thepressure and temperature dependenceof the latticethermal conductivity of enstatite,silica, K = 0.0062 + 4.0 X 10-sP and pyrophyllite, Eos Trans. AGU, 51, 420, 1970. Schloessin,H. H., and Z. Dvorak, Anisotropiclattice thermal con- The specificheat of Teflon decreaseswith pressureand is ductivityin enstatiteas a functionof pressureand temperature, discontinuousacross the TeflonII-11I phasechange. Geophys.J. Roy. Astron.Soc., 27, 499, 1972. Acknowledgments.We thank J. Yamane, L. Faus, and J. De- Shelley,D. L., andS. F. Huber,Thermal diffusivity of poly(tetraflu- Crossefor fabricatingthe many intricate parts for theexperiments. S. oroethylene)between -140 and 125øC,in ThermalConductivity, BaileyBrown aided with data reduction,and B. Cola providedin- Proceedingsof theEighth Conference, edited by C. Y. Ho and R. E. valuablecontinuous encouragement. This researchwas fundedby Taylor, p. 1067, Plenum, New York, 1968. National ScienceFoundation grant GA-31897. Publication1486 of Weir, C. E., Temperaturedependence of compressionof linearhigh the Instituteof Geophysicsand Planetary Physics, University of Cali- polymersat high pressure,J. Res.Nat. Bur. Stand.,53, 245, 1954. fornia, Los Angeles.

REFERENCES

Andersson,P., and G. Btickstr6m,Specific heat of solidsat high (ReceivedJuly 28, 1975; pressuresfrom simultaneous measurements of thermal conductivity revisedFebruary 6, 1976; and diffusivity,High Temp.High Pressures,4, 101, 1972. acceptedFebruary 20, 1976.)