IONIC DIFFUSION AND ELECTRICALCONDUCTIVITY IN QUARTZ

J. Vonuoocas, Uniaersity of Cali.Jornia, Berheley, CaliJornia.

ABSTRACT

The diffusion coefficients of Li+, Na+, K+ parallel to the c-axis of natural quartz

crystals in the range 300-500" C. are given by the following expressions:

Du,: 6.9 X 10-3 X d20'rot lnr cm.2/sec.

DN": 3.6 x 10-3 X e-24'000lRr cm.zfsec.

Dr : 0.18 I g3r,700lRi I cm.2/sec.

In the same temperature range the difiusion coefficients of Mg++, 6u++, Fe++, A+++

are considerably smaller, below the limit of experimental determination by the present

method.It is suggested that electrical conduction parallel to the c-axis of quartz results from

motion of Frenkel oxygen defects. On this hypothesis, the coefficient of self-difiusion ofoxygen in qtartzat 500" C. is found to beapproximately3XlO-tr cm.2fsec. It is suggestedalso that difiusion of foreign univalent ions occurs mainly through vacant oxygen latticepositions. Experimental values are compared with those predicted by the theory of abso-lute reaction rates.

Teslr' or CoN:rnrrsIntroductionAcknowledgments .Definition of Difiusion CoefficientExperimental ProcedureExperimental Results. . . .Electrical Conductivity of Quartz.Mechanism of Conduction and Self-Difiusion Rate of Oxygen. .Mechanism of DifiusionGeologic Implications.Appendix: Equations of Diffusion in an Electric FieldReferences

INrnouucrrow

The minerals and fabric of a rock, either igneous or metamorphic, de-pend to a large extent on the kinetics of the processes by which the rockis formed. Glassy rocks, for instance, form when the rate of cooling israpid compared with that of crystallization. Oscillatory zo\ing in plagio-clase presumably testifies to processes that were rapid in comparison withthe rate of the exchange reaction between crystal and melt. If mineralsA and B in a thermally metamorphosed rock react to produce mineral C,occurrence of C not only indicates that the rock was heated to theequil ibrium temperature for the reaction Al B:C;it indicates also thatthe rock was subiected to conditions under which the rate of this reaction

63763863863So+J

64647649651652655

oJ/

638 T. VERHOOGEN

became appreciable as compared to the length of time during which therock was held under those conditions. This may point to a temperaturemuch higher than the equilibrium temperature, or to operation of othercatalytic factors. It is possible, also, that C might form as a metastablephase below the equilibrium temperature, as when cristobalite forms wellbelow its own range of stability. Some knowledge of reaction rates, andof factors which may afiect them, is thus essential to correct understand-ing of the sequence of events recorded in igneous or metamorphic rocks.

The over-all rate of a heterogeneous reaction involving several succes-sive steps is determined essentially by the rate of the slowest step. Thereis little doubt that in processes involving solids, the rate-determiningstep, or "bottleneck," is the rate of diffusion of reactants and products inthe various solid phases. For instance, the rate of the reaction by whicha crystal oI quartz is converted to orthoclase, or andalusite is replacedby muscovite, depends mostly on the rate at which the alkali aluminatesdifiuse through the pre-existing crystal lattice. Knowledge of diffusionrates in solid silicates is thus of fundamental importance. The object ofthe present paper is to report on some experiments to determine theorder of magnitude of the diffusion coefficients of common ions (K+,Na+, Li+, Mg**, Ca++, Al+++) in quartz.

Acxxowlr'ocMENTS

This work was made possible by grants from the Institute of Geo-physics and the Board of Research of the University of California, whichare gratefully acknowledged. B. C. Morrisson, W. F. Burke, J. B. Farr,and P. I. Eimon assisted in the experiments. numerical calculations, anddrafting of figures.

DnrrNrrroN ol Dmlusrou ConlnrcrBwr

It is customary to define a diffusion coefficient in the foliowing manner:Let a particle with mass ln be subjected to a force F arising from a con-centration gradient or, as in the case of the present experiments, froman electrical f ield. Let it be assumed that this particle moves through asolid medium in which it encounters frictional resistance proportional toits velocity 2,. The equation of motion is then

*oo : r - L ,d t B

where the coefficient B is the "mobility" of the particle. If u:0 at timel:0, the solution of this equation is

1 : F B l t - e - t t B n l . (1)

The numerical values ol m and B are always such that the velocity

IONIC DIFFUSION AND ELECTRICAL CONDUCT-IVIT'Y IN QUARTZ 639

reaches an approximately constant value o: F B within a very small frac-tion of a second. The mobility B may thus be interpreted as the velocityof a particle under unit force.

Now if c is the concentration of diffusing particles (number of particlesper unit vohr.me), the flux, or number S of particles flowing across a unitsurface in unit time is

S : t c : FBc (2)

so that B may be found if F, S, and c are known. Where difiusion resultsfrom a concentration gradient, it is convenient to use, instead of B, a"diffusion coefficient" D defined as

D : B K T (3)

when fr is Boltzmann's constant, and T is (absolute) temperature. Withthis definition we have, very simply, that the flux S, in any arbitrarydirection r under a concentration gradient dc/d.r is

t": - o# (4)

(Turner and Verhoogen, 1951, p. 41).D has dimensions cm2. sec-l. B, themobility, has dimensions GM-l sec.

Diffusion coefficients are known to increase rapidly with temperature,following in general a law of the type

P: Pr;ElRr (5)

where Ds is independent (or nearly independent) of temperature. E is anexperimental "activation energy" for the process of diffusion.

Expnnrlrexrnr, PnocpounB

Equation (4) shows that D may be computed if we can measure theflow of particles under a given concentration gradient. The diffusion co-efficient of sodium in quartz at 500o C. is, however, of the order of1g-ro.p.r/sec. If we had a plate ol quaftz one millimeter thick in contacton one side with sodium chloride, say, the average gradient across theplate would be of the order of 2xl}zz (since the concentration of sodiumin sodium chloride is about 2.2X1022 particles per cm.3), and the resultantflux across the plate would be 2 X 1023 X 10-10: 2 X 10+13 particles per cm.2per second, or only 0.07 miliigram/day. The actual rate would be muchless because not all sodium ions in sodium chloride are free to migrate;the factor c which appears in (4) should really be the number of "free"ions, or the "activity" which is considerably smaller than the concentra-tion. Such small difiusion rates have been measured by using radioactiveisotopes, but unfortunately the radioactive isotopes of Li, Al and othergeologically important elements have very short half-lives and cannot be

J. VERHOOGEN

employed for this purpose. It is possible, however, to accelerate the ionsappreciably by using an electrical field rather than a concentration gradi-ent, and thus obtain a measurable flow.

The experimental technique is simple (Fig. 1). A thin layer of a non-volati le salt of the ion under study (e.g., NaCl, LiCl) is spread on aplatinum foil resting on a Iower massive copper electrode inside a furnace.A plate of quartz 2 or 3 mm. thick and 4 or 5 cm.2 in cross-section restson this layer of salt and is covered in turn by a second platinum foil. Aheavy upper copper electrode weighs down on the assemblage to main-tain good contact. The two copper electrodes are connected to a D.C.

l l 5 vo l ts A .G.

GolvonornGtet

Plot inum-Ouorfr plotc

Sol l loyer

Si lvcr Coulombmctcr

Frc. 1. Schematic drawing of difiusion assembly.

constant voltage supply adjustable in the range 100-350 volts, the upperelectrode being the negative one. The metallic positive ions are acceler-ated towards the upper electrode and move across the quartz plate,carrying a current which is measured by means of a sensitive galvanome-ter, this current being proportional to the concentration and to thevelocity of the difiusing ions, which are collected on the upper platinumfoil, usually as a mixture of hydroxide and hydrous carbonates. The totalamount transported in a given time may thus be determined by weighing.The total electrical charge which has flowed through the circuit in thissame time may be ascertained, if necessary, by inserting a silver coulomb-meter in the circuit.

An approximate value of D ntay be obtained in the following way:Using (3), equation (2) rr'ay be written

FcS : ,

* '

If an ion carrying a charge ez (zis valency) is placed in a field of intensityd'V/d.r in the r-direction, the force F in that direction is

dvF : - e z -

d'r

f o

opper elecfrodc

D.C. Rcgulotor35O vol ls

Coppcr e lecl rode+

(6)

IONIC DIFFUSION AND ELECTRICAL CONDUCTIVITY IN QUARTZ 641

the minus sign indicating that a positive ion difiuses towards the nega-

tive electrode. As an approximation let us take d'V /d'r: - (a'V /h) whete

A 7 is the difference in potential and h the thickness of the quart z plate.

Then the total number of ions diffusing in time d across a plate with

cross-section s is

0 : Ssa Q)

and

D - k r h Q ( 8 )

ez sc|LV

from which D lrray be determined if Q and c are known' Care must be

taken to express all quantities in the same units' If. e, the electronic

charge is expressed in c.g.s.e.u. (e:4.8X10-10), the potential drop AZ

must be expressed in the same units (1 volt:1/300 c.g.s.e.u.). The con-

stant fr is 1.38X10-r6 ergs/deg. The time 0 in these experiments was

usually of the order of 10 days:8.64X106 seconds.Alternatively, D may be determined from the electrical conductivity

o. According to Ohm's law, the current density i (current per unit sur-

face) is i. : - od.v/itr. (9)

If the current density results from a flux S of particles with charge ea,

i : S e z

dvS : - B e z c

d r

and S, as before, is

so that6

B : -e222c

ckTD - - .

e"z"e

(10)

( 1 1 )

The conductivity o is usually expressed in ohm-l cm.-l, so that, after

conversion to electrostatic units

D : 5.55 X 10* oz . 02)z"c

The conductivity o is easily determined for simple geometric shapes from

the potential drop A 7 and the total current ,I flowing through the circuit

o : Ih/sLV

where o will be given in ohm-l cm.-l if 1 is expressed in amps' and AZ in

volts.Equation (12), of course, is only applicable if it can be ascertained that

the ionductivity is due entirely to the difiusing ion. This, however, is

642 J. VERHOOGEN

rarely the case, for part of the current may be carried by electrons, or byions of opposite sign difiusing in the opposite direction. Occurrence ofsuch negative carriers appears indeed to be a necessity, for if only positiveions were to diffuse into and through the quartz plate, the plate wouldsoon acquire a very high electrostatic charge. This could be checked bycomparing the intensity of the current as it enters or leaves the crystal.The equation of continuity

d i v z :

where p is the density of electrical charges shows indeed that the currentmust decrease in the r-direction if positive space charges are being builtup. No differences in current could be detected that could not be tracedto leakage through faulty insulation.* Hence, it appears that the crystalplate as a whole maintains electric neutrality during the process of dif-fusion, which requires a motion of negative charges to balance advancingpositive charges. A comparison of the values of e and of the integraltpat (measttred with a coulombmeter or by graphical integration) showsthat usually no more than about 50/6 of the current is carried by thedifiusing ion. The exact proportion, however, is uncertain because ofdifficulty in determining exactly the quantity Q. what is collected on thenegative electrode is indeed usually found to consist of a mixture ofhygroscopic hydroxides and carbonates in various states of hydration (inthe case of lithium, for instance, at least three constituents were recog-nized optically in the deposit); and it is difficult to determine the exactproportion of each of these in a deposit weighing a few milligrams or less.This uncertainty on Q appears to be the main source of error in theseexperiments.

The question also arises as to whether difiusion occurs through thebody of the crystal or through mechanical defects such as invisible cracks,or grain boundaries, or twin planes. T]ne quartz crystals used in theseexperiments were natural crystals and were certainly not free from de-fects, as shown by the scatter in observed values of the conductivity ofthe quartz itself. rn general, quartz plates used for diffusion experimentswere selected on the basis of their low conductivity, which was inter-preted to indicate few mechanical defects. rn spite of this, difiusion alongcracls could be shown in the case of experiments with silver. Silver ap-

* Since the resistance of a quartz plate 2 mm. thick and 5 cm.2 in cross-section is about4X10E ohms at 500" C., careful insulation is necessary, particularl,rr where the wires enterthe furnace. Also, on account of the extreme resistivity of quartz, it was not found prac-ticable to place guard rings around the quartz plate, as the insulating material betweenrings and main electrodes could probably be a better conductor than quartz itself.

A ^

at

I ,NICDIFFUSI)NANDELECTRICALC)NDUCTIVITYINQUARTZ643

parently diffuses with extreme rapidity through such cracks which are

Iater found to be lined with a thin deposit of metallic siiver. This produces

a rapid, and very marked increase in conductivity, which soon reaches

values typical of conduction in metals. Such, however, was not the case

with the other ions, such as K, Li, Na, Mg, etc. The crystal remains per-

fectly clear a{ter the experiment and the cathodic deposit is usually

evenly distributed on the negative electrode, reproducing the outline of

the plate. It is thus likely that transport of ions occurred mainly through

the lattice itself. In any event, since the measurements were carried out

on natural crystals, present results should be applicable to processes of

diffusion in real crystals in nature.

ExpBnrlrnNt-qr RBsums

Using for c, the average concentration of the difiusing ion, a value the

determination of which will be discussed later, the following results were

obtained for the difiusion coefficient at 500o C., {or diffusion parallel to

the c-axis of qtartz.Du: I .1 X 10-8 cm.2 sec.-r

Dn. : 5.8 X 10-to cm.2 sec.-l

Dx: 2.O X 10-to cm.2 sec.- l

These values are not believed to represent more than the order of

magnitude of the diffusion coefficients; numerical values may be in error

by a factor oI 2 or 3.To obtain the temperature coefficient of difiusion rates, it was as-

sumed that since c changes only very slowly (due to the small value of the

diffusion coefficient), any rapid change in o with temperature would be

due essentially to changes in D itself (see equation 12).It was found that

by cooling or heating rapidly (i.e., in a matter of an hour) a quartzplate

in which difiusion had been going on at a steady rate for some time (of

the order of a few days), the conductivity changed exponentially, i 'e.,

could be represented as a function of temperature by the relation

q : 6 0 4 - E / R I : .

The value of -E appearing in this relation was taken to be the same as that

appearing in equation (5), and was determined by measuring the ratio

of the conductivities at two difierent temperatures, provided these two

measurements could be carried out within a short interval of time. The

following values were obtained, by averaging several determinations

Eri : 20,6O0 cal./mole

Ew, : 24,000 cal /rnole

Er : 31,700 cal./mole

with a probable error of + 1000 calories, approximately. Combining these

64 r. VERHOOGEN

results with the values of. D at 500o, the following values were obtainedin the range 300-500' C.

Du:6 .9 X 10 -3X e -2o .6oo lR r

DN, : 3.6 X 10-3 X e-2t,ooolnr

Dx :0 .18y -€ - s r , 7oo lR r .

Tests were also made with Mg++, Q4++, ps++ and Al+++. In the caseof Mg, a small amount of magnesium was detected spectroscopically onthe negative electrode after a few days, but the amount was too small tobe determined with accuracy; hence Dy" is probably less than 10-12 at500' c. The other ions gave completely negative results, and probablyhardly difiuse at all. The anomalous case of silver, using silver foil as asource' has been mentioned earlier. Similar tests with gold, platinum,aluminum and copper foils gave no result at all.

Erncrnrcar, Corqoucrrvrrv ol euanrzrn order to throw light on the mechanism of difiusion, experiments

were made on the conductivity of quartz in the absence of diffusing ionsfrom an external source. The conductivity of quartz has been investi-gated by many writers apparently because of its peculiar behavior (for asummary of earlier work see Sosman, 1927, pp.525-554). It is wellknown, for instance, that the conductivity parallel to the c-axis is con-sideratrly greater than the conductivity along any direction normal tothis axis. The c-conductivity at 5000 c. was found in our experiments torange from about 5X10-to to 5a1g-s ohm-l cm.-l, depending on thespecimen, whereas Rochow (1938) reports an equatorial conductivity of2.ly1g-tz at this same temperature. Rochow states that the equatorialconductivity is essentially electronic, whereas current in the c-directionis carried mainly by ions. Reasons for this difierence are not known.

An interesting feature of the c-conductivity is its dependence on time.rf a plate of quartz is placed between electrodes and a fierd applied inthe direction of the c-axis, the current flowing through the circuit de-creases at first very rapidly, then more slowly. Although the actual rateof decay varies somewhat from one sample to the next, the current after10 or 15 minutes is commonly less than l/10 of the current after 1 minute;indeed, the initial rate of decay (that is, during the first minute) is sogreat that it is difficult to determine experimentally the value of theinitial conductivity. conductivity may be observed to decrease continu-ously, although more slowly, over periods of many days. [This is in sharpcontrast to the behavior observed when foreign ions are made to difiusethrough the plate, in which case the conductivity increases slowly withtime as more and more ions move into the crystal.] (Fig. 2). rf the erectricfield is removed after a certain time, the quartz slowly regains its initial

IONIC DIFFUSION AND ELECTRICAL CONDUCTIVITY IN QUARTZ 645

conductivity, but if the field is actually reversed, the conductivity reverts

almost instantaneously to its initial value or a higher value. Apparently,

this is why the conductivity for alternating current is considerably higher

than for direct current. If the circuit containing t]ne quartz plate is short-

circuited after removal of the field a "recovery," or "relaxation," or

"inverse" current is observed to flow, the quartz plate acting somewhat

Iike a storage battery. Jofr6 (1929) claims that this recovery current

equals in magni tude the d i rect , or charging current ; in our own exper i -

ments the recovery current was always considerably less than the charg-

ing current, although it decays at about the same rate.

The conductivity of qrartz depends on temperature according to an

exPonent ia l law 6:6os-EtRr

where the activation energy is about 23,000 cal. in the range 300-500o C.

l o l o l cu r ren li n d i t f us i on exoe r imen l

l o l o l cu r ren lin conducl ion erper imenl(o) chorging current(b) recovery cunenlot ler removing f ic ld

Frc. 2. Current-time curves for conduction and for difiusion in quartz.

There is no sudden change in the conductivity at the a-B transition

point, although the activation energy seems to increase somewhat above

this point. If a specimen is heated slowly to 500o and allowed to cool, the

conductivity measured during cooling is less (by a factor of 2 to 3) than

the conductivity measured during heating; but this may be due partly

to the time dependence of conductivity since the cooling curve is de-

termined after the heating curve. Curie (1889) observed however, that

the conductivity may be greatly reduced by preheating the specimen.

This efiect is presumably similar to annealing, insofar as preheating re-

- i lmc

646 J. VERHOOGEN

moves lattice defects due to strain or acquired otherwise during theprevious history of the crystal, and which contribute to the conductivity.

Several explanations have been suggested for the time-dependence ofaxial conductivity. Curie (1889) suggested that it results from waterfilling channels of molecular size parallel to the c-axis; the conductivityof this water would gradually decrease as more and more of the saltsdissolved in it are removed by electrolysis. Warburg and Tegetmeier(1888) suggested that the conductivity is due to ions (mostly Na) presentinitially as impurities in the quartz lattice; again these foreign ions wouldbe gradually eliminated. There is little doubt that part of the initial con-

oCo)o(L

o

c.oo

co,Qco()

o

- \

q u \ p t o f e

\\

x=-h x : o l = hFrc. 3. rnitial distribution of potential and concentration in difiusion experiments.

ductivity of quartz may be due to such impurities, for in one of our ex-periments with a quartz crystal from Arizona, noticeable amounts ofboron were collected on the cathode. But such "impurity" theories failto explain why quartz may regain its initial conductivity merely bystanding for a few days.

The same time-dependence of conductivity has been observed for otherionic crystals, such as NaCl, and Beran and Quittner (1930) pointed outthat both the rate of decay of the recovery current and the absence ofany measurable change in dielectric constant of a crystal during an ex-periment indicate that the time-dependence of conductivity, or the exist-ence of a recovery current cannot be due only to displacement currentsand dielectric polarization. Jofi6 (1929) and von Seelen (1924) havesucceeded in measuring the potential distribution across crystals ofquartz and halite respectively, and from the curvature of the potential

s o l f l o y e r

IONIC DIFFUSION AND ELECT:RICAL CONDUCTIVITY IN QUARTZ 647

curve showed the existence of space charges. In NaCl these are mostlypositive charges accumulated on the anode-side of the crystal, whereasin quartz Jofi6 found practically equivalent amounts of positive andnegative charges, the positive charges residing on the half of the crystalplate in contact with negative electrode. It is apparently the gradualbuilding up of these space charges which produces the decrease in thecharging current, whereas a redistribution and gradual disappearance ofthese same charges explains the recovery current. Joff6 explained theoccurrence of these space charges by assuming dissociation of quartz intounits of opposite sign which migrate in opposite directions in the electricfield. He was vague as to the nature of these units, although he mentionedbriefly (p.92) that"a deposit of silicon and liberations of oxygen from anormal crystal lattice may be observed." He succeeded in measuring theaverage density of these positive and negative space charges and foundthem to be roughly equal, with a value of 2X10s electronic charges percm.3 at t7" C.

MBcrr'tNrsu ol CoNoucuoN AND Sell-DrrlusroN RarB ol Oxvcom

It is now well known that electrical conduction in ionic, or nearly ionic,crystals results essentially from the.motion of lattice defects, althoughelectronic conduction may also be present in some cases (e.g., Ag2S, FeO).The lattice defects usually considered in this connection are either of theFrenkel or Schottky type. In the former type, ions are assumed to movefrom a lattice to an interstitial position, leaving "holes" behind them;other lattice ions may then be expected to move into these holes, whichtherefore migrate in a direction opposite to that of the ions themselves.In the Schottky type of defect it is assumed that ions of both signsmigrate out of the lattice towards its surface, leaving equivalent numbersof holes of both signs which migrate in opposite directions when anelectrical field is applied. If er is the energy per ion to form a Frenkeldefect, i.e., to move an ion from a lattice to an interstitial position, thenumber zr of such interstitial ions in a crystal in equilibrium at tempera-ture ? is (Mott, 1948, p. 2l)

np : 71/1.1f I{t 6esl2kr

where /y' is the total number of lattice positions and ly'1 is the number ofpossible interstitial positions. The factor ! in the exponential arises fromthe fact that displacement of one ion creates lzrlo defects (an interstitialion and a hole). The factor "y arises from the change in vibrational fre-quencies, and hence in free energy, of the atoms and ions surroundingthe interstitial ion or the hole; it may be greater or less than 1. Thenumber z6 of Schottky defects, on the other hand, is

648 J. VERHOOGEN

n6 : ^yNdcslkr

being the energy to produce such a defect. The factory in this case maybe of the order of 103- 104 (see also Barrer, 1941,p.294). Thus if electricalconduction results from motion of Frenkel defects, its temperature co-efficient would be of the type erpl-(en,l2er')/2*?] where er' is theactivation energy for migration of a defect, whereas for conduction bySchottky defects the corresponding term would be erp[-(esle s')/kT]es'being the activation energy for migration of a Schottky defect.

As mentioned above, Jofi6 finds that the average density of charges ofone sign in quartz at 17" is 2 X 109 electronic charges per cm.3 If thiscorresponds to the density of Frenkel defects for oxygen ions, the corre-sponding number of interstitial oxygen ions is I X 10e/cm3. The totalnumber /y' of oxygen lattice position in one cm.3 of qtartz is approxi-mately 5.2XI022, so that assuming "y:1, N:Nr, we f ind ee:36,400cal./mol. [The same value would obtain if we considered silicon defects.]The activation energy for conduction in quartz is about 23,000 cal. Hence

(36,400 + 2e' F) /2 : 23"000

and er'r25,600 cal./mole. On the other hand, if conduction occurs bySchot tky defects we f ind, for 7:10a, eg: !J ,$00 cal . , es ' : -800 cal .nwhich is impossible. These figures are, of course, quite approximate, sincean exact calculation would require evaluation of 1 for the quartz lattice,which is not available. The suggestion is, however, that conduction oc-curs by Frenkel defects, which is what one would expect in a lattice withhigh evaporation energy (hence high es) and relatively high compressi-bility (hence low e7).

It is impossible to decide from Jofi6's experiments whether oxygen, orsilicon, or both, move into interstitial positions, but it seems likely thatit is oxygen. Removing an oxygen requires breaking 2 O-Si bonds,whereas removing a silicon requires breaking 4. In other cases it is usuallythe ion with the lesser charge (in this case O- -) which moves most readily.

If the hypothesis is accepted that conduction of electricity parallel tothe c-axis in qttartz results from motion of oxygen ions in one directionand simultaneous displacement of holes (bearing an efiective positivechange) in the opposite direction, it becomes possible to compute theself-diffusion coefficient of oxygen from the conductivity itself by usingequation (12). Joff6's measurements show that the mobil ity of oxygenions and oxygen holes is about the samel hence it may be assumed thateither one carries half the total current. From the number of defects at17o, measured by Joff6, the number of such defects at 500o C. is foundto be 3.6X1018/cm3. Then

D o : 5 . 5 5 X 1 0 u X 2 "

X - - ! : -

IONIC DIFFUSION AND ELECTRICAL CONDLICTIVITY IN QUARTZ 649

Thus for a crystal with an initial conductivity (obtained by extrapolation

of the readings after 1 minute) of 2X10-e, the diffusion coefiicient of

oxygen is 3X10-rr cm.2/sec.

M,ncn,qlrrsu or DrlrusroN

Very simple experiments with a scale model of the quartzlattice and

marbles of various sizes indicate that the maximum radius of a rigid

sphere capable of moving freely through the lattice is about 0.57 A;

larger ions cannot difiuse without making room for themselves by distor-

tion of the lattice, or availing themselves of preexisting holes. If the

diffusing ion makes room for itself by crowding the lattice ions aside, the

energy requirements, and hence the difiusion rates, would be very sensi-

tive to ionic size. While this is true to some extent, the fact remains that

the large K+ ion diffuses more rapidly than smaller bivalent or trivalent

ones. Ilence, foreign ions may diffuse mainly through preexisting holes,

such as vacant oxygen lattice positions. Once a foreign ion, say sodium,

has occupied a hole, it moves farther under the influence of the electric

field or of a concentration gradient by exchanging position with neighbor-

ing oxygens which migrate in the opposite direction, thus accounting for

the fact that the diffusing ion apparently carries only a fraction of the

total current.Since the oxygen holes are originally distributed uniformly through the

crystal, they cannot all become occupied at once. The number of ad-

mitted ions and occupied holes will thus increase gradually with time,

Ieading to a gradual increase in conductivity, as observed. (See Fig. 2')

When all the holes are occupied, the conductivity reaches a steady value,

which is usually found to occur after a few days. The concentration of dif-

fusing ions then presumably reaches a maximum value of 3.6X1018/cm.3

at 500o C. The values of the diffusion coeficients listed above were ob-

tained by substituting this value and the corresponding value of the

steady-state conductivity in equation (12), checking the result by means

of (8) to determine the fraction of the total conductivity due to migration

of the diffusing ions.An oxygen hole, formed by removing a negative oxygen ion to an inter-

stitial position, carries an efiective positive charge; it would then tend to

repel an invading positive ion. This repulsion would increase with in-

creasing charge of the diffusing ion. This may explain why bivalent ions,

even of relatively small size (..g., Mg**) diffuse more slowly than larger

univalent ones (e.g., Na+, K+).How the crystal manages to maintain electric neutrality during the

process of diffusion is difficult to understand. The charge corresponding

with a concentration of 3.6 X 1018 foreign univalent ions per cm'3 would be

650 J. VERHOOGEN

sumcient to raise the potential of the capacitance formed by the quartzplate and its metallic electrodes to a value of about 1010 volts, whichobviously does not happen. rt is thus necessary to assume that additionalnegative charges are also present. These might consist, for instance, ofnegative oxygen ions in excess over the stoichiometric proportion. Suchis the case, for instance, for cuprous oxide which shows a marked increasein conductivity when heated under high partial pressure of oxygen. Elec-tric neutrality in quartz could also be restored by electrons from theexternal source which become trapped in the crystal structure.

It is interesting to compare experimental values of the difiusion coeff-cients with values predicted from the theory of absolute reaction rates.According to Eyring's theory [Glasstone, Laidler and Eyring, 1941] theconstant D6 in equation (5) should be given by the relation

Ds:6szf f "ou,o

where fr and h are the Boltzman's and Planck constants, respectively, },is the average length of a jump, i.e., the distance between successiveequilibrium positions of the difiusing particles, and AS is the entropy ofactivation. ), is not likely -to be less than the distance between two oxygensites, which is about 2.6 A. substituting this value and the experimentalvalue of Do at 500" C., the following values of the entropy of activationare found:

for Li AS : - 1.45Rfor Na AS : - 21Rf o r K A S : + 1 . 8 R .

As we have chosen a. minimum value for ),, the above values of AS arethe maxima that are consistent with the experimental values of Do. rt isinteresting to notice that the value for Na is such as could have beenpredicted on the simplest possible hypothesis, namely, that the entropyof activation corresponds merely to a redistribution of degrees of freedom,the difiusing sodium ion exchanging its three vibrational degrees of free-dom in the NaCl lattice for one translational degree of freedom in thedirection of difiusion. This would correspond to a negative entropychange of about 2R at 500" C. This negative value of the entropy ofactivation indicates also that difiusion of sodium and lithium does norinvolve any considerable distortion or disordering of the quartz lattice,in agreement with our hypothesis that diffusion occurs through preexist-ing holes. This, however, is no longer true for potassium: the positivevalue of the entropy of activation indicates in this case that some dis-ordering does occur, presumably because of the larger size of the po-tassium ion. This disordering may well occur when oxygen and potassium

IONIC DIFFUSION AND ELECTRICAL CONDUCTIVITY IN QUARTZ 651

ions exchange position, and is consistent with the higher value of theactivation energy observed in this case.

GBorocrc Iupr,rcerroNS

To give a more pictorial representation of diffusion rates in quartz,let us assume that potassium ions diffuse from a plane boundary into aqtartz crystal under the influence of a concentration gradient. The aver-age distance r from the boundary reached by potassium ions after time Iis then given by the relation

i2 : 2Dt

so that, at 500o C., the average distance reached in one million years(3X1013 sec.) is

E : (2 X2 X 10-10 X 3 X 10ts;trz^z 110 cm.

which is very small compared to the dimensions of a batholith, say. Itshould be remembered also that these difiusion coefficients apply to dif-fusion parallel to the c-axis; difiusion normal to this axis is apparentlymuch slower. Admittedly, in large masses of rocks, difiusion will occurpreferentially along crystal boundaries rather than through crystals; yetmetasomatic replacement does involve at times penetration into thegrains themselves. Present results would indicate very slow rates ofmetasomatic replacement.

Are the present results at all applicable to processes of metasomaticreplacement? The popular view of "clouds of ions" diffusing into rocksand replacing them is, of course, completely inadequate unless these

"clouds" happen to be of such composition as to remain electricallyneutrall otherwise the difiusion cloud, if it has any appreciable concen-tration, would involve impossibly large electric potentials. Thus the

"cloud" which allegedly transforms great masses oI qttartz into greatmasses of orthoclase must contain, in addition to potassium and alumi-num ions, a proper number of oxygen ions, as shown already by thesimple fact that the chemical formula 3 SiOr+K++AI+++ does not addup to orthoclase KAlSi3O3. The problem of the source of this oxygen doesnot seem to have been given proper attention. It might be solved byassuming removal of some of the sil icon, i.e., 4SiOz*K++Al+#: KAlSiaOs* Si++++' but once again the problem of electric potential inthe cloud would present itself. Furthermore, since diffusion rates inqtrartz seem to be more sensitive to electric charge than to ionic size,difiusion of Si++++, which would be the rate-determining step of the re-action, must be extremely slow. Obviously, difiusion in the solid statedoes not have the answer to all our oroblems.

J. VERHOOGEN

APPENDIX

EquerroNs ol DrrrusroN rN AN Er,Bcrnrcar Frrr,r

Let us suppose that a positive ion with valence a difiuses from a layerof salt of thickness h into a qvartz plate of equal thickness under theinfluence of an electrical f ield (Fig.3). At t ime l:0, the concentrationsatisfies the conditions

c : c o f o t - h 1 r 1 0

c : 0 f o r 0 1 r 1 h

and may thus be represented by the function

co 2cn7 rx | 3nx I 5rr Ic : - - - l s i n . * - s i n . - f : s i n - + " ' l . ( 1 )2 n L h 3 h 5 h I

At t ime l:0, we apply a potential lVo at x:-h, the potential atr : lh being taken as zero. Hence, at t :0 , AV/0x:- (Vo/2h) ; andV :Vo fo r x : - h , V :0 f o r r :Z a t a l l t imes .

1. Let us assume first that only the positive ion moves (no negativecarriers). The equations to be satisfied for o1rlh are

(a) The equation of continuity

d i v s : - t ( 2 \dt

where .S, the flux in the r-direction, is

S : a c ( 3 )

the velocity v being given by the relation

, : - , r n ! . ( 4 )dfi

(b) Poisson's equationA2V 4r

d x c z : - - e z c ( 5 )

where e is the dielectric constant of quartz.Eliminating S, 7, and u leads to the equation

n*Xll"'*"-^f:-u# (6)where

A : L ( d ! \ B : . .

4rez \ 0x ,1":o' 4re2z2B

It appears simpler to eliminate c rather than V. The difierential equa-tion for the potential 7 is then

aV aBV , / arv \ " 1 aava- d.^" + ( * /:

-rrB-d*rdt' G)

IONIC DIFFUSION AND ELECTRICAL CONDUCTIVITY IN OAARTZ 653

Substituting Y:0V/6x, ezB:l/a the general solution of (7) is foundto be

where F is any arbitrary function of the variables indicated in paren-theses, f is an arbitrary function of I only, and rtt -- [Qd.t This function Fmust satisfy the boundary conditions for 7 given above, and in addition,

-L(!L\ must satisfy (1).4rze \ 6x2 ,l eo

2. It seems hardly worth looking for a suitable form of (8), since con-dition (5) is not likely to represent conditions accurately. Equation (5)implies indeed an unbalanced concentration of space charges, the crystalacquiring a total charge t! ezcrdx which would raise its potential to avery high value. We will certainly be closer to experimental facts if weassume a simultaneous motion of negative charges in the opposite direc-tion at a rate sufficient to maintain electric neutrality. Let us then as-sume first that there is a detailed balancing of charges everywhere, sothat d2V/0x2:0 and

6 V _ _ V o61 2h

at all times. The equations to be satisfied (2, 3, 4) become very simply

6c 6c (e)* a r - a r

where

p : (ezBVo)/2h. (10)

The solution of (9), subject to boundary condition (1), is

cn 2cnf r | 3 r - ' l

c : t

- - L ' . h @ - a ) *

; s i n 7 ( x - v t ) + . . .

1 .

To determine B, or D, we consider what happens at x:h. The concentra-tion there is

. o 2 c n 7 r p r 1 . , 4 + . . . 1 ( 1 1 ), o : t - - L t " , - r J s ' n - ,

I

and the total number Q of particles flowing across the plate boundarywith cross-section s in time d is

' 0

Q: Josua"d ' t ' ( r2)

5"1 p:p|/h,v:sh, u being the volume of the crystal. Substituting (11)into (12) and integrating, we obtain

n[(r+*), (, ***J*j)7:, (8)

J. VERHOOGEN

*: +- +lt*1*f * . - (.* *, * !cos3nr. )]The right-hand member of this equation is given approximately, for

large values oI n, by the expression t(n- 1), so that

n : 9 a 1 .C0't)

Hence, iI Q, cn and u are known, n may be computed. Since n:y?fh,p.:ezBVo/2h, we obtain finally

D : 2 n k r ( 1 3 )ezV o0

from which D may be calculated. A difficulty arises, however, in the de-termination ol cs. cs is not necessarily the total initial concentration ofpositive ions in the salt; i t is the concentration of free ions, i.e., of ionswhich are not so tightly held in the lattice that they cannot nIoV€. domight be taken, for instance, as the number of metallic ions which wouldparticipate in electrical conduction in the salt itself. As this number can-not be evaluated accurately at 500o C. for any of the salts used in theseexperiments, there is nothing to gain by using the exact relation (13)rather than the approximate method (also assuming 6Vf lcc:constant)outlined previously.

3. Finally, we might attempt a more exact solution by consideringthat there is not necessarily a detailed balancing of charges at everypoint in the crystal although its total charge should remain zero. Thisimplies two different carriers (the positive diffusing ion and a negativeone) with valencies 21, 22?,.nd concentratiorrs cy c2, respectively. Poisson'seouation is then

d2V 4r ,- € l h C t - S c b J

6x2 6

f k d 2 VI - d ' x : o

r o o r '

expressing neutrality as a whole. The equations of continuity for eachcarrier are

- 0V ]ct A2V 6cr€ Z t B t - - l c t e z t B ,- 6 x 0 x - - 6 x 2 A l

_ 6V 6cz _ A2V 6czez rB r - : * c , ez ,B ,- - d r

0 * - - - 3 x 2

d l

Let us consider the steady state in which the current remains constant,and in which presumably

subiect to the condit ion

(14)

(1s)

(16)

IONIC DIFFUSION AND ELECTRICAL CONDUCTIVITY IN QUARTZ 655

This state is usually reached after a few days. Eliminating 0V/0r inequations (16), we get

0c2/0x 62V d2V- " a r l a r - a * , + c z * : o

and since we assume 62V/0x2#0 locallv

which together with (14) and (15) leads to zp1:22c2,02V/3r2:0 every-where. This is contrary to our assumption, and we must conclude there-from that either there is detailed balancing everywhere, or that a steadystate cannot be reached. As it appears experimentally that a very closeapproach to a steady state may be reached after a few days, 02Vf0x2 cannever be very large, and may therefore be neglected. We revert thus tothe previous case.

Rnrnnrxcrs

Banr.on, R. M. (1941), Difiusion in and through solids. Cambridge University Press.

Bnnl.N, O., ,lNo QurrrNnn, F. (1930), Die Feldstiirkeabhiingigkeit von Gegenspannungund wahren Leitvermiigen in fonenkristallen: Zeits. Jiir Physih,64, 760-776.

Cunrt, P. (1889), Recherches sur la conductibilit6 des corps cristallis6s: Ann. Chimie

Phys., ser. 6, 18,202-255.Gressroxe, S., Lerllnn, K. J., aro Ei'trnc, H. (1941), The theory of rate processes.

McGraw-Hill, New York.

Jolrf, A. F. (1928), The Physics of Crystals, McGraw-Ilill Book Co., New York.Mom, N. F., aNo Gunxrv, R. W. (1948), Electronic Processes in Ionic Crystals, Oxford,

Clarendon Press.Rocnow, E. G. (1938), Electrical conduction in quartz, periclase and corundum at low

field strengths : J ourn. A p pl. P hysics, 9, 664469.Sosuew, R. B. (1927), The Properties of Silica, Chemical Catalog Co., New York.Turxnn, F. J., ,tNo VrnrroocEN, J. (1951), Igneous and Metamorphic Petrology, McGraw-

Hill, New York.vor Sner.nN, D. (1924), Ueber die elektrische Leitfehigkeit des Steinsalzkristalles: Zei'ts.

fiir P h ysi k, 29, 125-140.Wannrrtc, E., .alvo Tncrtmrrn, F. (1888), Ueber die elektrolytische Leitung der Berg-

krystalls: Ann. d. Physik u. Chemie, 35, 455-467 .

Monuscribt receiued. Not. 30, 1951

6c' 0c,- . - : - : 0 .a, at

L OC2 I dcl

cz 0x cr Ax