Solution and Diffusion of Helium and Neon in Tridymite and Cristobalite

BY R. M. BARRER AND D. E. W. VAUGHAN

Physical Chemistry Laboratories, Chemistry Dept., Imperial College, London, S.W.7

Received 8th May, 1967

The solution and trapping of helium and neon has been studied in cristobalite in the range 293- 857°K and in tridymite between 293 and 838°K. These temperatures cover regions in which (x- and P-cristobalite and a-, PI- and P2-tridymite are found. The solubilities are considerable, those in tridymite being close to solubilities in silica glass and those in cristobalite rather less. Heats of solution were evaluated from isotherms and had the same signs as those in silica glass. The results were discussed in terms of an oscillator model for dissolved atoms. For some of the systems approxi- mate diffusion coefficients were obtained which could be represented by the Arrhenius equation :

He in a-tridymite (236-336°K) : D = 7.7 x exp - 12000/RT cm2 sec-', He in a-cristobalite (244-335°K) : D = 2-0 x exp - 13800/RT cm2 sec-I,

Ne in a-tridymite (336-366°K) : D = 6.7 x exp -22500/RT cm2 sec-', Ne in P,-tridymite (418-474°K) : D = 9.9 x lo-' exp -24100/RT cmz sec-', Ne in p2-tridymite (513-577°K) : D = 4.5 x exp - 19100/RT cm2 sec-I.

Energies of activation are about twice their values in silica glass and diffusion coefficients are smaller.

There has been interest in the solution and diffusion of inert and of permanent gases in silica g l a~s , l -~ and also of helium and neon in germanium and silicon crystals,6 which provide relatively open diamond-type lattices. Helium loss from and diffusion in inaterials have likewise been inve~ t iga t ed ,~ '~~ and there have been a few studies of trapping of various gases in mineral structures, such as those of the zeolites l1 or of cordierite.

Two crystalline forms of silica which have relatively open structures are tridymite and cristobalite, which therefore might serve as traps for dissolved helium or neon. As part of a programme on encapsulation of gases in minerals, these two forms of silica have been investigated as solvent and diffusion media for inert gases. They have rigid three-dimensional networks of linked SiO, tetrahedra permeated by a channel system in which the windows leading from one interstice to others have free dimensions of 2.2-2.6 A. Although tridymite and cristobalite are easy to prepare and have good thermal stability and large potential capacity for sorption they do not appear to have been examined from this viewpoint. Moreover, the study of solution and diffusion in these crystals could help to clarify their relationships with silica glass. The similar densities of -2.2, 2.28 and 2.32 g C M - ~ for silica glass, tridymite and cristobalite, respectively, suggest that they have comparable free volumes.

The complex structures of the tridymites and cristobalites have been reviewed.l 3 9 l4

Our investigation covered the temperature range in which three tridymite and two cristobalite polymorphs could be involved, according to the inversion temperatures below :

a-tridymite + P,-tridymite + P,-tridymite

a-cristobalite + P-cristobalite. 2275

-117°C -163°C

- 2 50-27OOC

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2276 SOLUTION A N D D I F F U S I O N IN TRIDYMITE

These inversion temperatures can be modified by dissolved impurities, including the inert gases themselves.

EXPERIMENTAL

MATERIALS

Helium and neon were taken from cylinders with output pressures of 2000 and 1000 lb in.-2 respectively. The purity for helium was 99.99 %, and for neon >98 %, the main impurity being helium.

Cristobalite and tridymite were prepared from silicas. Syton 2X silica sol was pre- cipitated and the gel dried at 120°C for a day. It was then heated at 1490°C for 18 h to yield cristobalite and at 1100°C to give tridymite. Samples of pure cristobalite were also made from powdered Spectrosil and also from a pure natural quartz (99.95 % Si02) by heating for 18 h at 1490°C. Attempts to make tridymite from these materials by heating them for 24 h at 1100°C were unsuccessful, the only crystalline product being cristobalite. Analysis of the cristobalite and tridymite used for the experiments, and an analysis of

TABLE 1 . wt %of

polymorph starting material Fe A1 Na

tridymite Syton 2X 0.11 0.19 0.73 cris to bali t e Syton 2X 0.12 0.21 0.83 cristobalite Spectrosil 0.035 0-013 0

cristobalite from Spectrosil are given in table 1. In the original Spectrosil the only impurity reported by the makers is 0.1 wt % of OH. The Spectrosil was, however, ground to pass 120 mesh, and impurity could have been picked up at this stage.

The Na-impurity in the crystals formed from Syton 2X represents about 5 % filling of the interstices in tridymite or cristobalite. Attempts to prepare tridymite from Spectrosil in presence of a high pressure of neon were unsuccessful. It was hoped that dissolved neon would act as a structural support for the growth of the tridymite lattice. The conditions and products are given in table 2.

TABLE 2.-cRYSTALLIZATION OF SPECTROSIL IN PRESENCE OF NEON

temp. "C Ne press. (atm) (time h) spectrosil final product

934f2 15 falling to 7 24 120-mesh powder Spectrosil(-90 %)+ cristobalite

cr is t o bali te 1267f2 20 falling to 7 24 1 mmrod Spectrosilf outer layer of

PRESSURE LINE

The pressure line comprised a motorized pumping unit and oil reservoir of an Aminco hydraulic HP pump, coupled to one side of a Sprague 200 cubic inches intensifier unit (maximum output pressure 10,OOO lb. in.-2). High-pressure gas cylinders were suitably connected to this unit. The intensifier was connected to a bank of eight autoclaves, each with independent pressure and temperature controls. Pressures were measured with Bourdon tube gauges (up to 20,000 lb. in. -*), and temperatures with chromel-alumel thermocouples. Autoclave temperatures were controlled using a Pt-resistance thermometer as part of a bridge.15 The pressure line was made of stainless steel high pressure tubing, the different sections being isolated when required with needle valves. The autoclaves were provided with silver washers for better sealing, and at 60°C could be operated at a pressure of about 8000 lb. in.-2

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R. M . BARRER A N D D. E. W . V A U G H A N 2277 METHOD O F E X P E R I M E N T A T I O N

Weighed samples of cristobalite or tridymite of between 1 and 2 g were placed in capsules covered with fine wire baffles. They were outgassed overnight at 750°C and were then introduced into the autoclaves, where they were further outgassed for an hour at the desired experimental temperature. The autoclaves containing the crystals were then exposed at this temperature to appropriate pressures of gas for suitable periods. The length of time needed to establish the solution equilibrium was established by trial and differed according to circumstances (crystal, gas, and temperature). Some experiments lasted several weeks.

The equilibrium period was terminated by quickly lowering the furnaces on their guide rails and chilling the autoclaves (still under pressure) with ice water. This stage, in the experiments with helium, was followed at once by further cooling in liquid nitrogen, the gas pressures in the autoclaves being simultaneously released. The crystal samples in their capsules were then transferred to silica glass tubes cooled in liquid nitrogen and joined to an all glass apparatus. This apparatus comprised four units, each provided with mano- meters and calibrated for measuring volumes of evolved gases, so that four crystal samples could be degassed simultaneously. Before measuring the amount of dissolved helium or neon the silica tubes and their contents, still at liquid-nitrogen temperatures, were outgassed for 3 min to remove the ambient air, introduced during transfer from the high pressure to the low pressure system. The silica tubes and their contents were raised to 750°C and held at this temperature for 12 h. The evolved gas was passed through a U-tube cooled in liquid air which condensed traces of water or other vapours collected during the transfer from the high- to the low-pressure system. A cathetometer was used to read the final manometer pressures.

D E S O R P T I O N K I N E T I C S

Desorption kinetics were studied, after the crystals had been equilibrated with helium or neon, quenched and transferred to the desorption apparatus as described previously. The crystals vxre allowed to warm up to the experimental temperature as rapidly as possible, and were held at this temperature. During the warming process, gas is being evolved at an increasing rate, so that curves of the amount desorbed against time3 show an initial period of acceleration. The formal treatment of diffusion, when the diffusion coefficient, D, thus becomes a function of time, is given in appendix 1, which provides the information needed to interpret the kinetic runs to give D.

This interpretation requires a knowledge of the surface area of the crystal powder. The surface area was measured by the projected area method of Kendricks,16 Tooley and Parmelee,l' and Pidgeon and Dodds.l* The method is described in a British Standards Monograph. To carry out the area measurements a Vickers Projection microscope was used with a 4 mm (0.85 N.A.) achromatic objective lens, and x 6 compensating projecting lens. A graticule was placed in the projecting lens, and, after adjustments, the system was calibrated by photographing the graticule superimposed upon a calibrated 0.1 mm grating in position on the stage. The particles in the photographs were seen in projection, and the projected areas determined. Then the geometrical area of the particles, in absence of re- entrant angles are related to the total area by mean surface area per particle = (4x total projected area)/(number of particles). This method, which ignores fine-scale surface roughness, will be somewhat less than an area based upon adsorption measurements. From it, assuming spherical particles, the average particle radius may be derived. For tridymite this mean radius was 3.9 x cm, and the surface area was 7.7 x lo3 cm2 per cm3. Cristobalite gave values of radius and area respectively equal to 4.1 x loe4 cm and 7.2 x lo3 cm2 per cm3. If the particles were all assumed to be cubes the above areas should be multiplied by 1.38, but in the present work they have been treated as spheres.

RESULTS AND DISCUSSION SOLUBILITY

ISOTHERMS A N D HEATS

The amounts of helium or neon dissolved are shown as functions of pressure in fig. 1 and 2. The time required to reach equilibrium between neon and the low-

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2278 S O L U T I O N A N D D I F F U S I O N I N T R I D Y M I T E

temperature polymorphs of tridymite and cristobalite was > 50 days, and evaluation of complete isotherms was accordingly abandoned. Nevertheless a small number of very long experiments with these materials are included to indicate the approxi- mate sorptions.

Although both tridymite and cristobalite have structures of comparable openness, and similar total numbers of interstices per g, each potentially able to accommodate helium or neon, the results in fig. 1 and 2 show that the solubility of these two gases in tridymite is considerably greater than it is in cristobalite, at comparable temperatures and pressures. Also, the actual amounts of gas dissolved are much below the

I

pressure X lb in.-2 FIG. 1 .-Tridymite sorption isotherms.

(a) a-, P1-tridymite-helium ; (b) P2-tridymite-helium ; (c) &-tridymite-neon.

theoretical saturation values, which, assuming one dissolved He or Ne atom per interstice would be 187 cm3 at s.t.p. per g for tridymite and cristobalite assuming two sitcs per unit cell for tridymites, and four per unit cell for cristobalites. This indicates a small affinity, associated with a small heat of solution and a decrease in thermal entropy when the gas molecules are trapped in the crystals. The isotherms are all approximately in the Henry’s law range, which further indicates that the fraction of the total available interstices occupied is small. Finally, the amount of gas dissolved at constant pressure tends to decrease as the temperature is raised, so that the gases dissolve exothermally in the crystals.

The Clapeyron-Clausius equation in the form

(apIaT), = -AH/TAV (1)

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R . M . BARRER A N D D. E . W . V A U G H A N 2279

relates the molar enthalpy AH of solution to the rate of change of equilibrium pressure P with temperature T and the corresponding change AV in volume accompanying the solution process. Since solution is interstitial in character, AV = - Vg, where Vg is the molar volume of the gas in the gas phase. The subscript a in eqn. (1) denotes that the amount sorbed is to be kept constant as the system moves along the equilibrium P-T line. Because the pressure is high the virial equation is required, i.e.,

PVg = RT(I+BP+CP2+. . .).

pressure X lb in.-2

FIG. 2.-Cristobalite sorption isotherms. (a) cc-cristobalite-helium ; (b) fl-cristobalite-helium ; (b) fl-cristobalite-neon.

Eqn. (1) gives, on substituting for A V and integrating,

(3)

wherefdenotes fugacity and where B, c are mean values of B and C over the tempera- ture interval (T1--T2) for which the corresponding pressure interval is (Pl-P2). Virial coefficient values from ref. (20), (21), (22) were used, interpolated where necessary. Mean values of AH were calculated from each isotherm as follows. If the isotherms had respectively a and m points, n and m linear slopes were obtained, one for each point, on the basis of the applicability of Henry's law for isotherms of amount sorbed against5 and each slope on one isotherm was combined with all slopes for the other, giving nrn slope ratios and so nm calculations of AH. These were averaged to give

AH,,,,,, = XAH/nm. (4)

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2280 SOLUTION AND D I F F U S I O N IN TRIDYMITE

A t test was applied to these results to obtain a 95 % confidence limit arising from random but not from systematic errors. The rounded results are given in table 3. The energies AE of sorption are given by

where Vg is given by eqn. (2), and are therefore less negative than AH by a term of the order of magnitude RT. Accordingly some of the energies of sorption are endothermic.

AH = AE+PAV = AE-PV,, ( 5 )

TABLE 3.-HEATS OF SOLUTION OF He AND Ne IN CAL MOLE-’

helium neon 95 % 95 %

-AH confidence limit - A H confidence limit

cc-t ridymit e 160 f 120 I - P,-tridymite 8 10 A400 - I

&tridymite 1230 & 560 1990 f 3 5 0 a-cristobali te 640 f120 - - p-crist obali te 2340 f610 2630 k510

EQUILIBRIUM CONSTANTS

The solution process is interstitial in character so that if each interstice is regarded as a site, Langmuir’s isotherm equation should be obeyed :

where the fugacity and pressure are related by K = O/f(l -O), (6)

(7) In (f/P) = BP+ C P 2 + . . . .

TABLE 4.-vALUES OF K X lo7 (Cm Hg)-’ FOR He AND Ne IN CRYSTALLINE SILICAS

gas polymorph temp. (“K) Kexptx lo7

helium a-tridymite 293 5.1 361 5.0

P,-tridymite 410 4.6 433 4.2 533 3-7

P,-tridymite 670 4.6 777 4-1

helium a-cristobalite 293 1-2 416 0.77 537 0.6, 61 2 0.59

p-crist o balite 773 0.71 857 0.6’

P,-tridymite 675 3.9

/I-cristobalite 673 1.2

766 3.3 neon 838 2.9

767 0.83 836 0.70

Since the non-ideality correction is not large relative to other uncertainties, eqn. (6) may be approximated by

K = e/[p(i - e)(i + 13p)l. (8)

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R . M . BARRER AND D . E . W . V A U G H A N 228 1 Equilibrium constants evaluated from eqn. (8) are given in table 4, column 4, assuming that saturation represents one rare gas atom per interstice. This means that at saturation there would in tridymite, e.g., be as many helium or neon atoms in a given number of unit cells as there are Na+ or K+ ions in equilvalent amounts of the " stuffed tridymite " structures nepheline or kalsilite.

D I F F U S I O N

The kinetics of desorption, under conditions near to those described above and in the appendix, are shown in typical instances in fig. 3. If Q, and Q, denote respectively the quantities desorbed at time t and at infinite time, the curves of Q f / Q , against Jt are sigmoid. The period of acceleration corresponding with rising temperature is often followed, after the steady temperature has been reached,

0

FIG. 3.-Desorption kinetics. (a) a-tridymite-helium ; (b) a-cristobalite-helium ; (c ) PI-tridyrnite-neon ; ( d ) P2-tridymite-neon.

by an approximately linear section, and then by a period of deceleration as Qf approaches its final value. Helium escaped rapidly from the higher temperature polymorphs of tridymite and cristobalite, making it difficult to obtain reliable rate curves. Neon was well trapped by a-cristobalite but escaped readily from the p-form. In the latter case the small amount of the dissolved gas and the rapid release again prevented reliable kinetic measurements from being obtained. Neon was also firmly trapped in a-tridymite whereas the P-polymorphs released the gas readily.

From the slopes of the linear sections of the curves in fig. 3 the diffusion coefficients may be obtained in the manner indicated in the appendix, i.e.,

d( Qt / Q ,) /dt = 4S2D(T,) In V 2 3 (9)

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2252 SOLUTION AND D I F F U S I O N I N TRIDYMITE

13-

where S and V are the surface area and volume of the crystallites and D(T,) is the value of D when the temperature has reached its final value. Diffusion coefficients thus obtained are given in table 5, together with values for silica glass 2* for comparison.

v z

s e ,

TABLE 5.-DIFFUSION COEFFICIENTS OF He and Ne IN CRYSTALLM SILICAS AND IN SILICA GLASS

13

HELIUM u-tridymi te a-cristobalite silica glass

temp., OK D, cmz sec-1 temp., OK D, cm2 sec-1 temp., OK D, cmz sec-1

296 1 ~ 1 O - l ~ 296 1.3 x 297 2 . 4 ~ lo-' 273 2~ 1043 273 1 . 6 ~

336 1 . 2 ~ lo-'' 335 1 . 7 ~ 351 1x10-7

235 3 . 9 ~ 10-15 244 8 . 7 ~ 10-15

7'

NEON a-tridymite /71-tridymite Bz-tridymite silica glass 2. 3

temp., O K D, m z sec-1 temp., OK D, c m 2 sec-1 temp., OK D , cmz sec-1 temp., OK D, cmz sec-*

366 2 . 2 ~ 10-15 474 7.1 x 577 3 x 10-l1 577 4 . 5 ~ 10-7 354 8 . 7 ~ 444 1-1x10-12 543 1.1 x lo-'' 513 6 * 1 ~ 1 0 - ~ 336 1 . 2 ~ 418 2.3 x 513 3 . 9 ~ 10-l2 418 2x

336 5.9 x lo-''

13

14 a -tridymite-helium

I I I I I

t ridymites -neon '. 16 L-L- 2 0 2.5 30 3.5 4.0

1 O~/TOK FIG. 4.-Plots of diffusion coefficient ( D ) as a function of temperature.

Slopes of plots of log D against 1/T (fig. 4) were approximately linear so that estimates were made of the Arrhenius energies of activation E from D = Do exp (-EIRT). These are given in table 6. Again, for comparison purposes, values of E for diffusion in silica glass are included. Diffusion coefficients

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R . M . BARRER A N D D . E . W . V A U G H A N 2283

are considerably larger in glass than in either tridymite or cristobalite (table 5), and energy barriers for diffusion are nearly twice as high in the crystalline silicas as in the glass. On the other hand, energy barriers and diffusion coefficients are similar for helium in a-tridymite and a-cristobalite.

TABLE 6.-ACTIVATION ENERGIES E FOR DIFFUSION IN CRYSTALLLNE SILICAS AND IN SILICA GLASS (kcal/mole)

tridymite cristobalite silica glass ?,

13.8 5-6 to 6.6 gas U 81 82 U

- - helium 12 neon 22.5 24.1 19.1 - 11.3

DISCUSSION

Comparisons of interest include relations between solution and diffusion in crystalline silicas and silica glass, between experimental and theoretical heats of solution, and between experimental and calculated equilibrium constants.

CRYSTALLINE SILICAS A N D SILICA GLASS

The solubilities of helium and neon in tridymite, cristobalite, and also in silica glass,2* are shown as functions of temperature in fig. 5. The solubilities, in atoms/cm3 atm, are much closer between tridymite and glass than between cristo- balite and glass. Previous hypotheses on the glassy state assume that it is the cristobalite structure which approximates the more nearly to that of glass, but at least as far as the solubility measurements are concerned the reverse is the case.

.z 4 tr idynt les

* cristobalites --- silica glass

4 tr idynt les

* cristobalites --- silica glass

D

lo141 ,I 2bo 3bo 4bu 5b0 ti,',

temperature, "C

Neon I

A 400 500 600

FIG. 5.-Solubilities of helium and neon in silica polymorphs.

The random network of silica glass should contain, inter alia, 5, 6-, 7- and 8- membered rings of SiO, tetrahedra as units in the structure and the diffusion path of helium and neon in silica glass must take them through those rings which are wide enough to permit transit (i.e., 6-, 7- and 8-rings). However, diffusion paths involving as many as possible of the widest rings will be statistically favoured and so will contribute most to the transport. In cristobalite and tridymite the rings defining the d.iffusion paths are all 6-membered ones so that the average energy barrier in these crystals would be expected to exceed that in silica glass where the larger, more

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2284 SOLUTION A N D D I F F U S I O N I N TRIDYMITE

open rings also occur. We have also made some measurements with quartz powder (density 2-65), and although very small amounts of helium were dissolved, we were unable to measure diffusion or solution equilibria of helium or neon. Here the interstices are probably too small to accommodate the rare gas and the energy barriers for migration correspondingly high. If the diffusion coefficients are expressed in terms of the Arrhenius equation [D = Do exp (- E/RT)] their values are as follows :

He, a-tridymite (235-335°K) : D = 7.7 x exp - 12,000/RTcm2 sec-l, He, a-cristobalite (244-335°K) : D = 2.0 x exp - 13,800/RT cm2 sec-',

Ne, a-tridymite (336-366°K) : D = 6.7 x exp -22,50O/RT cm2 sec-l, Ne, p,-tridymite (418-474°K) : D = 9.9 x lo-' exp -24,10O/RT cm2 sec-l, Ne, P,-tridymite (513-577°K) : D = 4.5 x exp - 19,100/RT cm2 sec-l.

These expressions are approximate. Uncertainties in E strongly influence the magnitudes of the pre-exponential constants, but these are often comparable in order of magnitude with values of Do for diffusion of He and Ne in fused quartz 2* which varied from 2.1 to 7.4 x cm2 sec-' .

C A L C U L A T E D B I N D I N G ENERGIES OF DISSOLVED G A S

The Lennard-Jones 12 : 6 potential was used, with the lattice positions of oxygen in a-cristobalite 23* 24 P-crist~balite,~~* 26 and Pi-tridy~nite,~~ to calculate the energy of binding of inert gas molecules within the interstices of these crystals. The dispersion energy 4D and repulsion energy & are then,

where ri denotes the distance between the centres of the rare gas atom and the ith oxygen atom. ro is the distance between these centres for an isolated rare gas- oxygen atom pair when (a+/&) = 0. A is the dispersion energy constant, which has previously been evaluated for rare gas + oxygen pairs,28 according to the approxi- mations of London,29 Slater and Kirkwood 30 and Kirkwood and M~ller.~' , 32

In the present work, these values have been used taking the polarizability a as 1.65A3 for framework oxygens.28 The radius of oxygen was taken as 1-35A, and those of rare gases, except helium, are given elsewhere.33 The summations required in eqn. (10) were made using an Atlas computer.

For all crystalline silicas referred to above the energy contours were evaluated for movement of the rare gas atom within a cavity and in transit from one cavity, through a ring of six tetrahedra (a 6-ring) to an adjacent cavity. In a-cristobalite the sums in eqn. (10) were evaluated for the 38 nearest oxygens. The first axis considered was the line running from Si (-2, -j, -25) through the centre (-25, '5, -625) of a 6-ring and thence to the Si (*3, 1.3, 1) (fig. 6a). Due to the configuration of the lattice these two paths though joined are not collinear. The energy contours were also found along each of two new axes normal to, and passing through the energy minimum on the first axis. The two additional axes were chosen by trial calculation to give the greatest and the least vibration frequencies.

For /?-cristobalite 4 was evaluated for 38 oxygen atoms along the [ l l l ] axis passing through (OOO) and (1 11). It was also found for two additional axes normal to the first chosen just as for a-cristobalite (fig. 6b). Finally, in /?,-tridymite 4 was calculated for 57 oxygen atoms along [OOl], and along two other axes normal to [OOl] and chosenjust as for the cristobalites (fig. 6c).

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R . M . BARRER A N D D . E . W . V A U G H A N 2285

[ii)

FIG. 6.-Views of cavities with argon atoms (diam. 3.8 8,) on sorption sites.

(a) P2-tridymite : (i) [Ool] axis vertical in the plane of the paper, (ii) [Ool] axis normal to the plane of the paper. (b) P-cristobalite: (i) [ l l l] axis vertical in the plane of the paper, (ii) 11111 axis normal to the

plane of the paper.

(ii)

I I I 1

2 3 4 5 6 5 6 7 6 8 10

FIG. 7.-Potential energy contours. V, helium; m, neon; 0, argon; 0, krypton (using the London A constant).

(a) a-cristobalite contour following a line from (-2, *z, -25) through (-25, -5, -625) to (-3, 1-3, 1): abscissa in 8, units from (-2, -3, -25). (b) fkristobalite contour parallel to the [l 1 11 axis ; abscissa in A units from (OOO). (c) P,-tridymite contour parallel to c axis: abscissa in A units from (000). (d) P2-tridymite contour along an axis from the cavity centre through a six-ring side window : abscissa

in 8, units from (0, 0, -25).

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2286 S O L U T I O N A N D D I F F U S I O N I N TRIDYMITE

Some of the energy contours are shown in fig. 7a-c for movements of dissolved atoms along the first axes. In these calculations the positions of lattice atoms are necessarily assumed to be fixed, whereas thermal vibrations must cause perturbations of the curves from moment to moment. According to the energy contours helium could move freely along the [OOl] axis. However, thermal vibrations and collisions would soon terminate such flights. Moreover, the centre of the helium atom must move exactly along the [OOl] axis to be transported with such a low energy of activation, and the chance of this is very small. Glancing passages, in which helium strikes peripheral oxygens of the 6-ring, and in which considerable activation energies may thus be required, will be much more frequent. It is still more probable that the helium would be reflected back from oxygen atoms in its cavity and perform many oscillations before escaping to another cavity. For the reasons given above, it is most unlikely that the energy contours give any true picture of the activation energy for diffusion. They may, however, give reasonable approximations to the energy minima relative to the least energy of the gaseous molecules.

The energy contours for Ne in P,-tridymite also give an activation energy (- 3 kcal mole-') much below the experimental value ( N 19.1 kcal mole-') for movement precisely along the [Ool] axis. Clearly the arguments presented in the previous paragraph apply here, and to all such calculated energy barriers. A series of additional paths leads off from any [OOl] axis, at appropriate points along it for which the calculated energy barriers are likewise small (- 0.5 and 4.5 kcal mole-'). These paths lead, e.g., from (0, 0, 025) along an axis through (-5, a%, -25) (fig. 7d).

The trapped gas atoms can vibrate with many possible frequencies in any cavity, according to the different energy contours associated with movement in different directions. Some specimen calculations were made by a procedure already described 33 of frequencies vl, v2, v3 of the rare gas atom when it moves along any one of the three axes described, except where the barriers were less than RT in height. These calculations ignore effects of lattice vibrations upon the energy contours. They are, however, of interest in connection with the oscillator model for the dissolved atoms. Their values, and those of &, are given in table 7, using both the London and Slater-Kirkwood approximations for the dispersion energy constant, A.

OSCILLATOR MODEL FOR DISSOLVED GASES

If the dissolved atoms are considered as a set of Einstein oscillators of frequency v, then allowing for zero-point energy the equilibrium constant, K of Langmuir's isotherm in cm-1 (see table 4) is given by

S m 1 h3 ( 2 s i n h g ) - 3 exp-RT.

K = - - koT (2nmkT)+

Here ko is the value of the Boltzmann constant k when p in p = ckT is expressed in cm Hg and c in molecules cm3 and $+,, is the value of 4 (eqn. (10)) at the minimum in the energy well. Similarly the heat of solution is given by

5RT 3Rhv hv AH = +,,,--+- - coth -.

2 2 k 2kT Accordingly we have used the experimental values of K to calculate mean frequencies, using & evaluated in table 7 according to the London and Slater-Kirkwood approxi- mations for the dispersion energy. The frequencies calculated for each K and then averaged for all K are given in table 8. They were used in eqn. (12) to calculate AH, which is also tabulated.

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R . M . BARRER A N D D . E . W. V A U G H A N 2287

The frequencies are higher in cristobalite than in tridymite, reflecting a smaller thermal entropy in accordance with the smaller equilibrium constants. The calculated heats are not significantly different, where the comparison can be made, between tridymite and cristobalite. The mean frequencies in table 8 are in general of the

TABLE 7.-VALUES OF # m AND OF V IN CRYSTALLINE SILICAS

- dm, cal mole-1 vibration frequency, v (sec-1 X 10-12)

London Slatcr-Kirkwood London Slater-Kirkwood v1 y2 v j v1 v2 v3

He P,-tridymite 2060 3190 - 6.3 6.3 - 9-0 9.0 a-cristo balit e 2040 3160 4.0 3-7 1.9 4.5 4.4 1.9 p-crist o balite 2000 3090 2.0 7.1 6.1 1.9 7.8 5.5

gas polymorph

Ne P2-tridymite 2040 3810 0.94 2.8 0.9 1-3 2.9 1-2 P-cristobalite 1360 2540 3.1 3.1 2.6 3.4 3.7 3.3

same order as those in table 7 calculated from the energy contours of fig. 7, although the frequencies in table 7 refer only to specimen calculations and are not mean values over all possible energy contours. The largest difference is for helium in a-cristobalite.

The experimental heats of solution in table 3 and those calculated in table 8 all correspond in sign, and sometimes in magnitude. In some cases AH calculated

TABLE 8.-FREQUENCIES AND HEATS OF SOLUTION CALCULATED FROM #m AND K mean v x 10-12 (sec-1) from -AH (cal mole-1) from temp. OK

gas polymorph dm dm (Slater- dm dm (Slater- for calc (London) Kirkwood) (London) Kirkwood) AH

He P,-tridymite 4-8 6.3 1250 2360 770 a-cristobalite 11.2 15.8 1300 2100 41 5 P-cristobalite 9.4 11.7 1080 2100 800

Ne P,-tridymite 2.1 3.4 1290 3060 750 p-crist obali te 3.5 4.8 600 1700 750

using the London approximation is nearer the experimental values (He in P,-tridymite, a-cristobalite, and Ne in P,-tridymite) while in others the Slater-Kirkwood value is better (He and Ne in P-cristobalite). Thus, no choice between these two approxima- tions can be based upon the calculations.

CONCLUSION

The main interest in this study centres upon the solubility and diffusion measure- ments, and on the considerable capacity which tridymite and to a lesser extent cristobalite can show for encapsulating neon and helium. When argon was heated under pressure with these crystals there was evidence of solution but under the conditions employed here equilibrium times were in excess of three months. To trap the heavier rare gases more open crystals will therefore be needed. The theoretical calculations are also of considerable interest since they lead despite some unavoidable limitations to an encouraging measure of correspondence with experimentally determined heats. This is seen in the limiting case hv-4 kT for which eqn. (12) gives AH = #,++RT and so AE = +,+$RT; i.e., at T"K both AH and AE are less negative than &.

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2288 SOLUTION A N D DIFFUSION I N T R I D Y M I T E

APPENDIX The sigmoid kinetic curves of fig. 3 must be representable by an appropriate solution of

the diffusion equation, from which it may be possible to derive D, the diffusion coefficient. The solution required is that for a system in which the temperature is allowed to rise from an initial temperature (77, 273 or 293°K) to a selected final temperature, during and after which dissolved gas is being desorbed.

Since D depends on temperature T, and temperature on time t , D must also be a function of time. Considering the crystal particles to be spheres, the diffusion equation in a sphere of radius a can be written

aulat = D(tp2Upr2, (13)

auld7 = a2Ujar2, (14)

where u = cr, c being the concentration at a surface of radius, r within the sphere. If we set dz = D(t)dt then this equation reduces to

while the boundary conditions in our experiments are (i) for z = 0, C = Co for O<r<a

(ii) for z>O, C = 0 at r = a (15)

An appropriate solution of the above equation and boundary conditions gives for the amount Qt desorbed in time t ,

In this expression Q , is the total amount of desorbed gas after infinite time. For smaller values of z only the first term need be retained, so that

This latter solution does not require all the spheres to be of the same radius, only the total area S and volume V of all the particles being required for a powder composed of spheres of varying radii, or else an average radius, ii = 3S/ V.

We next require an expression for z. The diffusion coefficient can be assumed to obey the Arrhenius equation, D = Do exp (-E/RT), where E is the energy of activation and Do is a constant. Since T = 4(t)

D(t) = Do exp ( -E/R$(t)) . The simplest form for $(t) which has the required properties is

T = T,(l - A exp -kt) , (1 9) where Tis the temperature at time t and A and k are constants. Then the initial temperature is T,(1 - A ) and the final temperature is T,. Also

)dt . RT,[ 1 - A exp ( - kt ) ]

z = ex( -

If we set B = EIRTo and Z = -B/[1 - A exp ( - k t ) ] , then

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R . M . BARRER A N D D. E . W . V A U G H A N 2289 The integrals in the square bracket can be obtained by expanding the exponentials as series and integrating term by term. Then if the first and second integrals are denoted by I , and I,

(23) * "-l (-B~(B+Z)"-~]"

m = l s = o (m-S)!S!(m--) zo' ln(B+Z)exp-B+

In these expressions, 2, = - B / ( l - A ) , and, making all substitutions to obtain z in terms of A, B and t , one obtains

t = (Do exp-B)t+- DO In (l-exp-B)+ k (1-Aexp-Bt)

1 1 [l-Aexp(-kt)]"-(=}-

- A m - s I]. (24) ' [cxp (kt)-A]"-" (1 -A)"-'

When the final temperature T , has been achieved all the terms exp ( -kt) have decayed to zero, so that

Accordingly,

( m - 1 ) A"--"

s = o

z = D(T,)t + constant. (25)

4S2 36

This last relationship should be valid not only for the particular example taken above for 4 ( t ) , but for any 4(t) which changes from an initial to a final value during only part of the interval in QJQ, over which the d t law applies under isothermal conditions.

F. J. Norton, Gen. Elect. Co. USA, Res. Lab. Reprint, no. 4142, 1962. D. E. Swets, R. W. Lee and R. C . Frank, J. Chem. Physics, 1961, 34, 17. R. C. Frank, D. E. Swets and R. W. Lee, J. Chem. Physics, 1961,35, 1451. E. 0. Braaten and G. Clark, J. Amer. Chem. SOC., 1935,57, 2714. W. H. Urry, J. Amer. Chem. SOC., 1932, 54, 3887. A. van Wieringen and N. Warmoltz, Physica, 1956, 22, 849.

W. H. Urry, J. Amer. Chem. SOC., 1933, 55, 3242. Lord Rayleigh, Proc. Roy. SOC. A, 1936, 156, 350.

' N. B. Keevil, Proc. Amer. Acad. Arts Sci., 1940, 73, 311.

lo A. Holmes and F. Paneth, Proc. Roy. SOC. A , 1936, 157,412. l 1 G. A. Cook, Helium and the Rare Gases (Interscience, N.Y., 1961), vol. 1, p. 228. l 2 W. Schreyer, H. S . Yoder and L. T. Aldrich, Carnegie Inst., Washington, Year Book, 1959-60,

l 3 W. Eitel, Bull. Amer. Ceram. SOC., 1957, 36, 142. l4 R. B. Sosman, The Phases of Silica (Rutgers Univ. Press., N.J., 1965). l 5 M. H. Roberts, Electronic Eng., 1951, 23, 51. l 6 F. €3. Kendrick, J. Amer. Chem. SOC., 1940, 62, 2838. l7 F. V. Tooley and C. W. Parmelee, J. Amer. Ceram. Soc., 1940, 23, 304.

l 9 Optical Microscope Methods (British Standards Monograph 3406, part 4 ; 1963). 2o R. Wiebe, V. L. Gaddy and C . Heims, J. Amer. Chem. SOC., 1931, 53, 1724. 21 W. G. Schneider and J. A. H. Duffie, J. Chem. Physics, 1949, 17, 751. 22 G. Nicholson and W. G. Schneider, Can. J. Chem., 1955, 33, 589. 23 W. Nieuwenkamp. 2. Krist., 1935, 92, 82. 24 W. ,4. Dollase, 2. Krist., 1965, 121, 369.

59,94.

F. D. Pidgeon and C. G. Dodd, Anal. Chem., 1954, 26, 1823.

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2290 SOLUTION A N D D I F F U S I O N I N TRIDYMITE

25 W. Nieuwenkamp, 2. Krist., 1937, 96, 454. 26 W. G. Wyckoff, 2. Krist., 1925, 62, 189. 27 R. E. Gibbs, Proc. Roy. SOC. A , 1927, 113, 351. 28 R. M. Barrer and D. L. Peterson, Pro-oc. Roy. SOC. A , 1964, 280, 466. 2 9 F. London, Z. physik. Chern. B, 1930, 11, 222. 30 J. C. Slater and J. G. Kirkwood, Physic. Rev., 1931, 37, 682. 31 J. G. Kirkwood, Physik. Z., 1932, 33, 57. 32 H. R. Muller, Proc. Roy. SOC. A, 1936, 154, 624. 33 R. M. Barrer and D. J. Ruzicka, Trans. Furuduy SOC., 1962, 58, 2263.

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