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High pressure study of AgI: Diffusion in a pressure gradient
J. Phys.
Chem. Solids
HIGH
Pergamon
PRESSURE
STUDY
OF AgI:
PRESSURE ROBERT Department
Printed in Great Britain.
Press 1967. Vol. 28, pp. 1985-1993.
of Geology?
DIFFUSION
GRADIENT
N. SCHOCK*
and SAMUEL KATZ
and Materials Research Center, Rensselaer Polytechnic New York
(Received
15 December
IN A
1966; in revisedform
27 March
Institute, Troy,
1967)
Abstract-The
effect of a large pressure gradient on AgI is investigated using a diamond anvil high-pressure cell. In a pressure gradient AgI is observed to dissociate with metal ions collecting irreversibly in the low-pressure area and halide ions in the high-pressure area, where they combine to form molecular halogen. Ag+ IS . then reduced in incident light. To account for the observed migration, equilibrium and activated states are discussed separately so as to utilize equilibrium thermodynamics. For a multi-component system in a pressure gradient, a loss of energy and hence a more stable configuration, results from the movement of the most compressible substance to the highest pressure area and the least compressible substance to the lowest pressure area. The ratio of compressibilities of the components determines the direction of migration. For AgI, iodine is far more compressible than silver. A model combining rearrangement according to compressibility and the appropriate diffusion process is formulated and applied to the observations. The rate of migration is dependent on the energy change and on the activation energy for the process. Because of the low activation energy involved, migration along areas of misorientation, such as grain boundaries and zones of dislocation, is the preferred explanation for the observed migration.
INTRODUCTION THE PHASErelationships of AgI appear in Figure 1. BURLEY(~) showed that the stable form below 147” at 1 bar is a hexagonal wurtzite lattice (II). He also demonstrated the existence of a metastable f.c.c. (low) sphalerite structure (II’) below 120”, in agreement with MAJUMDAR and ROY.@) BASSETT and TAKAHASHI’~) have shown that the wurtzite form is stable to the 2.5 kb phase boundary at room temperature, when the specimen is gasketed to minimize shear stress. On the other hand, when large shear stresses are generated, as when the sample is not confined laterally, the wurtzite structure transforms to the sphalerite structure with increasing pressure. STROCK(~) suggested a disordered b.c.c. lattice above 147” (I). VAN VALKENBURG’~)identified a phase (IV), stable from about 2.5 to 4-O kb at room temperature, which DAVIS and ADAMS@) tentatively indexed as * Present Address: Department of Sciences, University of Chicago, Chicago,
Geophysical IL 60637.
* Contribution No. 67-S. Supported in part by NSF grant GP-980 and NASA grant NsG-100-60.
orthorhombic. Between 4.0 and 97 kb at 25”, a f.c.c. lattice (III) of the NaCl type is stable.“*s) Above 97 kb at 25”, a complex lattice’g*3’ is the stable form (V). BASSETT and TAKAHASHI’~)show the II-IV equilibrium line as vertical based on visual observations as temperature is increased. The present work indicates that this may be a result of extrusion induced by the temperature increase. DUECKER and LIPPINCOTT(~~) have suggested that AgI is unstable in a high-pressure anvil device, dissociating to free silver and molecular iodine in the region of phase IV. They note the presence, at reported pressures of 40-60 kb, of a dark area suggestive of free iodine. APPARATUS AND MATERIALS The pressure cell has been described in detail.(ll) The sample is typically 0.5 mm in dia. with a volume of about 0.01 mm3. The apparatus was modified to permit experimentation at high temperature through the introduction of a resistance coil around the cylinder wall which holds the
1985
ROBERT
f986
N. SCHOCK
diamond bearing pistons. Temperatures in the neighbourhood of 900” are obtainable with this method. Internal temperatures were calibrated using materials with known phase transformations. Microscope observations were made at magnifications up to 4.00 using a light source with a
/P t 0
I
2
I
h
I
3
y
80
P&JRElf kb)
*
loo
FIG. I. Phase diagram of AgI after D~vss and ADAMS.@)
tungsten filament at approximately 2700°K. Some observations were made with a minimum of illumination by using intermittently the lowest available light intensity for very short durations. No intensity effect was observed over periods substantially longer than those needed for adequate observation. For those experiments in which excess iodine was added to the specimen an iodine crystal was first pressed out as a thin layer partially covering one anvil face before inserting the specimen material. Powdered AgI, Agfir and CuI of unstated purity were supplied by Fisher Scientific Company. The Cu33r was specified as a reagent from the same supplier+
Numerous investigations have been made of the stress distributions in Bridgman anvils,(la - r6) The stress distribution in a specimen contained within a disc-shaped cavity, such as in the diamond cell, may be described by three principal stresses. According to widely-used criteria for plastic Row, the largest difference among these stresses is of the
and
SAMUEL
KATZ
order of the yield stress of the material. Above this, the three principal stresses differ from each other by no more than the yield stress. They are each approximately equal to the pressure, defined as the isotropic component of the stress tensur and given by the mean of the principal stresses. Above the yield stress, the specimen material flows radially outward, until the friction exerted by the diamond anvils and the internal friction of the specimen balance the radial stress. Since these frictional forces are a function uf the radius, as well as the thickness-to-diameter ratio and the properties of the materials, all such Bridgman anvil cells generate a radial gradient in the stress and the pressure. While such a non-uniform stress distribution is a disadvantage in some studies, it is a vital property of the apparatus in the present work, permitting the simultaneous observation of several phases, as well as the local effect of a radial stress gradient. As a result of this stress gradient, a line from the edge to the center of the cell represents an isothermal section on a P-T diagram. DACHILLE and Roy(17) have shown that polymorphic transitions of solids are determined largely by the pressure and temperature. On the other hand, shear stresses strongly affect transition rates. The difficulty in determining the radial distribution of stresses and the demonstrated dependence of transitions largely upon the pressure, justify an empirical calibration. Accordingly, in this study materials with suitable polymorphic transitions are observed as a function of applied force. As the force is increased, the transition boundary moves outward. Measurement of the radial position of transition boundaries for different materials as a function of the number of turns of the force-applying screw yields a pressure calibration (Fig. 2). In choosing materials for such a calibration care must be taken to use materials with similar plastic flow properties. Thus HgI,, a relatively hard, brittle substance, requires approximately one-fourth the nominal pressure to achieve the central transition pressure (13 kb). For the substances in Fig. 2, the nominal pressure is more nearly one-half the central transition pressure over the range O-20 kb. The central pressure, P,, is a linear function of the nominal pressure (force/ area), the proportionality constant being dependent on the anvil area and shape (square vs. octagonal), because of extrusion effects.
HIGH
PRESSURE
STUDY
OF
AgI:
DIFFUSION
A number of empirical relations were tried in fitting Fig. 2, including those of LIPPINCOTT and DUECKER(~~) and of TOWLE and RIECKER.~~)T~~ best fit was provided by P/PC = 1 -(Y/Y# with n = 1.4. Lippincott and Duecker found a similar relationship with n = 2.0. Elastic dishing of the diamond anvils and the resulting effective increase in friction may account for the edge effect (Fig. 2).
-KCI - KEr 5 lncremenfa
of
OPPiicdfwOc
- RbCl
+3bk&-
00
r/r0 2. Form of pressure gradient in diamond cell. r/r,, is the normalized radius from anvil center to edge. Curves represent equal increments of applied force. Experimental error for each transition point is shown. FIG.
The uncertainties in this calibration include the effect of cycling, which lowers the nominal pressure required to achieve a transition; the variability in the coefficient of friction among the different calibrating materials, which affects the extrusion of the specimen and the steepness of the pressure gradient near the edge; and the difficulty of controlling the quantity of initial material. EXPERIMENTAL
OBSERVATIONS
AgI DUECKER and LIPPINCOTT(~~) have described in part the sequence of events which are observed with AgI in the diamond cell, as pressure is
IN
A PRESSURE
GRADIENT
1987
The present increased at room temperature. description emphasizes the time-dependent effects. As the central pressure is increased to 2.5 kb, phase IV appears in the center, first as a dark, granular area which then rapidly crystallizes, forming clear, well-defined crystals, anisotropic in crossed polarized light. As the central pressure is increased to approximately 4.0 kb, phase III appears within IV, III being isotropic in crossed, polarized light in contrast to IV. As the pressure is increased still further, the central area is occupied entirely by phase III with phases IV and II surrounding phase III (Fig. 3). With time, a reddish-brown stain is observed to form at the center of the anvil faces within phase III (Figs. 4-7). This stain forms slowly and irreversibly in contrast to the phase transformations. The rate of formation increases with temperature up to loo”, where phase I first appears, with applied load, and with illumination. However, the stain forms only when phase III is present, at central pressures greater than 4-O kb. With a highintensity tungsten lamp illuminating the sample, the reddish-brown stain forms more rapidly and covers a larger central area. In addition, the outer two phases (II and IV) become darker with time. Continued illumination for long periods of time makes the central area increasingly opaque (Fig. 8). At the same time the outer area becomes increasingly opaque and reflective, suggesting the presence of a metallic substance. AgBr-CuBr-CuI Although AgBr undergoes no phase transformations to 90 kb, a yellowish stain collects with time at the center of the anvils (Fig. 9) and the low pressure areas darken, as in AgI. The central yellowish area appears anisotropic in crossed polarized light in contrast to the cubic AgBr (Fig. 10). Both CuBr and CuI show a darkening in the central area which increases with time. DISCUSSION
OF OBSERVATIONS
AgI The simultaneous formation of highly reflective material in phases II and IV at the periphery of the sample and of reddish-brown material at the center suggests pressure-induced dissociation and the migration of one or more dissociated species with the subsequent formation of iodine and metallic
1988
ROBERT
N. SCHOCK
silver. Although the metal halides undergo photochemical dissociation at atmospheric pressure,(l*) the above effects are observed in total darkness indicating that while contributing in some experiments, photochemical dissociation is not primarily responsible. Photolytic darkening in the outer two phases implies the presence of excess silver ions, which accept electrons produced by the absorption of photons by the lattice. This reduction process may be enhanced by the 4d95s1 electronic structure of the silver ion, forbidden in the free silver ion, but allowed in phases II and possibly IV due to the tetrahedral symmetry of the silver ion site.(19) In phases II, II’ and IV the silver ion would readily accept an electron, thereby completing the d shell. In any other phase the silver ion would already have a completed d sub-shell and a reducing electron would have to be added in the fifth shell. This requires more energy than addition to the fourth shell and explains the apparent preference for reduction of metallic silver in II, II’ and IV but not in III. BURLEY(~) has shown that the f.c.c. (low) phase is capable of accommodating more excess silver ions in interstitial sites than the hexagonal form. This favors the presence of interstitial silver ions, since the low pressure phase is predominantly f.c.c. (low), due to the high shear stresses in the present apparatus. DUECKER and LIPPINCOTT(~~) have attributed the reddish-brown area to the formation of iodine. The presence of iodine in the high-pressure area and metallic silver in the low-pressure area, both in amounts increasing with time, suggests strongly the migration of one or both of the dissociated species of AgI. To investigate further the processes involved in the above description, a sample of AgI was spiked with a small quantity of iodine. The iodine, originally present at the edge of the anvil, migrates rapidly inward toward higher pressures (Figs. ll14). The small amount of 12 remaining in the corner is due to the absence of a pressure gradient. In the subsequent discussion, the above observations will be accounted for by the following general mechanism. The energy of the system is lowered as a result of the movement of 12 inward and Ag outward because of the larger compressibility of 12 than of Ag, suggesting that the movement may proceed spontaneously. In addition, the activation energy for intergranular migration is sufficiently
and SAMUEL low to place.
allow
KATZ the proposed
migration
to
take
AgBr-CuBr-CuI Observations on these compounds also suggests the formation and separation of dissociated products, in a manner similar to AgI. The central anisotropic area and the darkening at the outer edge of the sample in AgBr suggests rhombic bromine and free silver, respectively. Similarly, the dark stains which form in the center with CuBr and CuI suggest the formation of bromine and iodine. As in AgI, the halogens appear to collect at the high pressure area, while metal ions collect at the low pressure area. In contrast to the formation of metallic silver by photoreduction, no formation of metallic copper was observed, perhaps due to the smaller reduction potential of the cuprous ion. DISSOCIATION
AND UNMIXING
The electrical conductivity of AgI was measured to 30 kbc20) to determine possible regions of dissociation. Observed increases in conductivity suggests dissociation in the regions of the 2.5 and 4-O kb transitions. During a transition, bonds are broken and reformed, resulting in highly mobile ions which then become available for migration. In addition, pressure favors dissociation above approximately 7 kb as a result of greater compression of the dissociation products than of AgI.‘21- 23) It is common in metal halides for the structure to be in equilibrium with a small amount of one or both dissociated products.(24*25) If these products are removed from the area of dissociation, the equilibrium is shifted in favor of dissociation. Dissociation proceeds until a new equilibrium is reached, either by reducing the thermodynamic activity of the metal halide or increasing that of the dissociation products. Hence the pressure gradient, by removing dissociation products, favors dissociation and contributes to the observed effects. Dissociation may thus take place throughout the system in response to the pressure gradient, as in AgBr, rather than in a localized area. The mobility of ions at phase boundaries may enhance the effect in AgI. As interpreted above, an irreversible accumulation is observed of I, at the central high-pressure
FIG. 3. Photomicrograph of AgI at room temperature in diamond anvil high-pressure cell, viewed through one anvil. The pressure at the center is approximately 18 kb. The central area is phase III, the ring shaped area phase IV, and the outer area the stable and metastable low pressure forms (II and II’). The dark specks throughout the sample are impurities. FIG. 8. AgI at 25” after 18 hr continuous illumination. Central pressure is approximately 36 kb. Large fissure is in diamond anvil.
FIGS. 4-7.
AgI at 25” and 31 kb central pressure, From left to right, top to bottom, times (min.) application of pressure are 0.5, 5, 20, 40. Dark particles are impurities.
from
the
FIG.
9. AgBr
at 25” and 28 kb central showing stain in center.
pressure
FIG. 10. Closeup of central area in Fig. 9 showing anisotropy between crossed polarizing optical prisms. Undulatory extinction results from stressed diamond anvils.
FIGS. 11-14.
AgI with single crystal of 1~ originally at edge. 25” and 22 kb central pressure. top to bottom, times (min.) are: 1, 20, 50, 230.
From left to right,
FIG. 15. AgI after heating at 200” for 24 hr and after illumination. pressure is 20 kb. Edge to edge distance is 0.27 mm.
Central
HIGH
PRESSURE
STUDY
OF AgI:
DIFFUSION
region and Ag at the peripheral low-pressure region when the central pressure exceeds 4 kb. This accumulation results from the pressure dissociation of a small proportion of AgI, from the subsequent migration of the resulting ions in response to the disequilibrium generated by the pressure gradient, and from their recombination in the regions of maximum and minimum pressures. In view of the qualitative nature of the observations, no rigorous solution was attempted. Instead a model, perhaps not unique, will be developed to account separately for the equilibrium and activated states. This model is based upon the observation that the IV-III phase boundary remains fixed during the experiment, suggesting that mechanical extrusion and the temperature change are negligible and that the radial pressure distribution remains unchanged. In addition, it is assumed that surface diffusion at the edge is insignificant and that the system approaches equilibrium. Under these conditions, a pressure-induced, irreversible process carried out in a closed system must be accompanied by a decrease in volume dV, an increase in entropy dS, and a decrease in the Gibbs free-energy dG = VdP& SdT. Here the system is visualized as composed of many adjacent sub-systems all in a pressure gradient, the difference in pressure between two adjacent sub-systems being dP. There is no mass flow at low pressure, because of friction against the diamond anvils and internal friction. Since G is an extensive property, it may be added for all the sub-systems to obtain the total change AGr resulting from the observed interchange.. With these considerations, perhaps the simplest model consists of a three-component system; AgI, serving as a framework, and smaller concentrations of I- and Ag+. These components are visualized in an isothermal, unchanging pressure gradient. If an arbitrary volume V* of component b at P moves to a lower pressure PO, the same mass of material occupies a new volume
V,,* N V*(l +g*AP)
(1)
where g* = (- 1/Vob)(i3Vb/8P),, AP = P-P,, and higher-order terms in AP have been neglected. To avoid an empty space at pressure PO, an equal volume of substance a (V,” = Vob) moves to
IN A PRESSURE
GRADIENT
1989
pressure P and occupies a volume
V*( 1 +g*AP)( 1 -g”AP)
or
Va 1: V*[l + AP(g” -g=)] The resulting fractional volume change
AV = AP(g*-g=) V* where AV = Vu- V*. Thus, if ga > g* there will be a volume decrease as a result of the rearrangement. The same conclusion is reached if V* = V’“, and P is used as the reference pressure. The total Gibbs free-energy change (AG,) resulting from the replacement of b by a at P and of abybatpais AGr = AGa + AG*
(4)
where AGa and AG* are the changes in the individual components, since G is an extensive property. Only ideal solution between components is considered. Rigorously, partial molal quantities determined from binary mixtures should be used. With above assumptions and (2), the identity dG = V dP- SdT reduces to AG” = VoaAP-Ff=(AP)”
(5)
and llO* AG* = - V,,bAP+2gb(AP)2
(6)
With (5) and (6) the total free-energy change per unit volume of substance a or b, assumed equal at PO, is (AP)2 -
2
(7)
From this model, ifga > g* the free-energy change is negative, the assumed interchange is thermodynamically possible, and will occur at a rate determined by the activation energy. If g* > ga, the assumed interchange increases the free energy of the system and can not take place. The detailed movements may be explained by several models. Among those involving the movement of the smallest amount of material, a model in which Ag + , formed by the pressure-dissociation at the center, moves outward leaving I- behind to form I,, is consistent with all the observations except the spiking experiment which suggests the
1990
ROBERT
N. SCHOCK
interchange of AgI and I,. The only internally consistent model which accounts for all observations involves the motion of all three components Ag+ , I -, and AgI : (a) to account for the net inward movement of I, relative to AgI, with I, formed by pressure dissociation, an Ag+ from the AgI lattice migrates outward to lower pressure combining with I, to form AgI and I -. This I migrates inward combining with the I- from the lattice to form I,. The molar volumes of AgI and I, are approximately equal ; (b) to account for the net outward movement of Ag+ relative to AgI, with Ag+ also formed by pressure dissociation, an Iion from the AgI lattice migrates inward to high pressure, combining with the Ag+ to form AgI. The requirement that Vsa = Vsb results in the repositioning of approximately four Ag atoms for every AgI molecule, three Ag+ ions migrating outward toward lower pressure. The net effect of (a) and (b) is to move Ag+ out and I- in, leaving the distribution of AgI unaltered. As a result of the different molar volumes of Ag and I,, slightly less than half of the I, produced is removed to maintain volume requirements. However, in the I, spiking experiment, only the movement of I, with respect to AgI is involved, and essentially all of the added I, is replaced by AgI. This model also applies to the observed movement in the other halides. A number of other models were considered in detail and were found to involve a large, positive free-energy change as calculated from (7), as well as the transfer of larger amounts of material than that envisioned above. Among these is outward Table 1. Free-energy
and SAMUEL
KATZ
movement of Ag+ with loss of material at the edge leaving I- behind. As previously mentioned, this also cannot account for the observed inward movement of I, in the spiking experiment. On a qualitative basis, if the added Is were combining completely with outward migrating Ag+, it would shrink away from higher pressures, resulting in a stain-free area between the center and the added I,. In addition, this inward migration of I, suggests the system is closed; otherwise I, would tend to escape to lower pressures. Values of AG, for 30 kb central pressure for the substances in this study are given in Table 1. In making these calculations for the cuprous halides, only movement of halogen with respect to metal halide is considered, since outward movement of metal ions can only be inferred. Consideration of metal migration would make the free-energy change more negative. It must be emphasized again that these calculations are based on a highly simplified model and that a more refined theory is needed to deal with the irreversible phenomena which have been observed. When such a theory is developed it is likely to require more quantitative experimental techniques than the present arrangement allows. MECHANISMS
OF DIFFUSION
The most important processes of diffusion in single crystals involve vacant lattice sites, interstitial atoms, and lattice atoms of different types exchanging positions. In a polycrystalline material, diffusion along grain boundaries and zones of dislocation are generally more important. In the
changes, per molecule of a, resulting from rearrangement over 30 kb
_____________~___ ~_~____~~~ Substance a
Substance b
Is Br, Is Brs
Ag Ag GUI CuBr
lOla g(cma dyn-I) a b 13.01”’ 49.0’3’ 13.01 49.0
______(1) RICHARLX.(~~) (2) BRIDGMAN.(~~) (3) International Critical Tables, V. 1. (4) BRIDGMAN.(~~)
0.94’2’ 0.94 2.7Sc4’ 2.87@’
lOI* AGr(erg molecule -I) from (7) -4.64 - 19.59 -3.95 - 18.81
HIGH
PRESSURE
STUDY
OF AgI:
DIFFUSION
silver halides at high temperatures (300400”) the interstitial mechanism predominates(27*28) as a result of the predominance of Frenkel defects.(2s) At lower temperatures the importance of diff~ion along areas of misorientation is predicted from the equation for jump frequency of a migrating atom between neighboring positions I’ = 2, exp( -AG,*/kT)
(8)
where D is the atomic vibration frequency, and AG,* the activation energy for migration. This energy is free energy rather than internal energy, since atomic movement through a lattice involves change8 in the lattice as a result of this movement; e.g., volume changes. (30) Activation energies for lattice diffusion are of the order of 1 eV(31*32) and 21 is of the order of 1013 vibrations per second in most solids.‘33) With these values at 298X, I’ 11 lo-* jumps per second. With an average jump distance of 2 A and all jumps in the same direction, a maximum velocity of (2) 10e4 A set-l is obtained. This value is far too small, in terms of the observed movement, indicating that diffusion through a lattice at room temperature is not consistent with the observed rapid movement of the order of 10” A set-r. A velocity consistent with the observations is obtained by increasing the temperature to 750” or lowering the energy of activation to 0.4 eV. While activation energies for diffusion through a zone of misorientation have not been determined with comparable accuracy, TURNBULL(34) found AG* for grain boundary diffusion in silver to be approximately one-half that for lattice diffusion. Hence, high-diffusivity paths, such as grain boundaries, provide an adequate explanation. In the diamond cell, a powdered sample contains numerous grain boundaries even after application of pressure. In addition, axial compression without confinement leads to significant radial flow and to the formation of zones of dislocation which radiate from the central area of the anvils. These serve as high-diffusi~ty paths for m~gra~ng atoms and molecules, and as an efficient path for movement in a pressure gradient. The existence of defects of this type may be tested by annealing the specimen (Fig. 1.5). This should result in significant recrystallization, reducing the size and number of misorientation areas. Therefore, a specimen of AgI, initially powdered and under compressive load, was heated
IN A PRESSURE
GRADIENT
1991
at 200” for 48 hr, recrystallizing all of the material to phase I. Temperature was then slowly lowered to room temperature over a period of 24 hr. After the central pressure was increased to 20 kb without heating, a small central iodine stain formed. With illumination for 18 hr but no heating, this stain grows to over half the field (Fig. 8). On the other hand, when the specimen is heated before illumination, the stain remains small and doe8 not grow with time, even with 24 hr of subsequent i~umination. A slight darkening in the outer two phases is observed, perhaps due to silver ions produced before annealing. These results suggest that the migration and accumulation of silver and iodide ions can be greatly inhibited by annealing and point out the importance of areas of misorientation as paths for the migrating material. DIFFUSION
IN A PRESSURE GRADIENT
In the absence of a pressure gradient the diffusion rate is given by I> = f&zl?
(9)
where f is a dimensionless geometric constant characteristic of the particular diffusion process, a the lattice constant for the substance through which diffusion is taking place, and n varies with temperature through the Boltzmann distribution exp( - AG~*~~~), where AG,* is the free-energy of formation of the defect. With (8) and (9) the diffusivity
D = fa”v exp( - AG*JkT)
(10)
where AG* = AG,*+ AG,*. Differentiation of (10) with respect to pressure at constant temperature and use of the thermodynamic identity d{AG*) = (AV*) dP-(AS*) dT yields Zalna =---.-+.---
8P
alnv
AV*
kT
ap
(11)
where AV* and AS* are the volume and entropy changes associated with the diffusion of an atom or molecule. Using Griineisen’s assumption that v is the same for all atoms (e.g., SLATER,(~~) p. 219) the constant y = d In v/d In V, and (10) one obtains a In
( “) 8~
2t!?lna
=-+yg T ap
-g
(12)
1992
ROBERT
N. SCHOCK
Using representative values for a, y, g, and AV* shows that the term AV*/kT in (13) predominates by a factor of about 100 and therefore
N --
AV* kT
(13)
While AV* is a function of pressure, available evidence(35) indicates that in most solids the dependence is small, at least to 12 kb. Thus, positive values of AV* result in the diffusion rate decreasing with pressure. On the other hand, diffusion rates increase with pressure for negative values of AV*. Although the diffusivity may vary with pressure, the energy of the present system is lowered as a result of migration in a pressure gradient. Following the treatment of EYRING(~~) for chemical reactions, KAUZMANN(~~) for flow in solid metals, and TURNBULL for crystal growth, this change in energy as a result of rearrangement is considered the driving force for migration, with the activation energy for diffusion controlling the rate. This formulation allows identical interpretations for AG* from reaction-rate theory and for AG from equilibrium theory. An atom a jumps between two adjacent stable positions, a distance h apart, with a frequency given by (8). In the presence of a pressure gradient dP/dr which leads to a change in free-energy AGa between high and low pressure, the resulting freeenergy change is Xd(AGa)/dr. The frequency of jumping back to the original position is z, exp( - AG,*/kT)exp[(
- h/kT)(d(AGa)/dr)]
the energy barrier being changed by hd(AGa)/dr. Thus the net frequency of jumping r NET = vexp[-AG,*/kT][l-exp[(-X/kT) x
(4~WlW)ll
and SAMUEL
KATZ
misorientation areas, M in (10) approaches unity and D 1: I?. (14) yields AG,* = 0.2 eV as a maximum value for a velocity of 0.1 mm mine1 and a jump distance of 2 a with AG” = (2.1) lo-l2 erg molecule-l from (5) for AP = 30 kb. This is consistent with Turnbull’s measurement on grain boundary diffusion. Similar results may be obtained for the movement of Br, and I, in AgBr, CuBr, and CuI. The observed increase in the rate of iodine formation with increasing applied load results from an increase in the pressure gradient and consequent increase in d(AG)/dr. In conclusion, the rapid movement of I, observed in this study is due to (1) a large value of the free-energy gradient and (2) a relatively small value of the activation energy. These result in a high frequency of jumping in the direction of the higher pressure. The large value of the free-energy gradient is a result of the large compressibility of I, and of the large pressure gradient. The small value of AG,* suggests that the dominant diffusion mechanism is through areas of misorientation. This model may be applicable to a system of two or more components of different compressibilities in a pressure gradient. They will tend to unmix as a result of a thermodynamic driving force, arising from the tendency of a system under pressure to reduce its volume. This is accomplished by a shifting of components, so that those with the largest volume change on migration occupy areas of highest pressure. The rate of migration is dependent on the free-energy gradient and on the availability of sites which the migrating atom or molecule can occupy. In the present system, grain boundary diffusion is favored over interstitial diffusion because of its lower activation energy. However, at high temperatures and/or over longer periods of time, the amount of material moved interstitially in a large pressure gradient could be appreciable.
(14)
Although rNET varies with pressure through the pressure dependence of AG,*, the direction of diffusion is independent of the diffusivity gradient
d(AG*)/dZ’. An estimate of the jump rate is obtained from the observation that visible movements occur on the order of minutes. Because of the large defect concentration resulting from the high density of
Acknowledgements-A. VAN VALXENBURGsupplied single crystals of AgI. His encouragement and counsel are appreciated. The interest and help of the following during various phases of the investigation is gratefully acknowledged: M. B. BAYLY, R. S. GILMORE, J. B. HUDSON, D. S. MILLER, H. B. REED JR., and G. L. SALINGER.One of us (RNS) acknowledgesthe donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support in preparing the manuscript for publication.
HIGH
PRESSURE
STUDY
OF AgI:
DIFFUSION
BURLFI G., Am. Miner. 48,1266-1276 (1963). MAJUMDARA. J. and ROY R., J. phys. Chem. 63, 1858-1860 (1959). BASSETT W. A. and TAKAHA~H~T., Am. Miner. 50,
1576-1594 (1965).
19. 20. 21. 22.
STROCK L. W., 2. Phys. Chem. B25, 441-459 (1934). VAN VALKEN~URGA., High Pressure Measurement, 87-94, edited by GIARDINI and LLOYD, Butterworth, Washington (1963). 6. DAVIS B. L. and ADAMSL. H., Science, N. Y. 146,
519-521 (1964). 7. BRIDGMANP. W., Proc. Am. Acad. Arts Sci. 51, SS-
23. 24. 2s. 26.
124 (1915).
8. JACOBS R. B., Phys. Reo. 54, 325-331 (1938). 9. RIGGLEMANB. M. and DRICKAMERH. G., J. them.
12. 13. 14.
IS. 16.
ii:
(1963).
10. DUECXER H. C. and LIPPI~XOTT E. R., Sci&ce, iV. Y. 11.
146,1295-1297 (1964). WEIR C. E., LIPPINCOTT E. R., VAN VALKENR~RGA. and BUNTING E. N., J. Res. natn. Bur. Stand. 64A, 55-62 (1959). JACKSON J. W. and WAXMAN M., High Pressure Measurement, 39-58, edited by GIARDINI and LLOYD, Butterworth, Washington (1963). BOBRO~~KY A., High Pressure Me~r~t, 172-185, edited by GIARDINI and L~oun, Butterworth, Washing& 11963). MONTGOMERYP. W.. STROMBERGH. D.. JURA G. H. and JURA G., High Pressure Measurement, l-16, edited by GIARDINI and LLOYD, Butterworth, Washington (1963). LIPPINCOTT E. R. and DUECKERH. C., Science, N. Y. 144,1119-1121 (1964). TOWLE L. C. and RIECKER R. E., J. geophys. Res.
71,2609-2617 (1966). 17. DACH~LLE F. and ROY R., J. Geol. (1964).
GRADIENT
1993
18. PAULING L., College Chemistry,
REFERENCES
Phys. 38,2721-2724
IN A PRESSURE
72, 243-247
29.
2nd edition, Freeman and Co., San Francisco (1955). SEITZ F., Rev. mod. Phys. 23,328-352 (1951). SCHOCKR. N. and KATZ S., to be published. RICHARDST. W., J. Am. &em. Sot. 27, 1643-1656 (1915). BRXDGMAN P. W., Proc. Am. Acad. Arts Sci. 67,345375 (1932). BRIDGMANP. W., Proc. Am, Acad. Arts Sci. 76, i-7 (1945). WAGENER K., 2. phys. Chem. B23,305-312 (1960). KR&XR F. A., J. Phys. Ckem. Solids 26, 901-909 (1965). BRIDGMANP. W,. Proc. Am. Acad. Arts Sci. n, 189234 (1949). ’ KURNICK S. W., J. them. Phys. 20,218-228 (1952). LAZARUS D. and NACHTRIEB N. H.. Solids under Pressure, 43-69, edited by W. PAUL and D. WARXHAIJER, McGraw-Hill, New York (1963). HOSHINO S. and MIYAKE S., Rev. mod. Phys. 30,
172-174 (1958). 30. RICE S. A. and NACHTRIEBN. H., J. them. Phys. 31, 139-14s
(1959).
31. GIRIFALCO L. A., Atomic
Migration in Crystals, Blaisdell, New York (1964). 32. KEYES R. W., Solids under Pressure, 71-99, edited by W. PAUL and D. WARSCHAUER, McGraw-Hill, New York (1939). to Chemical Physics, 33. SLATER J. C., Introduction
McGraw-Hill, New York (1939). 34. TURNBULL D., Atom Mooements, edited by J. H. HOLLOMAN. Cleveland, Am. Sot. Metals (1951).
3s. LAWSONA. W., High Pressure Physics and Chemistry, Vol. 1. 411-420. edited bv R. S. BRADLEY. Academic Press, New York (i963). 36. EYRING H., Ciben. Rev. 17,6S-77 (1935). 37. f(AUZMANN W., MetaZs Technet. 8,4, I-25 (1941). 38. TURNBULL D., Solid State Physics, Vol. 3, 225-306, edited by F. SEITZ and D. TURNBULL, Academic Press, New York (1956).
Chem. Solids
HIGH
Pergamon
PRESSURE
STUDY
OF AgI:
PRESSURE ROBERT Department
Printed in Great Britain.
Press 1967. Vol. 28, pp. 1985-1993.
of Geology?
DIFFUSION
GRADIENT
N. SCHOCK*
and SAMUEL KATZ
and Materials Research Center, Rensselaer Polytechnic New York
(Received
15 December
IN A
1966; in revisedform
27 March
Institute, Troy,
1967)
Abstract-The
effect of a large pressure gradient on AgI is investigated using a diamond anvil high-pressure cell. In a pressure gradient AgI is observed to dissociate with metal ions collecting irreversibly in the low-pressure area and halide ions in the high-pressure area, where they combine to form molecular halogen. Ag+ IS . then reduced in incident light. To account for the observed migration, equilibrium and activated states are discussed separately so as to utilize equilibrium thermodynamics. For a multi-component system in a pressure gradient, a loss of energy and hence a more stable configuration, results from the movement of the most compressible substance to the highest pressure area and the least compressible substance to the lowest pressure area. The ratio of compressibilities of the components determines the direction of migration. For AgI, iodine is far more compressible than silver. A model combining rearrangement according to compressibility and the appropriate diffusion process is formulated and applied to the observations. The rate of migration is dependent on the energy change and on the activation energy for the process. Because of the low activation energy involved, migration along areas of misorientation, such as grain boundaries and zones of dislocation, is the preferred explanation for the observed migration.
INTRODUCTION THE PHASErelationships of AgI appear in Figure 1. BURLEY(~) showed that the stable form below 147” at 1 bar is a hexagonal wurtzite lattice (II). He also demonstrated the existence of a metastable f.c.c. (low) sphalerite structure (II’) below 120”, in agreement with MAJUMDAR and ROY.@) BASSETT and TAKAHASHI’~) have shown that the wurtzite form is stable to the 2.5 kb phase boundary at room temperature, when the specimen is gasketed to minimize shear stress. On the other hand, when large shear stresses are generated, as when the sample is not confined laterally, the wurtzite structure transforms to the sphalerite structure with increasing pressure. STROCK(~) suggested a disordered b.c.c. lattice above 147” (I). VAN VALKENBURG’~)identified a phase (IV), stable from about 2.5 to 4-O kb at room temperature, which DAVIS and ADAMS@) tentatively indexed as * Present Address: Department of Sciences, University of Chicago, Chicago,
Geophysical IL 60637.
* Contribution No. 67-S. Supported in part by NSF grant GP-980 and NASA grant NsG-100-60.
orthorhombic. Between 4.0 and 97 kb at 25”, a f.c.c. lattice (III) of the NaCl type is stable.“*s) Above 97 kb at 25”, a complex lattice’g*3’ is the stable form (V). BASSETT and TAKAHASHI’~)show the II-IV equilibrium line as vertical based on visual observations as temperature is increased. The present work indicates that this may be a result of extrusion induced by the temperature increase. DUECKER and LIPPINCOTT(~~) have suggested that AgI is unstable in a high-pressure anvil device, dissociating to free silver and molecular iodine in the region of phase IV. They note the presence, at reported pressures of 40-60 kb, of a dark area suggestive of free iodine. APPARATUS AND MATERIALS The pressure cell has been described in detail.(ll) The sample is typically 0.5 mm in dia. with a volume of about 0.01 mm3. The apparatus was modified to permit experimentation at high temperature through the introduction of a resistance coil around the cylinder wall which holds the
1985
ROBERT
f986
N. SCHOCK
diamond bearing pistons. Temperatures in the neighbourhood of 900” are obtainable with this method. Internal temperatures were calibrated using materials with known phase transformations. Microscope observations were made at magnifications up to 4.00 using a light source with a
/P t 0
I
2
I
h
I
3
y
80
P&JRElf kb)
*
loo
FIG. I. Phase diagram of AgI after D~vss and ADAMS.@)
tungsten filament at approximately 2700°K. Some observations were made with a minimum of illumination by using intermittently the lowest available light intensity for very short durations. No intensity effect was observed over periods substantially longer than those needed for adequate observation. For those experiments in which excess iodine was added to the specimen an iodine crystal was first pressed out as a thin layer partially covering one anvil face before inserting the specimen material. Powdered AgI, Agfir and CuI of unstated purity were supplied by Fisher Scientific Company. The Cu33r was specified as a reagent from the same supplier+
Numerous investigations have been made of the stress distributions in Bridgman anvils,(la - r6) The stress distribution in a specimen contained within a disc-shaped cavity, such as in the diamond cell, may be described by three principal stresses. According to widely-used criteria for plastic Row, the largest difference among these stresses is of the
and
SAMUEL
KATZ
order of the yield stress of the material. Above this, the three principal stresses differ from each other by no more than the yield stress. They are each approximately equal to the pressure, defined as the isotropic component of the stress tensur and given by the mean of the principal stresses. Above the yield stress, the specimen material flows radially outward, until the friction exerted by the diamond anvils and the internal friction of the specimen balance the radial stress. Since these frictional forces are a function uf the radius, as well as the thickness-to-diameter ratio and the properties of the materials, all such Bridgman anvil cells generate a radial gradient in the stress and the pressure. While such a non-uniform stress distribution is a disadvantage in some studies, it is a vital property of the apparatus in the present work, permitting the simultaneous observation of several phases, as well as the local effect of a radial stress gradient. As a result of this stress gradient, a line from the edge to the center of the cell represents an isothermal section on a P-T diagram. DACHILLE and Roy(17) have shown that polymorphic transitions of solids are determined largely by the pressure and temperature. On the other hand, shear stresses strongly affect transition rates. The difficulty in determining the radial distribution of stresses and the demonstrated dependence of transitions largely upon the pressure, justify an empirical calibration. Accordingly, in this study materials with suitable polymorphic transitions are observed as a function of applied force. As the force is increased, the transition boundary moves outward. Measurement of the radial position of transition boundaries for different materials as a function of the number of turns of the force-applying screw yields a pressure calibration (Fig. 2). In choosing materials for such a calibration care must be taken to use materials with similar plastic flow properties. Thus HgI,, a relatively hard, brittle substance, requires approximately one-fourth the nominal pressure to achieve the central transition pressure (13 kb). For the substances in Fig. 2, the nominal pressure is more nearly one-half the central transition pressure over the range O-20 kb. The central pressure, P,, is a linear function of the nominal pressure (force/ area), the proportionality constant being dependent on the anvil area and shape (square vs. octagonal), because of extrusion effects.
HIGH
PRESSURE
STUDY
OF
AgI:
DIFFUSION
A number of empirical relations were tried in fitting Fig. 2, including those of LIPPINCOTT and DUECKER(~~) and of TOWLE and RIECKER.~~)T~~ best fit was provided by P/PC = 1 -(Y/Y# with n = 1.4. Lippincott and Duecker found a similar relationship with n = 2.0. Elastic dishing of the diamond anvils and the resulting effective increase in friction may account for the edge effect (Fig. 2).
-KCI - KEr 5 lncremenfa
of
OPPiicdfwOc
- RbCl
+3bk&-
00
r/r0 2. Form of pressure gradient in diamond cell. r/r,, is the normalized radius from anvil center to edge. Curves represent equal increments of applied force. Experimental error for each transition point is shown. FIG.
The uncertainties in this calibration include the effect of cycling, which lowers the nominal pressure required to achieve a transition; the variability in the coefficient of friction among the different calibrating materials, which affects the extrusion of the specimen and the steepness of the pressure gradient near the edge; and the difficulty of controlling the quantity of initial material. EXPERIMENTAL
OBSERVATIONS
AgI DUECKER and LIPPINCOTT(~~) have described in part the sequence of events which are observed with AgI in the diamond cell, as pressure is
IN
A PRESSURE
GRADIENT
1987
The present increased at room temperature. description emphasizes the time-dependent effects. As the central pressure is increased to 2.5 kb, phase IV appears in the center, first as a dark, granular area which then rapidly crystallizes, forming clear, well-defined crystals, anisotropic in crossed polarized light. As the central pressure is increased to approximately 4.0 kb, phase III appears within IV, III being isotropic in crossed, polarized light in contrast to IV. As the pressure is increased still further, the central area is occupied entirely by phase III with phases IV and II surrounding phase III (Fig. 3). With time, a reddish-brown stain is observed to form at the center of the anvil faces within phase III (Figs. 4-7). This stain forms slowly and irreversibly in contrast to the phase transformations. The rate of formation increases with temperature up to loo”, where phase I first appears, with applied load, and with illumination. However, the stain forms only when phase III is present, at central pressures greater than 4-O kb. With a highintensity tungsten lamp illuminating the sample, the reddish-brown stain forms more rapidly and covers a larger central area. In addition, the outer two phases (II and IV) become darker with time. Continued illumination for long periods of time makes the central area increasingly opaque (Fig. 8). At the same time the outer area becomes increasingly opaque and reflective, suggesting the presence of a metallic substance. AgBr-CuBr-CuI Although AgBr undergoes no phase transformations to 90 kb, a yellowish stain collects with time at the center of the anvils (Fig. 9) and the low pressure areas darken, as in AgI. The central yellowish area appears anisotropic in crossed polarized light in contrast to the cubic AgBr (Fig. 10). Both CuBr and CuI show a darkening in the central area which increases with time. DISCUSSION
OF OBSERVATIONS
AgI The simultaneous formation of highly reflective material in phases II and IV at the periphery of the sample and of reddish-brown material at the center suggests pressure-induced dissociation and the migration of one or more dissociated species with the subsequent formation of iodine and metallic
1988
ROBERT
N. SCHOCK
silver. Although the metal halides undergo photochemical dissociation at atmospheric pressure,(l*) the above effects are observed in total darkness indicating that while contributing in some experiments, photochemical dissociation is not primarily responsible. Photolytic darkening in the outer two phases implies the presence of excess silver ions, which accept electrons produced by the absorption of photons by the lattice. This reduction process may be enhanced by the 4d95s1 electronic structure of the silver ion, forbidden in the free silver ion, but allowed in phases II and possibly IV due to the tetrahedral symmetry of the silver ion site.(19) In phases II, II’ and IV the silver ion would readily accept an electron, thereby completing the d shell. In any other phase the silver ion would already have a completed d sub-shell and a reducing electron would have to be added in the fifth shell. This requires more energy than addition to the fourth shell and explains the apparent preference for reduction of metallic silver in II, II’ and IV but not in III. BURLEY(~) has shown that the f.c.c. (low) phase is capable of accommodating more excess silver ions in interstitial sites than the hexagonal form. This favors the presence of interstitial silver ions, since the low pressure phase is predominantly f.c.c. (low), due to the high shear stresses in the present apparatus. DUECKER and LIPPINCOTT(~~) have attributed the reddish-brown area to the formation of iodine. The presence of iodine in the high-pressure area and metallic silver in the low-pressure area, both in amounts increasing with time, suggests strongly the migration of one or both of the dissociated species of AgI. To investigate further the processes involved in the above description, a sample of AgI was spiked with a small quantity of iodine. The iodine, originally present at the edge of the anvil, migrates rapidly inward toward higher pressures (Figs. ll14). The small amount of 12 remaining in the corner is due to the absence of a pressure gradient. In the subsequent discussion, the above observations will be accounted for by the following general mechanism. The energy of the system is lowered as a result of the movement of 12 inward and Ag outward because of the larger compressibility of 12 than of Ag, suggesting that the movement may proceed spontaneously. In addition, the activation energy for intergranular migration is sufficiently
and SAMUEL low to place.
allow
KATZ the proposed
migration
to
take
AgBr-CuBr-CuI Observations on these compounds also suggests the formation and separation of dissociated products, in a manner similar to AgI. The central anisotropic area and the darkening at the outer edge of the sample in AgBr suggests rhombic bromine and free silver, respectively. Similarly, the dark stains which form in the center with CuBr and CuI suggest the formation of bromine and iodine. As in AgI, the halogens appear to collect at the high pressure area, while metal ions collect at the low pressure area. In contrast to the formation of metallic silver by photoreduction, no formation of metallic copper was observed, perhaps due to the smaller reduction potential of the cuprous ion. DISSOCIATION
AND UNMIXING
The electrical conductivity of AgI was measured to 30 kbc20) to determine possible regions of dissociation. Observed increases in conductivity suggests dissociation in the regions of the 2.5 and 4-O kb transitions. During a transition, bonds are broken and reformed, resulting in highly mobile ions which then become available for migration. In addition, pressure favors dissociation above approximately 7 kb as a result of greater compression of the dissociation products than of AgI.‘21- 23) It is common in metal halides for the structure to be in equilibrium with a small amount of one or both dissociated products.(24*25) If these products are removed from the area of dissociation, the equilibrium is shifted in favor of dissociation. Dissociation proceeds until a new equilibrium is reached, either by reducing the thermodynamic activity of the metal halide or increasing that of the dissociation products. Hence the pressure gradient, by removing dissociation products, favors dissociation and contributes to the observed effects. Dissociation may thus take place throughout the system in response to the pressure gradient, as in AgBr, rather than in a localized area. The mobility of ions at phase boundaries may enhance the effect in AgI. As interpreted above, an irreversible accumulation is observed of I, at the central high-pressure
FIG. 3. Photomicrograph of AgI at room temperature in diamond anvil high-pressure cell, viewed through one anvil. The pressure at the center is approximately 18 kb. The central area is phase III, the ring shaped area phase IV, and the outer area the stable and metastable low pressure forms (II and II’). The dark specks throughout the sample are impurities. FIG. 8. AgI at 25” after 18 hr continuous illumination. Central pressure is approximately 36 kb. Large fissure is in diamond anvil.
FIGS. 4-7.
AgI at 25” and 31 kb central pressure, From left to right, top to bottom, times (min.) application of pressure are 0.5, 5, 20, 40. Dark particles are impurities.
from
the
FIG.
9. AgBr
at 25” and 28 kb central showing stain in center.
pressure
FIG. 10. Closeup of central area in Fig. 9 showing anisotropy between crossed polarizing optical prisms. Undulatory extinction results from stressed diamond anvils.
FIGS. 11-14.
AgI with single crystal of 1~ originally at edge. 25” and 22 kb central pressure. top to bottom, times (min.) are: 1, 20, 50, 230.
From left to right,
FIG. 15. AgI after heating at 200” for 24 hr and after illumination. pressure is 20 kb. Edge to edge distance is 0.27 mm.
Central
HIGH
PRESSURE
STUDY
OF AgI:
DIFFUSION
region and Ag at the peripheral low-pressure region when the central pressure exceeds 4 kb. This accumulation results from the pressure dissociation of a small proportion of AgI, from the subsequent migration of the resulting ions in response to the disequilibrium generated by the pressure gradient, and from their recombination in the regions of maximum and minimum pressures. In view of the qualitative nature of the observations, no rigorous solution was attempted. Instead a model, perhaps not unique, will be developed to account separately for the equilibrium and activated states. This model is based upon the observation that the IV-III phase boundary remains fixed during the experiment, suggesting that mechanical extrusion and the temperature change are negligible and that the radial pressure distribution remains unchanged. In addition, it is assumed that surface diffusion at the edge is insignificant and that the system approaches equilibrium. Under these conditions, a pressure-induced, irreversible process carried out in a closed system must be accompanied by a decrease in volume dV, an increase in entropy dS, and a decrease in the Gibbs free-energy dG = VdP& SdT. Here the system is visualized as composed of many adjacent sub-systems all in a pressure gradient, the difference in pressure between two adjacent sub-systems being dP. There is no mass flow at low pressure, because of friction against the diamond anvils and internal friction. Since G is an extensive property, it may be added for all the sub-systems to obtain the total change AGr resulting from the observed interchange.. With these considerations, perhaps the simplest model consists of a three-component system; AgI, serving as a framework, and smaller concentrations of I- and Ag+. These components are visualized in an isothermal, unchanging pressure gradient. If an arbitrary volume V* of component b at P moves to a lower pressure PO, the same mass of material occupies a new volume
V,,* N V*(l +g*AP)
(1)
where g* = (- 1/Vob)(i3Vb/8P),, AP = P-P,, and higher-order terms in AP have been neglected. To avoid an empty space at pressure PO, an equal volume of substance a (V,” = Vob) moves to
IN A PRESSURE
GRADIENT
1989
pressure P and occupies a volume
V*( 1 +g*AP)( 1 -g”AP)
or
Va 1: V*[l + AP(g” -g=)] The resulting fractional volume change
AV = AP(g*-g=) V* where AV = Vu- V*. Thus, if ga > g* there will be a volume decrease as a result of the rearrangement. The same conclusion is reached if V* = V’“, and P is used as the reference pressure. The total Gibbs free-energy change (AG,) resulting from the replacement of b by a at P and of abybatpais AGr = AGa + AG*
(4)
where AGa and AG* are the changes in the individual components, since G is an extensive property. Only ideal solution between components is considered. Rigorously, partial molal quantities determined from binary mixtures should be used. With above assumptions and (2), the identity dG = V dP- SdT reduces to AG” = VoaAP-Ff=(AP)”
(5)
and llO* AG* = - V,,bAP+2gb(AP)2
(6)
With (5) and (6) the total free-energy change per unit volume of substance a or b, assumed equal at PO, is (AP)2 -
2
(7)
From this model, ifga > g* the free-energy change is negative, the assumed interchange is thermodynamically possible, and will occur at a rate determined by the activation energy. If g* > ga, the assumed interchange increases the free energy of the system and can not take place. The detailed movements may be explained by several models. Among those involving the movement of the smallest amount of material, a model in which Ag + , formed by the pressure-dissociation at the center, moves outward leaving I- behind to form I,, is consistent with all the observations except the spiking experiment which suggests the
1990
ROBERT
N. SCHOCK
interchange of AgI and I,. The only internally consistent model which accounts for all observations involves the motion of all three components Ag+ , I -, and AgI : (a) to account for the net inward movement of I, relative to AgI, with I, formed by pressure dissociation, an Ag+ from the AgI lattice migrates outward to lower pressure combining with I, to form AgI and I -. This I migrates inward combining with the I- from the lattice to form I,. The molar volumes of AgI and I, are approximately equal ; (b) to account for the net outward movement of Ag+ relative to AgI, with Ag+ also formed by pressure dissociation, an Iion from the AgI lattice migrates inward to high pressure, combining with the Ag+ to form AgI. The requirement that Vsa = Vsb results in the repositioning of approximately four Ag atoms for every AgI molecule, three Ag+ ions migrating outward toward lower pressure. The net effect of (a) and (b) is to move Ag+ out and I- in, leaving the distribution of AgI unaltered. As a result of the different molar volumes of Ag and I,, slightly less than half of the I, produced is removed to maintain volume requirements. However, in the I, spiking experiment, only the movement of I, with respect to AgI is involved, and essentially all of the added I, is replaced by AgI. This model also applies to the observed movement in the other halides. A number of other models were considered in detail and were found to involve a large, positive free-energy change as calculated from (7), as well as the transfer of larger amounts of material than that envisioned above. Among these is outward Table 1. Free-energy
and SAMUEL
KATZ
movement of Ag+ with loss of material at the edge leaving I- behind. As previously mentioned, this also cannot account for the observed inward movement of I, in the spiking experiment. On a qualitative basis, if the added Is were combining completely with outward migrating Ag+, it would shrink away from higher pressures, resulting in a stain-free area between the center and the added I,. In addition, this inward migration of I, suggests the system is closed; otherwise I, would tend to escape to lower pressures. Values of AG, for 30 kb central pressure for the substances in this study are given in Table 1. In making these calculations for the cuprous halides, only movement of halogen with respect to metal halide is considered, since outward movement of metal ions can only be inferred. Consideration of metal migration would make the free-energy change more negative. It must be emphasized again that these calculations are based on a highly simplified model and that a more refined theory is needed to deal with the irreversible phenomena which have been observed. When such a theory is developed it is likely to require more quantitative experimental techniques than the present arrangement allows. MECHANISMS
OF DIFFUSION
The most important processes of diffusion in single crystals involve vacant lattice sites, interstitial atoms, and lattice atoms of different types exchanging positions. In a polycrystalline material, diffusion along grain boundaries and zones of dislocation are generally more important. In the
changes, per molecule of a, resulting from rearrangement over 30 kb
_____________~___ ~_~____~~~ Substance a
Substance b
Is Br, Is Brs
Ag Ag GUI CuBr
lOla g(cma dyn-I) a b 13.01”’ 49.0’3’ 13.01 49.0
______(1) RICHARLX.(~~) (2) BRIDGMAN.(~~) (3) International Critical Tables, V. 1. (4) BRIDGMAN.(~~)
0.94’2’ 0.94 2.7Sc4’ 2.87@’
lOI* AGr(erg molecule -I) from (7) -4.64 - 19.59 -3.95 - 18.81
HIGH
PRESSURE
STUDY
OF AgI:
DIFFUSION
silver halides at high temperatures (300400”) the interstitial mechanism predominates(27*28) as a result of the predominance of Frenkel defects.(2s) At lower temperatures the importance of diff~ion along areas of misorientation is predicted from the equation for jump frequency of a migrating atom between neighboring positions I’ = 2, exp( -AG,*/kT)
(8)
where D is the atomic vibration frequency, and AG,* the activation energy for migration. This energy is free energy rather than internal energy, since atomic movement through a lattice involves change8 in the lattice as a result of this movement; e.g., volume changes. (30) Activation energies for lattice diffusion are of the order of 1 eV(31*32) and 21 is of the order of 1013 vibrations per second in most solids.‘33) With these values at 298X, I’ 11 lo-* jumps per second. With an average jump distance of 2 A and all jumps in the same direction, a maximum velocity of (2) 10e4 A set-l is obtained. This value is far too small, in terms of the observed movement, indicating that diffusion through a lattice at room temperature is not consistent with the observed rapid movement of the order of 10” A set-r. A velocity consistent with the observations is obtained by increasing the temperature to 750” or lowering the energy of activation to 0.4 eV. While activation energies for diffusion through a zone of misorientation have not been determined with comparable accuracy, TURNBULL(34) found AG* for grain boundary diffusion in silver to be approximately one-half that for lattice diffusion. Hence, high-diffusivity paths, such as grain boundaries, provide an adequate explanation. In the diamond cell, a powdered sample contains numerous grain boundaries even after application of pressure. In addition, axial compression without confinement leads to significant radial flow and to the formation of zones of dislocation which radiate from the central area of the anvils. These serve as high-diffusi~ty paths for m~gra~ng atoms and molecules, and as an efficient path for movement in a pressure gradient. The existence of defects of this type may be tested by annealing the specimen (Fig. 1.5). This should result in significant recrystallization, reducing the size and number of misorientation areas. Therefore, a specimen of AgI, initially powdered and under compressive load, was heated
IN A PRESSURE
GRADIENT
1991
at 200” for 48 hr, recrystallizing all of the material to phase I. Temperature was then slowly lowered to room temperature over a period of 24 hr. After the central pressure was increased to 20 kb without heating, a small central iodine stain formed. With illumination for 18 hr but no heating, this stain grows to over half the field (Fig. 8). On the other hand, when the specimen is heated before illumination, the stain remains small and doe8 not grow with time, even with 24 hr of subsequent i~umination. A slight darkening in the outer two phases is observed, perhaps due to silver ions produced before annealing. These results suggest that the migration and accumulation of silver and iodide ions can be greatly inhibited by annealing and point out the importance of areas of misorientation as paths for the migrating material. DIFFUSION
IN A PRESSURE GRADIENT
In the absence of a pressure gradient the diffusion rate is given by I> = f&zl?
(9)
where f is a dimensionless geometric constant characteristic of the particular diffusion process, a the lattice constant for the substance through which diffusion is taking place, and n varies with temperature through the Boltzmann distribution exp( - AG~*~~~), where AG,* is the free-energy of formation of the defect. With (8) and (9) the diffusivity
D = fa”v exp( - AG*JkT)
(10)
where AG* = AG,*+ AG,*. Differentiation of (10) with respect to pressure at constant temperature and use of the thermodynamic identity d{AG*) = (AV*) dP-(AS*) dT yields Zalna =---.-+.---
8P
alnv
AV*
kT
ap
(11)
where AV* and AS* are the volume and entropy changes associated with the diffusion of an atom or molecule. Using Griineisen’s assumption that v is the same for all atoms (e.g., SLATER,(~~) p. 219) the constant y = d In v/d In V, and (10) one obtains a In
( “) 8~
2t!?lna
=-+yg T ap
-g
(12)
1992
ROBERT
N. SCHOCK
Using representative values for a, y, g, and AV* shows that the term AV*/kT in (13) predominates by a factor of about 100 and therefore
N --
AV* kT
(13)
While AV* is a function of pressure, available evidence(35) indicates that in most solids the dependence is small, at least to 12 kb. Thus, positive values of AV* result in the diffusion rate decreasing with pressure. On the other hand, diffusion rates increase with pressure for negative values of AV*. Although the diffusivity may vary with pressure, the energy of the present system is lowered as a result of migration in a pressure gradient. Following the treatment of EYRING(~~) for chemical reactions, KAUZMANN(~~) for flow in solid metals, and TURNBULL for crystal growth, this change in energy as a result of rearrangement is considered the driving force for migration, with the activation energy for diffusion controlling the rate. This formulation allows identical interpretations for AG* from reaction-rate theory and for AG from equilibrium theory. An atom a jumps between two adjacent stable positions, a distance h apart, with a frequency given by (8). In the presence of a pressure gradient dP/dr which leads to a change in free-energy AGa between high and low pressure, the resulting freeenergy change is Xd(AGa)/dr. The frequency of jumping back to the original position is z, exp( - AG,*/kT)exp[(
- h/kT)(d(AGa)/dr)]
the energy barrier being changed by hd(AGa)/dr. Thus the net frequency of jumping r NET = vexp[-AG,*/kT][l-exp[(-X/kT) x
(4~WlW)ll
and SAMUEL
KATZ
misorientation areas, M in (10) approaches unity and D 1: I?. (14) yields AG,* = 0.2 eV as a maximum value for a velocity of 0.1 mm mine1 and a jump distance of 2 a with AG” = (2.1) lo-l2 erg molecule-l from (5) for AP = 30 kb. This is consistent with Turnbull’s measurement on grain boundary diffusion. Similar results may be obtained for the movement of Br, and I, in AgBr, CuBr, and CuI. The observed increase in the rate of iodine formation with increasing applied load results from an increase in the pressure gradient and consequent increase in d(AG)/dr. In conclusion, the rapid movement of I, observed in this study is due to (1) a large value of the free-energy gradient and (2) a relatively small value of the activation energy. These result in a high frequency of jumping in the direction of the higher pressure. The large value of the free-energy gradient is a result of the large compressibility of I, and of the large pressure gradient. The small value of AG,* suggests that the dominant diffusion mechanism is through areas of misorientation. This model may be applicable to a system of two or more components of different compressibilities in a pressure gradient. They will tend to unmix as a result of a thermodynamic driving force, arising from the tendency of a system under pressure to reduce its volume. This is accomplished by a shifting of components, so that those with the largest volume change on migration occupy areas of highest pressure. The rate of migration is dependent on the free-energy gradient and on the availability of sites which the migrating atom or molecule can occupy. In the present system, grain boundary diffusion is favored over interstitial diffusion because of its lower activation energy. However, at high temperatures and/or over longer periods of time, the amount of material moved interstitially in a large pressure gradient could be appreciable.
(14)
Although rNET varies with pressure through the pressure dependence of AG,*, the direction of diffusion is independent of the diffusivity gradient
d(AG*)/dZ’. An estimate of the jump rate is obtained from the observation that visible movements occur on the order of minutes. Because of the large defect concentration resulting from the high density of
Acknowledgements-A. VAN VALXENBURGsupplied single crystals of AgI. His encouragement and counsel are appreciated. The interest and help of the following during various phases of the investigation is gratefully acknowledged: M. B. BAYLY, R. S. GILMORE, J. B. HUDSON, D. S. MILLER, H. B. REED JR., and G. L. SALINGER.One of us (RNS) acknowledgesthe donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support in preparing the manuscript for publication.
HIGH
PRESSURE
STUDY
OF AgI:
DIFFUSION
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