J. Chem. SOC., Faraday Trans. I, 1983,79, 2043-2053

Hysteresis in Solid-state Reactions

BY SPENCER K. PORTER*

Chemistry Department, Arizona State University, Tempe, Arizona 85287, U.S.A.

Received 29th September, 1982

The problem of the hysteresis that accompanies reversible solid-state redox reactions of oxides has been studied from the point of view of classical thermodynamics. Use is made of the thermodynamic equation of state in its integrated form in order to show that such systems cannot be at equilibrium but must contain some energy of metastability. Data from the literature are used to calculate some of these energies, and it is shown that the energy of interface formation is a likely explanation for the metastability. Thermodynamic equations alone fail to account for the phenomenon of scanning, but a model is developed using thermodynamics plus the nucleation and growth model for solid-phase intergrowth. It is shown that the concept of domains is unnecessary for an explanation of the hysteresis or the scanning. This is in contrast to the hysteresis seen in gas/porous-solid systems for which the domain model works well.

The name hysteresis (from the Greek rjazkpqaz~, a coming late) is used for a variety of phenomena in which the reversal of some independent variable does not give a retracing of the path previously taken through the system's states. Hysteresis was first seen in magnetization curves of ferromagnets,l is seen in surface pressure-area curves of monolayers over liquid is found in gas/porous-solid adsorption ~ystems,~ has been studied extensively in redox reactions of solid o ~ i d e s , ~ - ~ and is seen in many other systems.

Interpretation has proved difficult, and a set of simple postulates that apply to all such systems is not available. It has been widely assumed for some time that systems with hysteresis cannot be at equilibrium,1° and whether thermodynamic or kinetic arguments are most appropriate is not sett1ed.l' The domain theory has been quite successful in interpreting ferromagnetisrnl2 but less so when applied to solid-state reactions. l3

The position of this paper is that hysteresis is a group of several phenomena which will require several explanations, including at least thermodynamics, kinetics and domains. Its purpose is to show that some of the solid-state equilibria which are affected by gas pressure or temperature changes may be explained by postulates from thermodynamics. The domain theory has been developed as a general postulate for hysteresis14-17 and applied to solid-state reactions. 1 3 9 The theory requires that we imagine that each phase is divided into domains, and that these domains have an intrinsic variability of response to external factors such as the pressure and temperature. As no independent verification of such variability was in hand, other explanations were sought.

The transitions of terbium oxide and praseodymium oxide as studied by Eyring and coworkers were chosen for ana lys i~ .~ -~ These are all reversible redox reactions among the several metal-oxygen phases of these oxides. We consider all such systems to be

* On leave 1981-1982 from: Chemistry Department, Capital University, Columbus, Ohio 43209, U.S.A.

2043

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online / Journal Homepage / Table of Contents for this issue

2044 HYSTERESIS IN SOLID-STATE REACTIONS

closed and of one component, the dioxide, with each containing one mole of metal atoms. With these conventions a typical reaction (terbium oxide 4 I)? is written

+ 0, +4Tb, 0, $3 Tb, O,, +to,. We may imagine that the system’s states are controlled by a large heat sink and a piston and that all the equilibrium states are accessible by the proper operations. All these states will be represented by points of the geometric surface obtained by integrating the thermodynamic equation of state

d U = TdS-pdV

in V-S-U space, as shown by Gibbs.,O This is not a trivial task, but it can be done if heat capacities and the equation of state are known.21 Gibbs called the geometric surface the surface of dissipated energy and showed that if the internal energy is plotted vertically no states below the surface could exist, and that points above the surface represent unstable or metastable states. The vertical distance from such a point to the surface itself may be thought of as ‘extra’ energy in the system from motion, stress, interfacial energy or other sources. If we are to decide whether a particular state is an equilibrium state or not, we must determine whether or not such ‘extra’ energy is in the system. The problem may be thought through by finding the state for given conditions that has the lowest conceivable energy.

We must determine the nature of the equilibrium states of the system shown by eqn (1) when the conditions are such that both solid phases exist. We also need to know the nature of possible equilibrium changes of state in the system. The equilibrium state must be a perfect single crystal of Tb,O,, a perfect single crystal of Tb7012 not touching the Tb,O,, and the gas. There will be gas/solid but no solid/solid interfacial energies. A typical equilibrium transition would go as follows: Say that the pressure is increased or that the temperature is lowered favouring the right-hand side of eqn (1). Terbium and oxygen atoms in the ratio 2:3 would be transported from the Tb,O, to the Tb,012, and sufficient oxygen atoms would move from the gas to the Tb,O,, to restore the stoichiometry. Such a process is not the same as the real process and is in fact impossible unless we could retrain Maxwell’s Demon! Nonetheless it is the equilibrium change represented by a series of points on the surface of dissipated energy. We take the gas/solid interfacial energy to be constant, set to zero, and not included in eqn (2). As will be shown below, the solid/solid interfacial energy, which necessarily appears in the real phase change, is the source of both the system’s metastability and its hysteresis.

The form of the Gibbs V-S-U surface is instructive in deciding whether or not hysteresis is possible with an equilibrium transition. U is plotted vertically and is a single-valued function of S and V. If a section is taken of any vertical plane, the surface is either concave upward or linear. It is linear only if representing a first-order phase transition as shown in fig. 1. The surface is concave upward in a region where no transition is occurring or where a second-order transition is occurring.

Equilibrium processes are represented by lines or paths on the surface, and the crucial question here is whether any path with a variable temperature or pressure constant may loop. Typical hysteresis loops are shown in fig. 2 and 3 ; if they represent series of equilibrium states, a closed loop on the V-S-U surface will result. Note that these loops may be traced with virtually infinite slowness, and they may be retraced.

The answer to the question must be rigourously no. The temperature increases

t 4 and z are standard phase definitions. See ref. (19).

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

S. K. PORTER 2045

T u

phase cx

S and/or - V - Fig. 1. Vertical section of V-S-U surface showing a first-order phase transition between phases.

The dotted lines are metastable states seen on superheating and the like.

1.7

D e 1.6

1.5 I I I I I I I I I I 0 10 20 30 800 900 1000 1100 1200 1300 lL00

PlkPa T/K

Fig. 2. Fig. 3.

Fig. 2. Isotherm for the system of eqn (1) as measured by microbalance.21 T = 98 1 K. Fig. 3. Isobar for the system of eqn (1) as measured by microbalance.s p = 51 kPa.

monotonically with increasing entropy, and the pressure decreases monotonically with increasing volume. To do otherwise would be to cause superheating or some other such phenomenon requiring metastability. An example would be to superheat phase a (fig. 1) at constant V such that the system would follow the dotted line of pure a from low S to high. At some point some would form, and the system would fall back to equilibrium (the solid line). The temperature would rise, fall, then rise again; and a particular temperature would appear more than once along the isochore. But this cannot happen if the system is always at equilibrium. Neither isobars nor isotherms will loop or cross. Hysteresis cannot occur as a succession of equilibrium states.

If on the other hand a series of metastable states which do not fall to equilibrium form, looping and branching can occur. The states will be represented by points

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

2046 HYSTERESIS IN SOLID-STATE REACTIONS

above the surface, and the plane representing the temperature and pressure will not be restricted to the slopes from eqn (2). Metastable states may be shown geometrically as in fig. 4. The concavity of rjs or lack of it is of no importance; either first- or second-order transitions may develop metastability.

The length hj is the maximum 'extra' energy which we will call U z needed during the course of the changes r to s and s to r. For reasons to be discussed later, the probable source of this energy is in the solid/solid interfaces which necessarily form during the real reactions. We write, therefore,

rr*m = YA, (3) where y is the specific interfacial energy and A 'is the area of the interface.

L S and/or - V -

Fig. 4. Vertical section of V-S-U space showing equilibrium states on the line rjs and metastable states on the line rhs. The states r and s are those that exist when an equilibrium system is shifted

completely to one side or the other. The line segment jh is Uz.

Consider first the isochoric reduction effected by reversible changes in temperature. Following the path rh (real process) on fig. 4 the heat capacity will be higher than it would be during the equilibrium process (rj), and

Teq d S < AUreal = Jrh Treal dS. (4)

Since A S is the same, Treal > Teq. T is increased monotonically from its equilibrium value at state r to its equilibrium

value at state s. Thus Teal - Gq reaches a maximum at h and thereafter declines to zero.

The oxidation path will now be analysed. On the path sh (real process) the heat gained, Q, will have two opposing parts, Q < 0 for the equilibrium process (sj), but Q > 0 for surface formation

Since A S is the same, Keal < cq. T will decrease monotonically on this path under ordinary experimental conditions. The upshot is that the temperature on the reduction leg will always be higher than the temperature of the equilibrium process, and the

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

S. K. PORTER 2047

T-

Fig. 5. Oxidation, reduction and equilibrium paths for an isochoric phase change represented by fig. 4. The arrows show directions of change. The outside loop is a typical hysteresis path.

P-

Fig. 6. Isentropic hysteresis loop.

temperature on the oxidation leg will always be lower than the temperature of the equilibrium path. The paths are shown in fig. 5 to give the usual hysteresis loop.

Consider next an isentropic cycle. The shape of the path in V-S-U space is still that of fig. 4 with the abcissa being - V. The logic is the same. On the reduction leg we find:

AUeq =- peqdV< AUreal =- Jrh Preal d v - J: (6 )

Since A V is the same, preal < pes. For the oxidation r i

Since A V is the same, preal > peq. This leads to the isentropic loop of fig. 6. We may cycle to the original state (path rhshr) adiabatically only if the energy from

the exothermic surface destruction processes (on hs and hr) is turned into work done on the piston. As this would violate the second law of thermodynamics, strictly adiabatic cycles are impossible.

67 F A R 1

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

2048 HYSTERESIS IN SOLID-STATE REACTIONS

When the V-S-Uequilibrium surface of the system is viewed as a whole, there is a region of finite area representing all the states in which the three phases Tb,O,,, Tb,O, and 0, coexist. A reaction cycle may be performed by crossing the three-phase region and returning by the same path. The initial state would be in a two-phase region (one of the oxides and 02), and the middle state would be in a different two-phase region (the other oxide and 0,). Any such path may be performed as a combination of isentropic and isochoric steps which are infinitesimal in length.

L I I I

0 5 10 15 V/m3

Fig. 7. Isothermal p against V plot for the system of eqn (1) from the data provided for fig. 2.

We therefore conclude that for any reduction path at all that either the real pressure will be less than the equilibrium pressure or the real temperature will be greater than the equilibrium temperature or some combination of the two. If the process is carried out isothermally, the pressure will be lower; if it is done isobarically, the temperature will be higher. Similar logic applied to oxidation processes dictates that the real temperature will be lower than the equilibrium temperature during isobaric processes and that the real pressure will be higher than the equilibrium pressure during isothermal ones. This is in accord with experience (fig. 2 and 3), and the equilibrium path on either graph will be through the middle of the loop. We could plot it if we knew the equation of the energy surface. Calculating the shape of the actual paths on these figures would require the equation describing the metastable surface of the real states, i.e. we would need to know U* at every point.

The isothermal changes studied by Eying and coworkers were chosen for analysis.22 These systems may be imagined to function as pressure-volume engines. The data used to make fig. 2 were used to make several plots like the one shown in fig. 7. The volume of the system is almost entirely that of the gas and is assumed to be such. The area of the plot (which is negative) is the work done by a cycle and is also the heat absorbed. It is 2Vm or 2yAm; i.e. the solid/solid interfaces form twice, and when they are destroyed the energy becomes heat.

A number of these areas were calculated by a curve-fitting technique (Simpson’s rule modified for uneven values of Ax), and the results are shown in table 1. For a

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

S . K . PORTER 2049

Table 1. Heats from the p against V plots from isothermal redox cycles in

rare earths

T / K - Q/kJ(mol Tb)-l

(Q) iO2 + iTb203 + +Tb7012 + 30, 98 1 0.47

1013 0.63 1045 0.58 1079 0.47

(b) 40, + :Tb,012 * hTb1102,, + h0, 760 0.35 773 0.23 789 0.52 804 0.38

T / K - Q/J (mol Pr)-l

(c ) &02 + iPr,O, + 8Pr60,, + &O, 708 62 723 86 738 76 753 70

(6) go, + QPr9016 * kPr,Og +&O, 737 63 750 48 764 51 776 59

given reaction the energies are close to constant in spite of the fact that areas in plots like those of fig. 2 increase greatly with temperature. It is also noteworthy that the energies are small when compared with the reaction energies, i.e. the bump on the surface (like hj in fig. 4) is not very tall. There is scatter in the data, and it is believed that problem arises from the fact that the distribution of data points taken is much more suitable to the axes of fig. 2 than it is to those of fig. 7.

We must now ask whether the amounts of energy found are reasonable for interfacial energies or whether a better explanation is available. Another strong candidate is stress-strain; however, if this were present to any degree we would probably see gradual changes in unit-cell parameters. However, the available X-ray data show abrupt changes.

Nonetheless, we ask what energy would be needed to squeeze a less dense phase into the volume of a more dense phase. The volumes of 3Tb,012 and +TbllO,, are 20.7 and 22.4 cm3, respe~tive1y.l~ The compressibilities have not been measured, but work done on similar substances suggests that they are not far from 10 TPa-1.24-28 If this is correct, we may estimate the pressure needed and the work which must be done : (1.7 cm3)

= 7.6 GPa - AV /3V (10 TPa-l) (22.4 cm3)

Ap = -- -

W = p A V = (7.6GPa)x(l.7cm3) = 13 kJ. (9) 67-2

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

2050 HYSTERESIS IN SOLID-STATE REACTIONS

This is larger than the average of energies in table (b) by a factor of 35, so the explanation seems unlikely.

Quantitative calculation of possible surface energies is not easy, but some reasonable assumptions can be made. If Tb1,02, is ground to powder, a specific surface of ca. 5 m2 g-l is ~ b t a i n e d . ~ ~ &TbllOzo would therefore have a total surface of 940 m2. The interfacial energy is not known, but a value of 0.2 J m-2 is reasonable for two solids. Combining these gives an estimate for yA,,, of 188 J. Twice this is close to the average energy from table 1 (b), and the agreement is clearly better than that obtained by the s tress-strain assumption.

We pursue the consequences of this hypothesis by reasoning on a slightly simpler system than the ones studied so far. Consider a two-phase closed system with an interface between phases in which one component may pass from phase to phase. For the system as a whole the thermodynamic equation of state, eqn (2), holds for the actual states. For each phase a and /? we have

(10) d Ua = TdSa -pd Va + padnu + yadAa

dUB = TdSB-pdv+pBdnB+ yBdAB.

The extensive variables must add, so padnu+ yadAa+pBdnB+ yBdAB = 0. (1 1)

We set the surface energy for the outer surface to zero just as we did when we described the equilibrium system of eqn (1) and note that dna = - dnp. Then

(12) 2ydA = (,up -pa) dna

or

The effect of the geometry of the interface is now studied as follows. Assume that

(14) where p is the density of atoms per unit volume. Applying the formula for surface area and differentiating gives dna pr

(15) dA 2 '

phase a is a growing sphere of radius r within /3: na = - nr3 34P

- -_ -

Combining eqn (13) and (15) gives

There is nothing in these equations suggesting that the growth process could not be reversed at any stage. The radius of curvature, r, of the growing surface may be of either sign depending on the chemical potential difference, and either a convex or concave surface is capable of growing.

When, however, the processes under discussion are reversed in mid-change, the phenomenon of scanning is seen. Typical beginnings to scans are shown in fig. 8 as the arrows inside the loop. By suitable series of reverses all the states within the loop are accessible. If the growth processes can be described as strictly reversible by eqn (1 5 ) and (1 6) scanning should not occur. The paths ded and cfc should be seen in fig. 8, but they are not.

The domain model for hysteresis with a two-component system13 also fails at this

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

S. K. PORTER 205 1

independant variable - Fig. 8. Hysteresis loop with scanning behaviour shown as arrows in the interior of the loop.

The states marked by letters are referred to in the text.

b

I

composition - Fig. 9. Three-dimensional energy surface from domain theory of hysteresis. The independent variable (7' or p ) is the third axis. The valleys form a loop when the entire surface is seen.I3

Fig. 10. Schematic of the reversal of a phase intergrowth process: (a) a growing into p; (b) reversal: nucleation of /? and growth into a.

point. The energy surface derived by these authors has a double minimum like the one shown in fig. 9. The states between a and b are inaccessible, and scanning is not predicted.

An explanation for scanning must come from our knowledge of how phase intergrowth occurs. If phase 3 grows into phase p, and if the process is stopped halfway through, we will have a surfaces of some geometry that are convex outward, as shown schematically in fig. lO(a). Now if the direction of energy transfer is reversed, the process will not reverse, but B will nucleate and grow with a new surface convex

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

2052 HYSTERESIS IN SOLID-STATE REACTIONS

outward into the a, as shown in fig. lO(b). Thus at the point of reversal interfacial area will increase, at least for a time. Then when the direction of change in the system at state e (fig. 8) is reversed, the process will resemble the process b to c, and the same energy arguments will apply. The path eg will parallel bc, and fh will parallel ad at least at the beginning of the scanning processes.

It is of interest to try to apply the ideas of this paper to other processes that show hysteresis. Domain theory has been applied with success to hysteresis in ferromagnets. l 2

If B is plotted against m, the areas of the loops are of the order of 5 mJ per mole of iron,30 several orders of magnitude smaller than those found in the redox cycles under discussion here. A different sort of process is occurring: domain boundaries move through the magnet, but there is no mass transport.

Hysteresis in surface pressure-area curves of monolayers2* should be amenable to the approach of this paper. Surface energy becomes edge energy, but the logic is the same.

The hysteresis seen when porous solids adsorb gases4 would seem to be closely related to the systems discussed here, as we may set up the same kind of experiment and see the same kinds of loops. The explanation must, however, be different. The equilibrium state of such a system could not include any pores! It would have the solid formed into a perfect single crystal with adsorbed gas on the surface. The cause of hysteresis in the actual system is surely the energy needed to move a gas molecule through the bottleneck at the mouth of the pore. A domain model in which the properties may vary works well here, and the theory is well d e ~ e l o p e d . ~ ~

The main conclusions of this article may be summarized as follows: (1) Hysteresis in solid-state reactions is possible in a one-component closed system. (2) Hysteresis in such systems cannot occur by a path through equilibrium states but must go through a series of metastable states. (3) The shape of the equilibrium path may be predicted by thermodynamics. It will go through the middle of the hysteresis loop. Neither kinetics nor variable domains is necessary. (4) The shape of the real path (the loop) could be predicted from the shape of the geometric surface in V-S-U space representing the metastable states that include the ‘extra’ energies U*. (5) The best available hypothesis for the ‘extra’ energy of the metastability is the interfacial energy between phases. (6) Pressure against volume plots of cyclical changes in isothermal systems give energies on the order of twice the maximum interfacial energy. (7) Neither thermodynamics alone nor the variable domain model predicts scanning; however, thermodynamics with the nucleation and growth model does. (8) These ideas are probably applicable to surface pressure-area changes in monolayers but not to ferromagnetism or to porous-solid/gas systems.

Further experiments are suggested. (1) Energy changes from plots of paired energy variables on a variety of systems would be quite interesting. Some of this would require calorimetry, and cycles could be difficult to do for kinetic reasons; however, the results would surely be useful. (2) The data needed to plot the equilibrium surface itself are not available. If they were, it would not be necessary to do cycles to measure the ‘extra’ energies. (3) The effect of crystal size ought to be studied further. It has been shown that large single crystals have narrower loops than masses of fine p o ~ d e r , ~ and it may be possible to relate this to the surface area required. (4) High-energy electron microscopy might be used to watch phase-boundary movements. Ambient oxygen pressures of a few kilopascals would be needed.

In conclusion we may say that hysteresis is not due to a lack or want on the part of any system’s preparer or manipulator. Nature is driven by the necessity of her own energy laws as always.

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online

S. K. PORTER 2053

The author is grateful to Prof. LeRoy Eyring for encouragement and support through NSF grant DMR 8108306 during a sabbatical year at Arizona State University, 198 1-1982. The American Lutheran Church provided a Faculty Growth Award. Several stimulating discussions were held with Prof. Eyring and members of his research group, and with Prof. S. H. Lin and Prof. H. F. Franzen (who was on leave from Iowa State University).

A. Ewing, Proc. R. Soc. London, 1881, 33, 22. H. L. Rosano, D. Yin and C. J. Cante, J. Colloid Interface Sci., 1971, 37, 706. A. W. Neumann and R. J. Good, J. Colloid Interface Sci., 1972, 38, 341. K. S. Rao and B. C. Nayar, J. Colloid Interface Sci., 1972, 38, 45; see also many other papers. B. G. Hyde, D. J. M. Bevan and L. Eyring, Philos. Trans. R. Soc. London, Ser. A, 1966, 259, 583. A. T. Lowe and L. Eyring, J. Solid State Chem., 1975, 14, 383. A. T. Lowe, K. H. Lau, and L. Eyring, J. Solid State Chem., 1975, 15, 9. K. H. Lau, D. L. Fox, S. H. Lin, and L. Eyring, High Temp. Sci., 1976, 8, 129. H. Inaba, S. P. Pack, S. H. Lin, and L. Eyring, J. Solid State Chem., 1980, 33, 295.

lo P. W. Bridgman, Rev. Mod. Phys., 1950, 22, 56. l 1 C. N. R. Rao and K. J. Rao, Phase Transitions in Solids (McGraw-Hill, New York, 1978), chap. 2. l2 C. Kittel, Introduction to Solid State Physics (John Wiley, New York, 4th edn, 1971), chap. 16. l3 D. R. Knittel, S. P. Pack, S. H. Lin and L. Eyring, J. Chem. Phys., 1977, 67, 134. l4 D. H. Everett and W. I. Whitton, Trans. Faraday Soc., 1952, 48, 749. l5 D. H. Everett and F. W. Smith, Trans. Faraday Soc., 1954, 50, 187. l6 D. H. Everett, Trans. Faraday Soc., 1954,50, 1077. l7 D. H. Everett, Trans. Faraday SOC., 1955, 51, 1551. l8 A. Guha and S. H. Lin, The Chemistry of Extended Defects in Non-metallic Solids, ed. L. Eyring and

lCI L. Eyring, The Binary Rare Earth Oxides, in Handbook on the Physics and Chemistry of Rare Earths,

2o J. W. Gibbs, Trans. Connecticut Acad., 1873, 2, 382; reprinted in The Scienti3c Papers of J . Willard

21 S . K. Porter, J. Chem. Ed., 1971, 48, 231. 22 T. Sugihara, S. H. Lin and L. Eyring, J. Solid State Chem., 1981, 40, 189. 23 L. Eyring, unpublished results. 24 F. Levy and P. Wachter, Solid Sfate Commun., 1970, 8, 183. 25 B. Morosin and J. E. Schriber, Phys. Letts., 1979,73A, 50. 26 W. Manning and 0. Hunter, J. Am. Ceram. Soc., 1969, 52,492. 27 W. Manning and 0. Hunter Jr, J. Am. Ceram. SOC., 1970, 53, 279. 28 B. R. Powell Jr, 0. Hunter Jr, and W. R. Manning, J. Am. Ceram Soc., 1971, 54, 488. 29 S. K. Porter and B. C. Gerstein, unpublished results. 30 C. Kittel, Introduction to Solid State Physics (J. Wiley, New York, 2nd edn, 1953), chap. 15. 31 D. H. Everett in The Solid-Gas Interface, ed. E. A. Flood (Marcel Dekker, New York, 1967), vol.

M. O’Keefe (North-Holland, Amsterdam, 1970), pp. 444 ff.

ed. K. A. Gschneidner and L. Eyring, (North-Holland, Amsterdam, 1979), chap. 27.

Gibbs (Dover, New York, 1961), vol. 1, and elsewhere.

2, chap. 36.

(PAPER 2/1676)

Publ

ishe

d on

01

Janu

ary

1983

. Dow

nloa

ded

by U

nive

rsity

of

Illin

ois

at C

hica

go o

n 28

/10/

2014

09:

37:2

8.

View Article Online