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The Journal of Physical Chemistry, Vol. 82, No. 2, 1978
this would indicate a sixfold greater turnover number for Pt-in-zeolite than for Pt-on-alumina. This high turnover number is undoubtedly associated with the greater adsorption of ethylene on the zeolite than on the alumina, a point of view supported by the observation that water vapor completely suppresses the hydrogenation of ethylene but does not affect the chemisorption of hydrogen.
Conclusion It, was found that platinum particles supported on the exterior of the zeolite crystals lose activity a t about 180 “C while those with platinum dispersed in zeolite crystals are stable to about 500 “C. Furthermore, the latter show much lower activity for liquid phase reactions such as hydrogen peroxide decomposition than for gaseous reactions such as hydrogenations. The reduction of a platinum complex dispersed in a zeolite at 25 or 65 “C with hydrazine hydrate produces a preparation with much lower dispersion than gas phase reduction. This is undoubtedly due’to diffusion control of the reactants in the liquid phase reduction using hydrazine hydrate. This reducing agent is on the outside of the zeolite crystal. The reduction takes place at the pore entrance, forming small platinum particles which further hinder the interaction of the platinous complex inside the pore, the hydrazine hydrate outside the crystal, and the platinum particles between them. The activity of Pt-in-zeolite for hydrogenation of hexene is fivefold smaller at lower temperature than that of Pton-alumina. However the Arrhenius plot is regular and at the higher temperatures (>225 “C), it is greater than that of Pt-on-alumina. The latter catalyst shows an unusual behavior with the conversion of hexene to hexane leveling off at about 60% from 170 to 252 “C. Pt-in-zeolite shows good double bond isomerization activity at the lower
temperatures while Pt-on-alumina has no isomerization activity (4). A low value (ca. 5%) for dispersion of Ptin-zeolite was obtained both by hydrogen chemisorption and by poison titration, using ethylene hydrogenation. The value was sixfold smaller than that obtained for Pt-onalumina. In spite of this the reactivity for ethylene hydrogenation using the pulse technique was the same. This is undoubtedly due to greater adsorption of ethylene in the zeolite as compared to the alumina and points out the danger of comparing by the pulse technique catalysts of radically different adsorptive power for the organic reactant.
Acknowledgment. We acknowledge the hospitality and scientific support given to one of us (R.S.M., Jr.) at the Institute of Catalysis at Novosibirsk and particularly to express our gratitude to its Director, Academician G. K. Boreskov and also to V. Romanikov, whose catalytic test unit was used. We also acknowledge the support given to the US-USSR cooperative Program in Chemical Catalysis by the National Science Foundation, and two of us (S. Namba and J. Turkevich) acknowledge support from the U.S. Energy Research and Development Agency. References and Notes (1) J. A. Rabo, P. E. Picket?, D.N. Stamires, and J. E. Boyle, Proc. Int. Cong. Catal., 2nd, 2, 2055 (1961). (2) J. A. Rabo, V. Schomaker, and P. E. Picket?, Roc. Int. Cong. Catal., 3rd, 2, 1264 (1965). (3) M. Boudart, Adv. Catal., 20, 153 (1969). (4) R. A. Dalla Betta and M. Boudart, Proc. Int. Cong. Catal., 5th, 1, 1329 (1973). (5) P.Gallezot, A. Alarcon-Diaz, J. A. Dalmon, A. J. Renouprez, and B. Imelik, J . Catal., 39, 334 (1975). (6) K. Aika, L. L. Ban, I. Okura, S. Namba, and J. Turkevich, J. Res. Inst. Catal., Hokkaido Univ., 24,54-64 (1976). (7) L. Gonzalez-Tejuca, K. Aika, S. Namba, and J. Turkevich, J. Phys. Chem., 81, 1399 (1977).
Stabilization of the Tetragonal Structure in Zirconia Microcrystals R.
C. Garvle
CSIRO, Division of Tribophysics, Melbourne, Australia (Received March 4, 1977) Publication costs assisted by CSIRO
The occurrence of “metastable” tetragonal zirconia as a crystallite size effect is reviewed in the light of recently published experimental evidence. Evidence is presented to show this effect may be general and is a necessary consequence of a structural phase transformation associated with an endothermic heat effect during heating. Hydrostatic and nonhydrostatic stresses influence the microcrystal size-transformation temperature relationship profoundly, Consideration of the combined effects of these variables can account for all the experimental observations reported in the literature.
Introduction Zirconium dioxide is normally monoclinic at room temperature but undergoes a reversible martensitic phase transformation at about 1200 “C to a tetragonal structure;l the high temperature phase cannot be quenched. However it has been known for some time that the tetragonal structure exists at room temperature in microscrystals. Garvie2 earlier advanced the hypothesis that the tetragonal form had a lower surface free energy than the monoclinic, thereby accounting for the spontaneous occurrence of the former structure at a critical crystallite size at room temperature. Filipovich and Kalinina3 independently 0022-3654/78/2082-0218$01 .OO/O
showed in a more general way that the high temperature polymorph of a crystal could be stabilized at temperatures below its normal transformation temperature at some critical crystallite size if the high temperature polymorph had a reduced surface free energy with respect to the low temperature structure. This effect of crystallite size may be more widespread than realized. Takada and Kawai4 observed the existence of “metastable” cubic barium titanate in 10-nm crystals at room temperature. Normally the room temperature structure is tetragonal; this phase is ferroelectric with a Curie point of 120 “C, above which the cubic phase is 0 1978 American
Chemical Society
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978
Tetragonal Structure Stabilization in Zirconia Microcrystals
stable. Meguro et al.5 also found large differences in the room temperature heats of immersion of polymorphs of nickel phthalocyanine in a solution of ethyl alcohol. For the unstable a phase it was 3.08 X 10-1 J / m 2 compared with 4.15 X 10-1J / m 2 for the stable /3 phase. Mitsuhashi, Ichihara, and Tatsuke6 have criticized the concept of a crystallite size effect on the following grounds. X-ray diffraction evidence allows one to postulate a critical size of about 10 nm to stabilize tetragonal zirconia in powders at room temperature. Mitsuhashi et a1.6 cite the following observations which are difficult to explain in terms of a critical crystallite size: (a) Tetragonal microcrystals about 200 nm were observed. (b) Polydomain tetragonal crystallites transformed to monoclinic only with severe grinding. (c) Tetragonal microcrystals annealed under hydrothermal conditions transformed readily to monoclinic with only light grinding. This series of observations led Mitsuhashi and his colleagues to believe that strain generated at domain boundaries was responsible for the existence of metastable tetragonal zirconia. There is now sufficient experimental evidence about the zirconia transformation in the literature, as affected by crystallite size, nonhydrostatic and hydrostatic stresses to make an in-depth discussion possible. 1. The Crystallite Size Effect Following the formalism of Filipovich and Kalinina,3 the free energy of a spherical microcrystal, G, may be expressed as
G = 4/3nr3$ t 4nr2u
(1)
where r is the radius of a microcrystal, rc/ the free energy/unit volume of a crystal of “infinite” radius, and u the surface free energy of the crystal. The difference in free energy of two polymorphs is then given by
A G = 4/37rr3($’ - $ ) t 4nr2(o’ - u )
(2) where the primed symbols refer to the high temperature phase. If the particle size is now reduced to some critical value, r0 where AG is zero a t some temperature, T , below the normal transformation temperature, the high temperature form can exist and one can write:
rc=-3(u’-u)/($’- $) (3) If the free energy/unit volume is expanded in a Taylor’s series about the transformation temperature of an infinite crystal, eq 3 can be expressed in a more useful form, as follows:
(4) where q is the heat of transformation/unit volume of an infinite crystal, and Tb the transformation temperature of an infinite crystal. Filipovich and Kalinina3 commented on eq 4 stating that, “...prolonged and intensive subdivision of the crystals may result in a polymorphic transition to a structure giving a lower surface energy, d,than the structure of the original crystals.” Holmes, Fuller, and Gammage7obtained strong evidence that the surface free energy of tetragonal zirconia (a’) is less than that of the monoclinic structure (u) by studying the heats of immersion of the zirconium oxide-water system. They prepared finely divided samples of the oxide by calcining zirconium hydroxide in the range 600-1000 “C. The results of Holmes et aL7are summarized in Figure 1which is a plot of the heat of immersion as a function of the calcining temperature for a series of
0 Tetragonal 8 Tetragonal
+
21Q
monoclinic
0 Monoclinic
ot
400
I
I
600
CALCINING
I
I
800
I
I
f000
T E M P E R A T U R E , ‘C
Figure 1. Heat of immersion of finely divided powders of zirconia as a function of calcining temperature (Holmes et al.’).
powders outgassed at 300 “C, prior to measurement. In the range 600-800 “C, the powders are tetragonal and the crystallite size increased from 8 to 13 nm. The heat of immersion decreased sharply from 1.1to 0.55 J / m 2 in this range owing to the formation of a less energetic surface at the higher calcining temperatures. In the range 800-900 “C, the crystallites grew to 22 nm and the heat of immersion did not change significantly. The sample calcined at 900 “C contained a trace amount of monoclinic because some of the crystallites had a size greater than a critical value. In the range 900-1000 “C, the crystallites grew to 26 nm while the heat of immersion increased markedly to 0.9 J/m2. The reason is that the size of all the crystallites exceeded the critical value so that the entire sample was monoclinic; this structure must have a higher surface energy than the tetragonal phase. A striking confirmation of the prediction of Filipovich and Kalinina,3 is found in the work of Bailey and c o - w o r k e r ~who ~ , ~ milled course monoclinic zirconia powders and examined them by x-ray diffraction. They observed a gradual decrease of peak intensities and an increase in broadening of the monoclinic (111)and (111)lines. After several hours milling a very diffuse (101) line of tetragonal zirconia appeared between the two monoclinic lines. This line increased in intensity as the milling time increased. The existence of tetragonal zirconia was confirmed by annealing the milled sample at 600 O C ; certain broadened x-ray diffraction lines sharpened to reveal the characteristic tetragonal doublets. The size of the tetragonal crystallites was about 10 nm. The hypothesis that the high temperature structure has a lower surface free energy in a reversible polymorphic transformation can acquire more rigor by appealing to the “dangling band” model of the surface energy. This model assumes that the surface energy arises from broken chemical bonds protruding into space from the surface atoms. The model supposedly only applies to materials bonded by short-range forces and therefore is considered unsuitable for ionic crystals.1° This restriction is unwarranted because long-range forces are known to exist in
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R. C. Garvie
covalent elements and metals also; for example, force constants measured in metals extended to 15 neighbors.ll Moreover, Bruce12 calculated the surface energies of many ceramic oxides using the dangling bond approach and obtained results as accurate as those obtained by other methods. When the surface free energy of a solid is plotted as a function of temperature a discontinuity occurs at the melting point; the magnitude of the discontinuity is proportional to the latent heat of fusion.12J3 The situation is expressed quantitatively by the following relation:
(u' - U ) = BLdAN
(5)
4.0
3.0
n0 Q
a ' w 2.0 P
where B is the ratio of dangling bonds to normal coordinate bonds for one surface atom, Lf the latent heat of fusion, A the area occupied by one atom, and N Avogadro's number. The surface energy of a solid or liquid arises from the fact that surface ions or atoms experience forces of attraction on one side only. Consequently the strength of the bond which bonds the surface atom asymmetrically to the material is the important parameter which governs the magnitude of the surface energy of a solid. If a material undergoes an endothermic process so that the chemical bonds absorb energy (become weaker) then, necessarily, the surface energy must decrease. This is what happens during melting and is the physical basis of eq 5. Surely this state of affairs extends to any solid state transformation accompanied by an endothermic heat effect. Wawra'l measured the surface energies of silver bromide and chloride as a function of temperature using a sonic technique. The expected discontinuities at the phase transformation temperature were detected in both cases. There is a definite possibility then that the hypothesis of Filipovich and Kalinina3 and of Garvie2 can be consequences of a physical law expressed as follows: for any solid state transformation associated with an endothermic heat effect during heating there must exist critical crystallite sizes below which the high temperature structure is stable (not metastable) at temperatures much lower than the bulk transformation temperature. There remains to discuss the best room temperature values of rc, q, AB, and Tb to insert in eq 4 for the case of zirconia. No account of strain has been taken in eq 4. Therefore the values chosen for the parameter should refer to the strain-free condition. At room temperature the work of Bailey et a1.,8 Garvie,2and Mitsuhashi et ala6suggest a value of about 5 nm for rc. Mitsuhashi and Fujiki15 prepared pristine crystals of monoclinic zirconia (i.e., crystals that had never been cycled through the transformation) and studied their transformation temperatures. The kinetics of the transformation were very fast and independent of the heating rate. Also the transformation temperature (1175 & 3 "C) was sharp. This behavior is in strong contrast to the behavior of cycled crystals. It is assumed that pristine crystals are strain free and that 1175 "C is a suitable value for Tb, to be inserted in eq 4. The heat of transformation, q , has been measured by Coughlin and King16as 1420 cal/mol (equivalent to 2.82 X lo8J/m3). The difference in surface free energy of the two forms, AB, is estimated to be 0.36 J / m 2 from the heat of immersion data.7 To test eq 4, select a variable (the heat of transformation, q , e.g.), calculate its value, and compare it with the experimental value. From eq 4, q = (3 X 0.36)/(5 X W9)(l- 293/1448) = 2.71 X lo8 J/m3. The calculated value of q is in excellent agreement with the experimental value and we can place
3
m
2 a n
\
\ /
10
/ \
\
\
\ I
200
I
\
600 1000 TEMPERATURE, K
I
1400
Flgure 2. Pressure-temperature diagram for bulk (Whitney17*18) and microcrystalline zirconia.
confidence in the thermodynamic approach embodied in eq 4 and also the experimental values of the various parameters determined experimentally for zirconia.
2. Effect of Hydrostatic and Nonhydrostatic Stresses on the Crystallite Size. Temperature Relationship (a) Hydrostatic Stress. Whitney17J8studied theoretically and experimentally the thermodynamics of the monoclinic-tetragonal phase transformation in zirconia. The results are summarized by the full line in the pressure-temperature diagram of Figure 2. The full line is assumed to be the equilibrium line of the transformation for the strain-free condition. This cannot be strictly true because internal strains due to the transformation would generate a coexistence region, as discussed below. There will be a similar pressure-temperature curve for each crystallite size, r,, because each microcrystal of a given size has its own transformation temperature, T , given by eq 4. The curve for microcrystals 25 nm in diameter is shown by the dashed line of Figure 2. It was assumed that d P / d T is not size dependent. This assumption can be criticized on the grounds that the elastic properties of microcrystals are known to be size dependent;lg but there is no established procedure for handling this problem. For the purpose of the present semiquantitative discussion the situation depicted in Figure 2 is probably sufficient. The pressure-temperature curves of Figure 2 are expressed analytically as follows:
P = T(dP/dT)+ Po
(6)
where P and T refer to pressure and temperature, respectively, and Po is the intercept on the ordinate at T = 0. Equations 6 and 4 can be combined to yield a relationship between the applied hydrostatic stress (pressure) and the critical crystallite size:
(7)
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978 221
Tetragonal Structure Stabilization in Zirconia Microcrystals
1
4 4
“’/
.A
%w
zJ
08.
0
Figure 3. Influence of hydrostatic stress on the critical crystallite size at room temperature.
, /B ,
w
~ R ~ N S F O R P W T ~rN.*pTRRruRE. DN K
Figure 5. Influence of internal strain on the phase transformation in zirconia. The arrows denote heating (4)and cooling (+).
\
w (I
3 v)
3 e
P
\
\ \
\
\
TEMPERATURE
Figure 4. Influence of nonhydrostatic stress on a polymorphic solid.
The hydrostatic pressure is plotted as a function of the critical size a t a constant temperature of 293 K in Figure 3. The abscissa is the crystallite diameter rather than its radius. The effect of applying a hydrostatic pressure is to increase the critical crystallite size. (b) Nonhydrostatic Stress. Kumazawa” analyzed the influence of nonhydrostatic stress on phase transformations occurring in solids. The general effect is to cause the pressure-temperature transformation curve to broaden into a band in which both polymorphs coexist. The existence of nonhydrostatic stress introduces an extra degree of freedom into the phase rule thereby allowing the coexistence of phases. The effect is depicted schematically in Figure 4. The situation is beautifully illustrated in the case of zirconia. Mitsuhashi et al.15 determined the sharp transformation temperature of pristine monoclinic crystals that were strain-free, as described above. Ruh and his colleaguesz1studied the temperature range of the transformation (in both directions) of crystals that had been cycled through the inversion previously. Such crystals transformed over a temperature range of more than a hundred degrees; also considerable hysteresis was present between the forward and reverse direction. The change in the nature of the transformation between pristine and cycled crystals is due to the presence of internal non-
hydrostatic stress in the latter material generated by strains between the somewhat different structures, as predicted by Kumazawa.20 The mechanism whereby the equilibrium P-T line is broadened into a band is as follows. Consider a cycled crystal of monoclinic zirconia which is heated to within the transformation range. Tetragonal domains are formed randomly throughout the crystal generating nonhydrostatic stresses; when the stress energy equals the chemical free energy driving the reaction, the reaction comes to a halt although it is only partly completed. To drive the reaction further, the crystal must be heated to a higher temperature. In this way the coexistence of phases is observed over a considerable temperature range. The hysteresis arises because the stresses produced by monoclinic material growing in a tetragonal matrix are not the same as stresses generated by tetragonal material growing in a monoclinic matrix. UbbelohdeZ2has given an illuminating, qualitative discussion of this problem. Buljan and co-workers16investigated the linear strains generated during the transformation by x-ray diffraction line profile analysis. As expected, maximum strains occurred when the reaction was one-half completed. Their x-ray estimate of the maximum strain (0.2%) is open to question because, if the value AVf V X 1f 3 (where V is the molar volume) is used in the calculations, the result differs by an order-of-magnitude from the x-ray value. A recent estimate of the interfacial strain from the volume of the phases is about 1.3%.23The combine results of Mitsuhashi et al.,15 Ruh et and the strain data are given in Figure 5. The ordinate in Figure 5 is the level of internal, linear strain in crystals of zirconia. The abscissa is the phase transformation temperature. Pristine crystals are single crystals with zero strain; the transformation spreads rapidly through these crystals along a single front. The transformation temperature is sharp and is represented by a point ori the abscissa. Crystals of zirconia that have been repreatedly cycled through the inversion consist of small domains (diameter about 10 nm) slightly misoriented with respect to one another, Cycled crystals acquire considerable internal strain and the transformation then occurs at random points throughout the material and generates further internal strain, thus causing the transformation to occur over a range of temperatures. Hysteresis is also present as discussed above so that the temperature band obtained with heating is different from
222
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978
TRANSFORMATION
R. C. Garvie
TEMPERATURE, K
Figure 6. The transformation temperature of zirconia as a function of Crystallite size and nonhydrostatic stress.
the band obtained with cooling. The curves in Figure 5 were constructed as follows: It was assumed that the width of the transformation temperature band is proportional to the square of the internal strain level:20
AT= Ce2
(8)
where AT is the width of the band, C is a constant, and E is the internal strain. Ruh et alnZ1 accurately established AT for both heating and cooling in cycled crystals; such crystals had estimated linear strain levels of 1.3%. From these data two values of C were calculated, one for heating and one for cooling. With these values of C, the two transformation bands shown in Figure 5 were constructed. Consider the band for heating; to the left of the band the system is monophase and has the monoclinic structure; within the band both the monoclinic and tetragonal phases coexist; to the right of the band the system is again monophase and has a tetragonal structure. The tetragonal to monoclinic transformation temperature is considerably lowered by the internal stresses; that is, tetragonal material is observed (during cooling) where one would expect only monoclinic, if the crystals were strain-free. Conversely, during heating the monoclinic phase appears where only tetragonal existed, in the strain-free condition. Bailey et ala8 observed this phenomenon in microcrystals of monoclinic zirconia that were severely strained due to prolonged milling. When the strains were reduced by low temperature annealing, tetragonal zirconia spontaneously appeared. 3. Combined Effects The crystallite size, hydrostatic, and nonhydrostatic stresses all interact to alter the transformation temperature. At room temperature both types of stress tend to increase the critical Crystallite size at which the tetragonal structure can become stable. An attempt has been made to show the combined effect of the nonhydrostatic stress and crystalline size on the transformation temperature in Figure 6, for the monoclinic to tetragonal transformation only. The basal plane is the crystallite size-transformation temperature plane, in which the critical crystallite size, rc, is plotted as a function of the transformation temperature, T , at the zero strain level. The functional relationship between rc and T i s given by eq 4. It was assumed for that equation that the surface energy and the heat of transformation were both independent of temperature. It was also assumed that for a crystallite radius of 500 nm, size effects substantially disappeared and T Tb. The vertical axis is the internal strain level. The effect of nonhydrostatic stress is to generate a wedge-shaped transformation zone. To the right of the wedge, tetragonal exists,
-
within the wedge is a two-phase region, and to the left, the monoclinic phase is stable. At a given level of nonhydrostatic stress it was assumed that the width of the transformation temperature range remains constant as the crystallite size changes. The effect of hydrostatic pressure would be to shift the basal transformation temperaturecrystallite size curve to the left, in accord with the data plotted as in Figure 3. The observations cited above by Mitsuhashi et ala6which apparently are not in accord with a crystallite size effect can now be discussed. First tetragonal microcrystals were observed with diameters greatly in excess of 10 nm. This can be explained by the presence of hydrostatic and nonhydrostatic stresses in polydomain material. Consider a tetragonal microcrystal, about the critical size of 10 nm, coherently bound to neighboring microcrystals. When the crystallite transforms it would experience considerable stresses at its boundaries which would have the effect of increasing the critical size at a given temperature, and so the transformation would be inhibited. Second, polydomain tetragonal microcrystals only transformed with severe grinding while similar tetragonal crystals annealed under hydrothermal conditions transformed readily with light grinding. The action of grinding is to increase the internal stress in the crystallites, which tends to preserve the tetragonal phase. The transformation to monoclinic in strained material can only be effected by fracturing a crystallite at its boundaries (stress relief) or by causing appreciable grain growth due to local heating; both of these processes require severe grinding. On the other hand, an initially strained tetragonal microcrystal with a radius appreciably greater than 10 nm and subjected to hydrothermal annealing would be truly metastable at room temperature. The situation is depicted by the dashed line AB in Figure 5 for strained tetragonal crystals. At A, owing to the presence of internal strain, the crystal is in the tetragonal field. However at B the strains have been removed and the crystal is now in the monoclinic field. Similar conditions would prevail for microcrystals except that the transformation band would be shifted to lower temperatures. It is likely that the phase transformation is governed by a soft lattice mode as discussed by Burke and G a r ~ i e . The ~ ~ soft mode in annealed metastable tetragonal crystallites would be excited when the crystals were lightly ground, which would trigger the transformation. 4. The Critical Size in Thin Films and Bulk Material El-Shanshoury, Rudenko, and IbrahimZ5observed that tetragonal domains in thin films of pure zirconia transformed at room temperature to the normal monoclinic form when the crystallite diameter was about 80 nm. Garvie, Hannink, and PascoeZ6determined the critical diameter of tetragonal precipitates in partially stabilized zirconia to be about 90 nm. Hannink, Johnston, and P a ~ c o also e ~ ~noted that the critical size for transformation coincided with loss of coherency of the precipitate. It is argued that the following factors combine to increase the critical size for transformation: (a) The matrix imposes constraints on the transforming, expanding particle. (b) The surface free energy difference of the polymorphs as precipitates may be larger than in the case of unconstrained particles. (c) The transformation needs to be nucleated. Each of these points will be discussed in turn. The thermodynamic criterion for the transformation is satisfied when the chemical free energy released is just balanced
Tetragonal Structure Stabilization in Zirconia Microcrystals
The Journal of Physical Chemistry9 Vol. 82, No. 2, 1978 223
TABLE I: I n t e r f a c i a l Energies (J/mZ) of Metals and
A likely reason for the discrepancy noted above is that the thermodynamic criterion for the tetragonal to monoclinic transformation (embodied in eq 9) is a necessary but not sufficient condition; the transformation must be nucleated. The concept of the strain spinodal is useful in understanding the nucleation of martensitic phase transformation^.^^ According to this theory the free energy of a solid which can undego a martensitic transformation when plotted in strain space has a shape similar to that of the free energy curve of a solid which undergoes a spinodal decomposition. The implication is that martensitic transformations are triggered by a strain-induced lattice instability in special regions of the lattice involving soft phonons. That is, special localized regions of strain caused by free surfaces, dislocations, etc., induce soft phonon modes in the lattice which trigger the martensite transformation. These comments are particularly relevant to zirconia precipitates because it is likely soft modes are involved in the tran~formation.~~ Further, it is known that tetragonal domains are at the limit of coherency before they transform;23the loss of coherency provides a nucleating site. Thus the domains must grow beyond the size required to satisfy the thermodynamic criterion (the necessary condition) to that required for the loss of coherency (the sufficient condition). The coherency limit of the radius of tetragonal precipitates in a calcia-stabilized zirconia cubic matrix is calculated to be about 30 nm using Brooks’ criterion3I but this is known to be an u n d e r e ~ t i m a t e . ~ ~
Zirconia Polymorphs Monoclinic
Tetragonal
ZrO
ZrO, a
0.5-1.0 0.2-0.5
1.46 0.73
1.100 0.55
0.025-0.2
0.29
0.22
Interface Incoherent Partially coherent Coherent a Reference
Metals
29.
by the changes in surface free energy and mechanical energy, as shown in the following equation:
A G = 4/37rrc”($- $’) t 47rr,2(o - o ’ ) * 4/37rr,3(~- E’)= 0
+
(9)
or
r c = -3Aa*/(A$
+ AE)
(10) where A6 is the change in strain energy/unit volume for a particle. The asterisk denotes that we are concerned with precipitates and not unconstrained particles. The strain energy contribution due to the volume change on transformation can be estimated from an equation developed by Davidge and Green2’ for the case of spherical particles in a matrix of different thermal expansion. The thermal expansion term in their equation is replaced by one-third the volume difference associated with the phase transformation. When due account is taken of the initial small strain arising from the misfit of tetragonal particles in the cubic matrix, this strain energy term amounts to 0.46 X lo8 J/m3. It can be argued that the surface free energy difference of the polymorphs increases when they exist as precipitates, as follows: when the powders are unconstrained both polymorphs have free surfaces, but when they exist as precipitates, those with the monoclinic structure are likely to be incoherent while those that are tetragonal are partially coherent at the moment of transformation, which will change the relative values of the surface free energy. A range of values for the types of interface possible with precipitates in metals is known.28 From these data a table of values for the interfaces of monoclinic and tetragonal zirconia precipitates was calculated (Table I). The maximum value for incoherent boundaries in metals is close to the surface energy value calculated and measured for oxide ceramics; thus only the maximum values in the various ranges for metals was used to calculate the ratios of the various types of interfacial energies, viz., incoherent:partially coherent:coherent = 1:0.5:0.2. These ratios were then used to estimate values for the polymorphs of zirconia. The incoherent interfacial value for tetragonal zirconia is taken from the l i t e r a t ~ r eheat ; ~ ~ of immersion data were used to obtain the value for monoclinic zirconia. Therefore from Table I one obtains
A o * = 1.46 - 0.55 = 0.91 J/m2 with this reasoning the surface free energy difference of the polymorphs has increased 153% compared to the case of the unconstrained crystals. The critical size can now be estimated from eq 10 upon inserting appropriate values for the variables, as follows: “= -2.82 X
- 3 x 0.91 108(1- 293/1448) + 0.46 X lo8 -
15.3 nm The calculated critical diameter has increased to 31 nm but this is much below the observed value of 90 nm.
5. Conclusions (1) The existence of tetragonal zirconia in microcrystals at temperatures well below the normal transformation temperature can be explained by a crystallite size effect. (2) The critical crystallite size at any given temperature is strongly dependent on any hydrostatic and/or nonhydrostatic stresses present. (3) The requirement that the transformation must be nucleated governs the critical size of coherent tetragonal precipitates in bulk material.
Acknowledgment. I thank Drs. J. R. Anderson, M. J. Bannister, R. H. J. Hannink, K. A. Johnston, and the late R. T. Pascoe for discussions and criticisms. I also thank Dr. N. A. McKinnon for encouragement and support. References and Notes E. C. Subbarao, H. S. Maiti, and K. K. Srivastava, fhys. Status Solidi A , 21 (1975). R. C. Garvie, J . fhys. Chem., 69, 1238 (1965). V. Filipovich and A. Kalinina, Struct. Glass, 5, 34 (1965). T. Takada and N. Kawai, J. fbys. Soc., (Jpn.), 17, Suppl. B-1, 691 (1962). K. Meguro et al., Kogyo Kagaku Zasshi, 69, 1724 (1966). T. Mitsuhashi, M. Ichihara, and U. Tatsuke, J . Am. Ceram. SOC., 57, 97 (1973). H. Holmes, E. Fuller, Jr., and R. Gammage, J. phys. Chem., 76, 1497 (1972). J. Bailey, D. Lewis, E. Librant, and L. Porter, Trans. J. Brit. Ceram. Soc., 71, 25 (1972). J. Bailey, P. Bills, and D. Lewis, Trans. J . Brit. Ceram. SOC.,74, 247 (1975). J. Brophy, R. Rose, and J. wulff, “Structure and Properties of Materials”, Vol. 2, Wiley, New York, N.Y., 1964. W. Cochran, J. Phys. Chem. Sollds, Suppl. 1, 75 (1965). R. Bruce, Sci. Ceram., 2, 359 (1965). U. Ahmad and L. Murr, J. Mater, Sci., 11, 224 (1976). H. Wawra, Radex Rundscb., 4, 602 (1973). T. Mitsuhashi and Y. Fujiki, J . Am. Ceram. Soc., 56, 493 (1973). J. P. Coughlin and E. G. King, J. Am. Cbem. Soc., 72, 2262 (1950). E. D. Whitney, J . Am. Ceram. Soc., 45, 612 (1962). E. D. Whitney, J . Nectrocbem. Soc., 112, 91 (1965). K. Reider and E. Horl, fbys. Rec. Lett., 20, 209 (1968). M. Kumazawa, J . Earth Sci., Nagoya Univ., 11, 145 (1963). R. Ruh, H. Garrett, R. Domagala, and N. Tallen, J. Am. Ceram. Soc., 51, 23 (1968).
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The Journal of Physical Chemistry, Vol. 82, No. 2, 1978
(22) A. Ubbelohde, Q. Rev. Chem. Soc., 11, 246 (1957). (23) R. H. J. Hannink, K. A. Johnston, and R. T . Pascoe, to be submitted
(27)R. Davidge and T. Green, J. Mater, Sci., 3 , 629 (1968). (28) J. Burke, "The Kinetics of Phase Transformations in Metals", Pergamon, Oxford, 1965. (29) D. Livey and P. Murray, J . Am. Ceram. Soc., 39, 363 (1956). (30) P. C. Clapp, Phys. Status Solidi 6 , 57, 581 (1973). (31) H. Brooks, "Metal Interfaces", ASM, Cleveland, Ohio, 1952. (32) P. B. Hirsch et al., "Electron Microscopy of Thin Crystals", Butterworths, London 1965,Chapter 14.
for publication.
(24) S. Burke and R. Garvie, J. Mater. Sci., 12, 1487 (1977). (25) I. El-Shanshoury, V. Rudenko, and I. Ibrahim, J. Am. Ceram. SOC., 53,264 (1970). (26) R. Garvie, R. Pascoe, and R. Hannink, Nature (London), 258, 703 (1975).
Empirical Correlations between Solvent Properties and the Optical Excitation Energies of Solvated Electrons Peter R. Tremaine and Robert S. Dlxon" Atomic Energy of Canada Limited, Whitesheil Nuclear Research Establishment, Pinawa, ManiYoba, ROE 1LO Canada (Received June 16, 1977) Publication costs assisted by Atomic Energy of Canada Limited
The correlation between the maximum of the optical absorption spectrum of the solvated electron, E,,,, and the function cdg3ap is reexamined. A new correlation is presented in which E,,, is related to the classical semicontinuum charge-solvent interaction energy, U , calculated by assuming that the electron cavity radius is defined by the functional group of the solvent molecule. This treatment is consistent with current theoretical models for electrons solvated by simple molecules and the correlation holds well for a variety of solvents over wide ranges of temperature and pressure.
Recently, Freeman1 found an empirical correlation iphatic alcohols at room temperature.6 We have calculated between the maximum of the optical absorption spectrum values of d g 3 a Pcorresponding to the new data, and have of the solvated electron, E, and the function cdg3a (see also recalculated the earlier r e s u l t ~ using , ~ ~ ~the ~ more below for description of terms). Theoretical moders for commonly accepted Kirkwood-Frolich expressionll electrons solvated by simple, nearly spherical solvent g = - 1 (e - nD2)(2e f nD2)9kTM molecules do not require the use of Kirkwood's constant, (1) g , yet no relationship has been reported which yields an PO2 e(nD2 2)' 4nNd equally comprehensive correlation between E,,, and the instead of Kirkwood's original equation12 (results of these electrostatic and molecular solvent properties used in these calculations are available as supplementary material, see models. Here we comment on the relevance of g to models paragraph at end of text). Although values of g3 from the for solvated electrons and present a new correlation which two expressions differ by up to 2070,the plot of E,,, vs. is consistent with the current models. cdg3ap follows the trend described in ref 1. The only Definitions and source data for the symbols used in this serious exceptions to the correlation are dimethylacetwork are as follows: po, solvent gas phase dipole moment;' and some of the higher alkanols which deviate d, solvent d e n ~ i t y ;E~, static - ~ dielectric c o n ~ t a n t ; l ~n, ~ ~ ~ - lamide, ~ substantially at low values of cdg3ap. In this region, the refractive i n d e ~ ;M~ ,?molecular ~ weight; N , Avogadro's correlation is extremely sensitive to small differences in number; h, Boltzmann's constant; and e, electronic charge. the value used for g , because of the g3 term. Molecular polarizabilities, a , were calculated from nD using Although the empirical correlation between E,, and the Lorenz-Lorentz expression.ll The polarizability, ap, cdg3aP is followed approximately by many solvents, its of the polar group of solvent molecules in a homologous theoretical significance is questionable. Microwave and series was taken to be that of the lowest molecular weight far-infrared contributions to the p~larizability,~~ anisotropy species. The value of po for 1-butanol was used for the in the molecular shape,l8Jgand anisotropic polarizabilityz0 larger 1-alcohols for which data were not available. can all markedly affect g. For example, g values calculated Kirkwood's constant, g , describes the effect of solfrom eq 1 can differ significantly from unity without vent-solvent interactions on the orientational polarization implying specific solvent-solvent interactions.l89" The of a liquid.11J2 Values of g greater or less than unity are factor g is not required in Newton's recent modelz1 for attributed, respectively, to preferential parallel or antielectron solvation in HzO and NH3, or in earlier semiparallel alignment of the solvent molecules, due to specific classical, semicontinuum treatment^.'^-^^ Moreover, the solvent-solvent interactions such as hydrogen bonding. attemptz5to introduce the term g1/2poas the permanent Although the correlation between E,, and tdg3aP held dipole moment of solvent molecules in the first solvation reasonably well for a variety of liquids under different shell of a semicontinuum model is not consistent either conditions,l subsequent values of E,,, for dimethylacetwith classical models for ionic hydrationZ6pz7or with the amide and dimethylformamide were reported to deviate Kirkwood-Frolich definition of g~o.11~12~28 Clearly, the from it.13 Additional measurements of E,, have now been theoretical significance of the relationship between E,,, made in l-propanol,14 three primary amines,16 and four and g is not obvious, and a correlation which is more ethers16 a t various temperatures, and in a number of al-
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0022-365417812082-0224$01 .OOfO
0 1978 American
Chemical Society
