2976
I. ALTUGAND M. L. HAIR
The Ion-Selective Properties of Sintered Porous Glass Membranes by I. Altug and M. L. Hair Research and Development Laboratories, Corning Glass Works, Corning, New York 14830 (Received March 6,1.988)
A study of ion-selective properties of sintered porous glasses has indicated that the hydrated pore volume is closely related to the selective behavior. When the membrane matrix is relatively porous and hydrated, the ion response is nonideal and the degree of ion selectivity is small. As the porosity is decreased by sintering, the membrane becomes highly specific to K+, with a gradual decline in the Ca2+and Naf selectivities. The K + response also approachesideal values. One porous glass membrane has been sintered to such a degree that its Kf selectivity is comparable to that of the normal potassium glass electrode. Infrared spectra indicate that this membrane is nonporous, although ion exchange still proceeds and NH4+ ions can be detected in the membrane after immersion in ammonium chloride solutions. Some significance is attached to these results as indicating a possible analog for biological membrane function.
Introduction One of the most remarkable properties of membranes is their usefulness as electrodes in ion detection. The glass electrode is well known in this respect and is defined by Eisenman‘ as being a thin glass membrane which gives rise to an electrical potential when it is interposed between two solutions. Eisenman considers these glass membranes to be nonporous ion exchangers, and his theory dealing with the origin of electrode potentials and the selectivity of glass electrodes explains the experimental data. Glass electrodes exhibit resistances in the order of megohms or higher. The literature2also contains references to “membrane electrodes” which can be employed in the detection of ion activities in solution. These membranes are again of the ion-exchange type. They have somewhat lower resistivities and rather porous structures, and they can be made from both organic and inorganic materials. No satisfactory theory has yet been proposed that explains the behavior of all “membrane electrodes,” perhaps owing to the varying nature of ion-exchange groups and membrane matrix. Porous ion-exchange membranes may be adequate to detect the changes in ion activity in pure solutions, but their inability to distinguish between different ions is the main limitation in their use as electrodes. The nonporous glass membranes which do show this specific ion selectivity are therefore preferred to the porous membranes. Recently it has been shown that porous glasses can behave as membrane electrodes and that their behavior is adequately accounted for by the Teorell-MeyerSievers the0ry3-~in which the fixed charges in the membrane are treated as a quasi-homogeneous electrolyte. These porous glasses offer certain advantages for study as their pore structure and surface properties are reasonably well defined and the rigidity of the material minimizes effects due to swelling. Moreover, under certaiq circumstances the porous glass membranes show enhanced selectivity to the Ca2f As both porous The Journal of Physical Chemistrg
and nonporous glass membranes can be used for ion detection in aqueous solution, it is of some interest to explore the changes in ion selectivity that occur as a porous glass is sintered to become gradually nonporous. This report therefore describes a study of the ionselective properties of a series of porous glasses that have been subjected to various degrees of sintering. In order to examine the effect of varying the nature of membrane fixed charges (surface properties) as well as porosity, the selective properties of two different types of porous glass have been included. In the experimental work, the permselectivity and the ion selectivity have been electrochemically determined. The permselectivity is an indication of counterion permeability (q., cation permeability in a negatively charged membrane) and is reflected in the membrane potential measurements, while the ion selectivity, which is expressed in terms of selectivity constants, is related to the ability of the membrane to distinguish among ions of the same charge.
Review of the Literature Although some theories are in the process of development, no work has been found in the literature dealing with the relationship between the selectivity and porosity of membranes. Information does exist on the water content (Le., porosity), membrane potentials, and selectivity constants of some organic and inorganic systems, but the correlation of such data becomes rather difficult and often impossible, owing to different or un(1) G . Eisenman in “The Glass Electrode,” Interscience Publishers, Inc., New York, N. Y., 1965, p 215. (2) F. Helfferich, “Ion Exchange,” McGraw-Hill Book Co., Inc., New York, N. Y., 1962, p 376. (3) I. Altug and M. L. Hair, J . Phys. Chem., 72,599 (1968). (4) N. C. Hebert and I. Altug, Second International Biophysics Conference, Vienna, Sept 1966. (6) T. Teorell, Progr. Biophya. Biophys. Chem., 3 , 305 (1953). (6) K. H. Meyer and J. F. Sievers, Helv. Chim. Acta, 19,649, 987 (1936).
ION-SELECTIVE PROPERTIES OF SINTERED POROUS GLASSMEMBRANES known experimental conditions. A review of some points of interest is covered below. Carr7 and his coworkers have reported on the concentration potentials of poly(styrene) sulfonic acid membranes. The water contents of these membranes were varied (15-75% by volume) by swelling them in solutions containing different amounts of ethanol. The membrane potentials were measured (0.1-0.01 N KC1 solution) as a function of water content. The measured potential varied from 51 mV for a membrane having 23% water content to 19 mV for one having 75% water content. One membrane subjected to no alcohol swelling showed a 55-mV response, an almost ideal permselective behavior. The smallest molecule retained by membranes of varying water content was also reported, and these data indicated that decreasing water content corresponded to small pore sizes. This is good evidence of a relationship between the measured membrane potentials (as an indication of the degree of permselectivity) and the pore size. As the pore size gets smaller, better permselective behavior is observed. Marshall and Bergmans have shown that the drying of electrodialyzed bentonite suspensions a t different temperatures can give membranes of varying electrochemical properties. The membrane potentials measured at a given concentration range of KC1 increased in magnitude with the drying temperature. The increasing resistance of the membranes during the heating process is again evidence of a structure becoming smaller in pore size. They also reported that the charge on the membranes, estimated by comparing the experimental curves to the theoretical curves, increased with the drying temperatures. The results of Marshall, Bergman, and Carr, in the present authors’ view, indicate that the pore geometry (Le., pore radius, pore volume) plays an important role in the permselective behavior of the membranes. A system with smaller porosity gives rise to higher concentration potentials and thus exhibits better permselectivity. A great deal of information exists in the literature on the selectivity of ion exchangers. For wide-pore membranes, in general, the counterion with smaller solvated volume is preferred, the more hydrated ions being less strongly bound by the e ~ c h a n g e r . ~Boyd, et al.,1° found that the selectivity order observed for an ion exchanger (Amberlite IR-1, phenol-formaldehyde) is mainly governed by the hydrated radius of the ions in solution. The expected selectivity order for monovalent cations can be written as Cs+ > Rb+ > K + > Na+ > Li+. A similar order of selectivity was reported by Reichenberg and McCauley,g and a relationship between the degree of cross-linking and the magnitude of the selectivity was indicated. The higher magnitudes of selectivity corresponded to increasing cross-linking of the ion exchanger (sulfonated polystyrene). Bon-
2977
ner11112 in his equilibrium studies of monovalent ions on Dowex 50 has indicated that the selectivity of the ion exchanger is related to its water content, which varies with the degree of cross-linking. If the pores in the membrane are large enough, a hydrous medium is provided, such that the cations can be just as hydrated within the pores as in free solution. I n this case the ability of the exchanger to distinguish among ions will be governed by free-solution mobilities (e.g., minimum selectivity for I(+over Na+) following the selectivity sequence given above. I n a highly hydrated medium, the magnitude of selectivity for one ion over the other may be so small that the exchanger appears to have no specific selectivity. Walton13 reports data on two sulfonated polystyrene exchangers which had different degrees of swelling. The one which swelled to the greater extent did not seem to distinguish between ions, within experimental errors, whereas the one which swelled to a lesser degree was Na+ selective over Li+. I n view of this, one could expect to find a higher selectivity for Na+ over Li+ with an even less swollen exchange medium. The hydrated pore volume of membranes is also related to the monovalent-divalent selectivity. Marshall14in a study concerning the effect of heat treatment on the ion selectivity of montmorillonite membranes reported that, on heating to between 400 and 500”, the resistance of his membranes (Wyoming bentonite in H+ form) increased up t o 1-10 megohms. The one prepared at 490” practically lost its divalent sensitivity, such that potassium could be determined in the presence of moderate amounts of calcium and magnesium. The same membranes16 subjected to a heat treatment at only 350” were sensitive t o both monovalent and divalent ions. Another study by Gregor and WetstonelBhas shown that sulfonic and carboxylic interpolymer membranes both exhibit divalent response. The calcium selectivity of carboxylic membranes was reported to be an order of magnitude higher than that of sulfonic membranes. The authors attributed this effect to the specificity of the ion-exchange sites, but it is interesting (7) C. W. Carr, R. McClintock, and K. Sollner, J . Electrochem. SOC., 109,251 (1962). (8) C. E.Marshall and W. E. Bergman, J. Am. Chem. Soc., 63, 1911 (1941). (9) D.Reichenberg and D. J. McCauley, J . Chem. SOC.,2741 (1955). (10) G.E.Boyd, J. Schubert, and A. W. Adamson, J . Am. Chem. SOC., 69,2818 (1947). (11) 0. D.Bonner, J . Phys. Chem., 59,719 (1965). (12) 0.D.Bonner and W. H. Payne, ibid., 58, 183 (1954). (13) H.F.Walton in “Ion Exchange,” F. C. Nachod, Ed., Academic Press, Inc., New York, N. Y., 1949,pp 3-28. (14) C. E. Marshall, “The Colloid Chemistry of the Silicate Minerals,’’ Academic Press, Inc., New York, N. Y., 1949,pp 172-181. (15) C.E.Marshall, J . Phys. Chem., 48,67 (1944). (16) H. P. Gregor and D. M. Wetstone, Discussions Faraday Soc., 21,162 (1956). Volume 7.9, Number 8 August 1068
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I. ALTUGAND M. L. HAIR
to note that higher water contents were reported for the carboxylic membranes. It appears that the divalent-over-monovalent ion selectivity is enhanced (up to a certain point) by increasing hydration of the membrane. Similarly, nonporous exchange media can be expected to have little or no divalent response as exemplified by the normal glass electrodes which have resistances of the order of 100 megohms. As these examples from literature indicate, the hydrated pore volume is a parameter which has to be considered in viewing the selective behavior of membranes. If the membrane matrix is porous and highly hydrated, it is subject to anion invasion and therefore exhibits poor permselectivity. Such membranes respond to most ions of the same charge, showing little or no s p e cific ion selectivity. With a decrease in pore volume, the membrane starts distinguishing among ions more, and its permselectivity is improved. It may become highly specific toward an ion when the membrane matrix is considered no longer porous.
+50t7 0-
E(mW
-50-
0
I
2
3 pK or pNa
4
5
Figure 2. Membrane response upon titration of 10-8 M CaClz with KC1 or NaCl.
Experimental Section ( 1 ) Measurement of Membrane Potentials. The total membrane potential was measured by means of the cell
1
1
Ag-AgC1 reference membrane test solution /solution
1
saturated s oKC1 hti on IHg,CI1!Hg
The reference solution used throughout the experiments was 0.1 N KC1. A calomel electrode with a saturated salt bridge was used as a reference electrode. The emf change of this electrode due to the concentration changes was assumed to be negligible. The leads of the reference electrodes were connected to a Corning Model 12 p H meter, used as a potentiometer. The glass membrane was prepared as described below and glued to a glass stem with Silastic RTV 732. A diagram of the experimental equipment is shown in Figure Potentiometer
E l 3 slomel kctrode
1 '
-
& Test Solution
Figure 1. Cell for measuring potentials of sintered disks. The Journal of Physical Chemistry
1. The experiments were conducted a t room temperature which varied from 23 to 26". The use of a glued electrode is open to some criticism owing to possible participation of the glue in the electrode performance. Control experiments, however, indicated that this interference did not occur, and experiments using tubular membranes gave similar results. The disk-type construction was necessary to obtain the infrared data on the water content of the membranes during the latter stages of the sintering. (8) Membrane Preparation. The glass membranes studied in this work were prepared by conventional leaching techniquesO17 In order to examine the effect of varying the nature of membrane-fixed charges as well as porosity, two widely differing types of porous glass were selected. These were the standard Corning Code 7930 glass, which has a low charge density (about 0.01 N ) , and a leached lead borosilicate glass whose preparation has been described e l ~ e w h e r e . ~ ,The '~ charge density of the second glass was about 0.06 N . These glasses will be referred to as types A and B, respectively. In both cases the glasses were ground into round disks of 0.75411. diameter and 0.5-mm thickness, before heat treatment and leaching. After leaching, the disks were sintered for various times a t temperatures up to 1020". (3) Calculation of the K N ~ K Selectivity Constant. The KNaK selectivity constant was determined by comparing the ion activities required to produce the same electrochemical effect. The membrane potential was recorded as a function of K + or Na+ ion activity in KCl or NaCl test solutions, and the membrane po(17) w.Haller, J . Chem. Phys., 42,686(1966). (18) L.Hersh, J . Phy8. Chem., 72, 2196 (1968).
2979
ION-SELECTIVE PROPERTIES OF SINTERED POROUS GLASS MEMBRANES tential E (mV) was plotted against the negative logarithm of K + or Na+ ion activity in the solution, pK or pNa (see Figure 2.). Because of the ion selectivity of the membrane, different amounts of K + and Na+ ions are required to produce the same emf. The ratio of the E(+ and Na+ activities a t a given emf value expresses K as defined by the K N ~constant, KNagPot
a K+ = -
(1)
a N &+
Activity coefficients were taken from ref 19. (4) Calculation of the K C a K and K C ~ N *Selectivity Constants.20 I n calculating the divalent-monovalent selectivity constant, the above method was again used. First, the emf of the membrane in equilibrium with a CaClz solution was recorded. The change in emf was noted as a function of K + or Na+ ion concentration as the test solution was titrated with known amounts of KC1 or NaC1. Thus, as shown in Figure 2, the equilibrium CaC12 value of -59 mV changes with the added Na+ until the titration curves overlap the monovalent, response curves, which are shown as straight lines. For example, the K + activity which is as effective as lo-$ M CaC12in producing the equal membrane potential is 1.41 X lowaN (given as -log U K + = 2.85). By the use of eq 2 the K C ~constant K is calculated to be 690 for pK 2.85 and pCa 2.86. It is of some interest to note (2)
a K t 2 = (KKCaPot)UCa2+
that when K C ~ K = 1 (&e., membrane is not selective toward either ion), the membrane would olsppear to be more selective toward divalent ions for a monovalent concentration range less than 1 M . Obviously, smaller amounts of Ca2+ are required to hold the relationship a~ t 2 = acaa t I
Results and Discussion The results obtained for the two series of sintered glasses are shown in Tables I and I1 and Figure 3. ( 1 ) Membrane Potential. When an ionic membrane separates two solutions of different composition, an electrical potential is developed across the membrane Table I : Selectivity Constants, Cation Responses and Sintering Conditions of Type B Glass Membranes
Meabrane
1 2 3 4
5 6 7
Sintering --conditions-Temp, Time, “C hr
750 750 770 770 7 75 820 800
0.5
...
... 1 1.33
... ...
Cation response, mV --Selectivity constants(0.1-0.01 K N ~ K K C ~ K Kc&N& N KCl)
1.8 2 2.1 3.6 4.5 7.1 8
15 28 110 690 1780 8700
...
6 8 25 55 63 170
...
44 48 50 53 53 54 56
Table 11: Selectivity Constants, Cation Responses, and Sintering Conditions of Type A Glass Membranes
M a brane
8 9
Sintering --conditione---. Temp, Time, O C hr
1006 1018 1006 1006
10 11
3 3 5.5 21
Cation response, mV (0.1-0.01)
-8oleotivity oonstantsKN~K KC~K K c ~ N N ~ KCI)
2 2.2 2.2 10
70
11 18
42 51
180
35
53
28
...
I
.
55
.
owing to the unequal distribution of ions in the membrane and solution phase. The total membrane potential is given by the sum of the phase boundary potentials and the diffusion potential2I
.E =
u -.
u
- v RT
+ v -In[ F
+ +
al[r~u (v/rI)]] a~[r2u (v/r2)]
+ RT - ln-F
r2
(3)
r1
where u and v are the mobilities of the cations in the membrane (usually assumed to be the same as in free solutions), a is the ion concentration in t,he solution, and r is the Donnan ratio a t the membrane-solution interfaces. If the solution on one side of the membrane is kept constant, then the measured membrane potential will be a function of the ion activities in the solution facing the other side. If the membrane shows any specific ionic selectivity, equal concentrations of ions Na+ and K+ will not result in the same electrochemical effect, which will be controlled by the potential selectivity constant of eq 1. The Donnan ratio in eq 3 is given by Teorells as
(4) where X represents the fixed-charge concentration. As has been discussed previously,a in the case of a truly porous solid such as porous glass, the fixed-charge concentration must be some function of 2n/r, where n is the number of charges per unit area of the glass surface and r is the pore radius. As the porous glass is slowly sintered, its volume is reduced. At the temperatures used here it is anticipated that the pore will be reduced in size, and, providing there is no change in the concentration of fixed surface charges (n),a resultant increase in the (W)term should be observed. This in turn should result in a higher degree of anion exclusion and an increased membrane potential. The results obtained indicate that an enhanced improvement in cation permselectivity is indeed observed on sintering the porous glass membranes. The con(19) W. M. Latimer, “Oxidation Potentials,” Prentice-Hall, Inc., New York, N. Y.,1961, p 364. (20) B.B.Hanshaw, Clays Clag Minerals, Proc. Natl. Conf.Clays Clay Minerals, 12,397 (1964). (21) G.Eisenman, Biophys. J.,2,269 (1962).
Volume 72, Number 8 August 1968
2980
I. ALTUGAND M. L. HAIR
7.
0
x 11
I
I
I
I
I
2
4
6
8
IO
KPot
NaK
Figure 3. Change in selective behavior of the leached glass membranes after sintering.
starts differentiating between K + and Na+ ions more than before, along with further decrease in Ca2+ sensitivity. For example, the membrane sintered a t 820" is 8700 times more K + selective over Ca2+and 7.1 times more K+ selective over Na+. Finally, on further sintering, the divalent response is completely lost, and such is the case when optimum K+-Na+ selectivities are reached. Sintering the type A porous glasses resulted in a similar behavior (Table 11). In this case, sintering a t 1006" for 21 h r caused the membrane to become 10 times more selective to K + over Na+. This value is to be compared with the maximum of 8 which was found for the sintered type B porous glasses. The optimum selectivity so far observed in potassium glass electrodes corresponds to K N ~ = K 10. The potassium electrode is a sodium aluminum silicate glass (NAS2~--4) and the hydrated layer on the surface of the electrode is believed to give rise to the K + selectivity. It is noteworthy that a sintered porous glass has now been shown to exhibit a comparable K + selectivity. This strongly suggests that optimum potassium selectivity may be dependent on a charged and hydrated medium having small pores, rather than on a specific surface characteristic. EisenmanZ2has shown that the potential selectivity constant is related to the equilibrium selectivity constant by the equation UK
KNaKPot = -KN~K~~ UNa
(5)
where U K and UN&are the ion mobilities. From titration studieszaon the type B glass it is known that the glass surface shows only slight preference for K + over Na+ centration potential, taken as a measure of permselecand K N ~ KN " 1. 1. Thus, in the unsintered membrane, tivit!y, increases with sintering time or temperature. the potential selectivity is controlled by the free-soluThus, as indicated in Table I, the porous glass memtion mobility ratio u K / u N ~ N 1.5, and therefore K N ~ K ~ ~ brane B, when heated a t 750", shows only a 48-mV N 1.65, in good agreement with observation. If the response to a 10-fold change in KC1 concentration surface properties of the porous glass membranes are (0.14.01 N ) , whereas the one sintered a t 770" for 1 hr unchanged during these experiments, then we can shows a 53-mV response. Comparable results have assume that the equilibrium constant K N a K e q remains also been obtained by sintering the type A glasses the same. An improvement in the observed K+-Na+ (Table 11). The fact that the K + response approaches selectivity with sintering corresponds to increasing ideal values at later stages of sintering indicates an KNaKPotvalues. Equation 5 therefore implies that a insignificant amount of anion invasion. change in the mobility ratio U K / U N ~directly affects the (2) Ion Selectivity. In Figure 3 the K N ~constants K degree of ion selectivity. It is reasonable to expect that obtained for a series of sintered type B glass membranes have been plotted against the corresponding K c a ~ dehydration of a porous medium by a process such as sintering will cause a decrease in the ion m o b i l i t i e ~ . ~ ~ constants. These results show that the divalent selecMoreover, because of the highly hydrated nature of tivity is gradually lost as the sintering temperature or Na+, a decrease in the hydrated pore volume is extime is increased ( L e . , increasing koa^ value). In the pected to affect the movement of Naf ions to a greater first stages of sintering, the Ca2* response is affected extent than the K + ions. A decrease in the mobility more than the Na+ response. For example, a change from 6 to 110 in KCaK iS observed, whereas KNaK changes from only 1.8 to 2.1. This implies that a rela(22) G. Eisenman in "The Glass Electrode," Interscience P u b lishers, Inc., New York, N. Y., 1965, p 217. tively small decrease in volume hydration barely af(23) I. Altug and M. L. Hair, J. Phys. Chem., 71,4260 (1967). fects the K+-Na+ selectivity but gives rise to a con(24) It has been reported that the hydration of a normal glass gives siderable decline in Ca2+-K+ selectivity. When the rise to an increase in the mobility of Na+ by about 4 orders of magnihydration state reaches a certain point, the membrane tude: G. Eisenman and F. Conti, J. Gen. Physiol., 48,65 (1965). The Journal of Physical Chemistry
ION-SELECTIVE PROPERTIES OF SINTERED POROUS GLASSMEMBRANES
%T
Em-'
Figure 4. Infrared spectra of some sintered membranes after rehydration.
of the Na+ ion relative to that of the K f ion then results in a higher I(+selectivity as implied by eq 5. The enhanced K + selectivity observed in sintered porous glasses supports the above argument. The mobility of the divalent calcium ion is more likely to be affected by a smaller pore of increased charge density owing to its higher charge and the tenacity with which it is adsorbed on silica surfaces. Thus, although its mobility in aqueous solution approximates that of the sodium ion, it is the first to be affected by the silica pores. The changes in water content of these porous glass membranes with increased sintering are revealed by the spectra shown in Figure 4. The initial porous glass (membrane) is completely hydrated and causes complete absorption of radiation over the range 3750-3000 cm-l. As the sintering proceeds, the volume of membrane available to water is reduced, but membrane 3 still contains sufficient water to cause complete absorption over the spectral range shown. No major change in the K+-Na+ selectivity has occurred a t this point, although the Caz+ selectivity has been reduced and the charge density of the membrane increased to 0.13 N . Assuming a linear correlation between charge density andopore radius, the pore diameter at this stage is about 15 A. On sintering to membrane 4,however, a radical change occurs. Only a small quantity of water can enter the membrane, and a broad band at 3400 cm-l, typical of molecular water, is seen superimposed upon the hydroxyl groups of the glass (3660 cm-l). Further sintering of the glass causes dehydration until, with membrane 7 (KrqaxPot= S), no molecular water can enter the membrane, and a spectrum similar to that of a fused silica is obtained.2s Although water is effec-
2981
tively excluded from this material, ions may still enter it from solution, and large quantities of NH4+ ions can be readily detected in the glass after it has soaked for short periods of time in NH&l solutions. The "pore size" of this glass sample is therefore estimated to be about 3 d. I n view of the apparent discrepancies in some of the sintering conditions employed in Table I, some comments on the sintering of porous glasses seem in order. The sintering of many ceramicsz6and porous glass27~z8 proceeds via a viscous-flow mechanism, the rate of which is dependent upon the degree of hydroxylation of the surface and the quantity of impurities on the surface. The degree of sintering is therefore a function of the atmosphere in which the sintering occurs, the rate a t which the sample is brought to temperature, and the chemical constitution of the surface. It has been shown that considerable quantities of boric oxide can be present on the surface of porous glasses.29 The amount of this impurity varies with exact leaching procedure, and thus accurate reproduction of sintered samples is difficult and tends to be a hit-or-miss procedure. Data reported on the length change of porous glasses, as a function of temperature using a constant heating rate, reveal a sharp discontinuity around 1000" in the case of type A glass.28 This discontinuity occurs a t a much lower temperature for the type B glass. These temperatures also approximate those a t which the gross changes in K+-Na+ selectivity occur and a t which negligible water can enter the sample (ie., pore sizes of about 3 dl. It is suggested, therefore, that these temperatures correspond to minimum pore size; above these temperatures, the glass yields, and all traces of pores are lost in the final stages of sintering. Direct experimental evidence has been provided illustrating t;he effect of controlled pore size on the permselective and ion-selective properties of ionic membranes. In particular, the emergence of a relatively high E(+ ion selectivity in the presence of Na+ by a control of pore size is of some importance in biological applications. It provides direct evidence of a mechanism whereby biological membranes could produce changes in ionic specificity by changing only their physical shape and without invoking major chemical changes. (25) M. L. Hair, "Infrared Spectroscopy in Surface ChemistryqJ Marcel Dekker, Inc., New York, N. Y., 1967. (26) W, D. Kingery and M. Berg, J . A w l . Phys., 26, 1205 (1955). (27) T. H. Elmer, I. D. Chapman, and M. E. Nordberg, J. Phys. Chew., 66,1517 (1962). (28) T. Takamori and K. Iriyama, Ceram. Bull., 46,1169 (1967). (29) M. L. Hair and I. D. Chapman, J. Am. Caram. SOC.,49, 651 (1966).
Volume 78, Number 8 August 1068
