Equilibria, reactions and sintering in systems with iron oxide as one of the components

il i i i l i l l i i ^ l h i i i - i l n ^ i t 'till! i l l o rst O -4 O -o CD » -v ^ fs» / BIBLIOTHEEK TU Delft P 1224 8652 329278

EQUILIBRIA, REACTIONS AND SINTERING IN SYSTEMS WITH IRON OXIDE AS ONE OF THE COMPONENTS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL DELFT,OP GEZAG VAN DE RECTOR

MAGNIFICUS, DR.IR. C . J . D . M . VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP

WOENSDAG 2 JULI 1969 TE 1400 UUR.

door

PETRUS JOSEPHUS LOUISA REIJNEN Chemisch doctorandus

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR P R O F . DR. J . L . MEIJERING.

1

Contents Abstract

1. Introduction.

2. Equilibria between solids and oxygen in the t e r n a r y system MgO-FeO-Fe^O^.

3. The foi'mation of f e r r i t e s from the metal oxides. 4. Non-stoichiometry and sintering of ionic solids. Samenvatting, p^^^^^ , ^ . ^ ; . f ^ s ^ . ^ ^

Reprint " The-Termtt'y SystbirrMgO-FeO-Fe^Og, Philips R e s . Rpts. 23, 151 - 188, 1968.

Reprint " The Formation of F e r r i t e s from the Metal oxides " Science of C e r a m i c s (ed. G.H. Stewart ) 3, 245 - 261,

1967.

Reprint " Sintering Behaviour and M i c r o s t r u c t u r e s of Aluminates and F e r r i t e s with Spinel Structure with Regard to De-viation from Stoichiometry " . Science of C e r a m i c s (ed. G.H. Stewart ) 4, 169 - 188, 1968.

Reprint " Non-Stoichiometry and sintering of Ionic Solids " . Reactivity of Solids ( E d . J . W . Mitchell,

R . C . DeVries, R.W. Roberts and

Abstract 1. Introduction

During a chemical reaction, atoms molecules and ions a r e brough t o -gether. When a reaction takes place in the gaseous or liquid phase, this m a t e r i a l t r a n s p o r t on an atomic scale is easily to visualize since in these phases the a t o m s , molecules and ions a r e in random motion. In solid crystalline p h a s e s , however, the component particles vibrate around fixed positions and if these vibrations a r e the only modes of motion, m a t e r i a l t r a n s p o r t through a crystalline solid i s impossible. Assuming for a m o -ment that this is the c a s e , then a chemical reaction can take place only at the surface of a solid, provided that no protective product layer is formed. These surface reactions a r e generally not'considered a s solidstate r e a c t i o n s . A real solidsolidstate reaction proceeds by m a t e r i a l t r a n s -port through the solid. Schottky and Wagner ' in 1930 w e r e the first t o introduce the point defect concept to explain diffusion in crystalline solids. In this theory it is submitted that a small proportion of the lattice sites is not occupied and diffusion is explained by successive atom jumps into the vacant s i t e s or v a c a n c i e s . The vacancy type of disorder in a c r y s t a l -line solid i s usually r e f e r r e d to a s a Schottky d i s o r d e r . Previously in

2)

1926 it was suggested by Frenkel that in s u v e r bromide some of the silver ions occupy positions between the normal sites leaving an equal number of normal sUver s i t e s vacant. This Frenkel disorder gives r i s e to two modes of solid-state diffusion: jumps of interstitial ions or atoms into other interstitial positions and jumps of ions or atoms into vacan-c i e s . The theory of the defevacan-ct solid-state h a s been widely extended and applied to semi-conductors at the Philips Research Laboratories in the 3) first decade after the second world war by Verwey, Kroger, Vink et al . The formation of point defects i n c r e a s e s the energy or enthalpy of the solid but a l s o introduces a certain disorder which means an increase of entropy. The equilibriumconcentrationof vacancies is reached when (with constant T and P) the Gibbs free energy function has minimum value. Consider, for example, a crystal of N identical atoms and n vacancies

( N+n )' (n « N). The entropy of mixing is equal to k In '-JT^—/-' and the enthalpy of formation of the n vacancies is n < ^ h in which Ah is the

3

enthalpy of formation of one vacancy.

The dependence of the Gibbs free energy on n is thus expressed by

G(n,N) = G ( n = o , N ) + n - A h - k T In ^^,"^^/ " (eqn. 1)

The condition for equilibrium is ( O G/ O n) = 0 from which it is calculated that n/(N+n) = exp (-A h/kT)

The concentration of a t o m s , ions and defects is most adequately e x p r e s -sed by

Y = number of Y species _{number of sites available}

We therefore have C =exp(A h/kT)= exp(AH/RT) in which C is the v a -cancy concentration and A H is the enthalpy of formation of N vacan-c i e s . The thermodynamivacan-c potential of the vavacan-canvacan-cies ti follows from eqn. 1

v » o ( - ^ ^

*-* V V V

P . T , N (N is Avogadro's number).

In a s i m i l a r way it has been proved by Schottky and Wagner and m o r e 3)

recently in a m o r e extended form by Krbger and Vink ' that thermody-namic potentials can be given to all structuretflements of ionic c r y s t a l s including point defects. The law of m a s s action can therefore be applied t o equilibria between s t r u c t u r e elements if point defects a r e taken into consideration. As an illustration of such an equilibriimi let us consider a solid solution (s. s.) of F e „ 0 „ in MgFe.O . .

P a r t of the f e r r i c ions a r e reduced to f e r r o u s ions upon dissolution of Fe O and if the t e m p e r a t u r e i s increased or the p r e s s u r e of the oxygen is lowered further reduction takes place. This is expressed by

i F e g O g ( s . s . ) : ^ FcgO^ ( s . s . ) + i O^ (reaction 1). The thermodynamic potential of dissolved F e „ 0 in the spinel phase i s found from

''^^2°3 = ^"Fe^Og ^ ^T in [Fe^Ogj (eqn 2)

Upon dissolution Fe„0 is dissociated into independent moving ions and defects and since therefore Fe„0„ 1 is zero, eqn 2 makes no sense. In the spinel phase there are 3 cation sites available for every 4 anion sites. The dissolution of 4 molecules Fe_0„ into the spinel phase there-fore gives rise to one cation vacancy.

For ideal behaviour of the solid solution we have "^«203 = ^'^Fe^^ ^ 5^V + ^^O" =

c

2( o r +++1 1 o r -\ o

M j,g+++ + RT In [Fe J ) + i ( My + ^'^^'^ T c 1 ^ ^ % "

The concentration of oxygen ions is equal to unity as practically aU available oxygen sites are occupied by oxygen ions. Likewise we have '^FegO^ = 2 (/^°^+++ + RT In j r e " ^ ! ) + M°g++ + RTln JFe^^J + 4 ^ ° -The equilibrium concentrations of reaction 1 are determined by the condition M I M Q - 3 ^ = 0

*®3 4 4 " 2 2 *^2 3

Because /u„ = jip, + RT In p_ it follows for the equilibrium concen-" 2 concen-" 2 2

trations at a given temperature V4

RT In,;—.. .,r 2 - T 3 / = constant

[Fe--][vJ/8

This result can also be abtamed by applying the law of mass action to the equilibrium equation :

F e * ^ + I V + \ O"^ Fe~^ + ^ „ (reaction 3)

which is in fact the same as reaction 1, except that it is now given in terms of the free-moving structure elements. Reaction 3 satisfies three conditions

5

1) Conservation of atoms and ions (mass balance) 2) Conservation of charge (charge balance)

3) Conservation of the c h a r a c t e r i s t i c ratio (3/4) of cation and anion s i t e s (site balance).

The third condition is n e c e s s a r y because point defects contribute to the entropy of mixing and enthalpy of the system and therefore cannot be omitted in thermodynamic calculations on crystalline solids.

The role of point defects in t h r e e main a r e a s of solid-state chemistry is discussed in this a b s t r a c t . These a r e

1) Equilibria between oxides and oxygen ( section 2) 2) Reactions between oxides (section 3)

3) Sintering of oxides ( section 4)

2 Equilibria between solids and oxygen in the t e r n a r y system MgO-FeO-Fe„Og.

In the a r t i c l e , " Phase equilibria in the t e r n a r y system MgO-FeO-Fe 0 „ ", (Philips R e s . Repts. 23, 151 - 188, 1968) a t h e r m o g r a v i m e t r i c study of this system is described. The main r e s u l t s of this study a r e a detailed phase diagram and some conclusions regarding the defect s t r u c t u r e of the cation deficient magnesio-wustite phase and the spinel phase.

The experimental technique and the way in which the experimental data have been evaluated will now be shortly outlined. P r e - r e a c t e d powder m i x t u r e s of MgO and Fe^O with the composition (l-x)MgO +xFeO.. ^ a r e heated stepwise up to 1400 C on a "Stanton" thermobalance in the constant oxygen partial p r e s s u r e of a i r . In this way the oxygen l o s s e s of samples which have reached chemical equilibrium can be determined a s a fimction of t e m p e r a t u r e . The composition after a certain weight l o s s i s represented by

(1-x) MgO + (x-s) FeO. c + s FeO. The composition p a r a m e t e r s c a n be calculated from the weight l o s s of the sample: s = A g / g ' M / 8 . In this f o r m u l a e g is the weight l o s s , g the initial weight of the sample and M i s the weight of 1 mole of (1-x) MgO +xFeO j . . The weight of a sample is approximately 3 g r a m s and the weight l o s s e s a r e determined with an accuracy of 0.1 m g .

The relation between weight l o s s and t e m p e r a t u r e m e a s u r e d at constant oxygen partial p r e s s u r e , for samples with various x values can m o s t adequately be represented within an equilateral triangle. In this way an isobaric c r o s s - s e c t i o n of the complete three-dimensional phase diagram is obtained. In such a c r o s s - s e c t i o n each point r e p r e s e n t s a composi-tion with a c e r t a i n x and s value. The locus of points for composicomposi-tions with the same X value is a line parallel to the base of the triangle and the l o -cus of points for compositions having the same s value is a line parallel to one of the sides of the triangle (fig. 1).

FeO VaFeFejO^ i^l^Oj F i g . 1 Compositional diagram of the t e r n a r y system M g O F e O F e „ 0 . To fix a composition two c o m -position p a r a m e t e r s a r e required, e . g . x and s in

(l-x)MgO+s FeO + (x-s) FeOj^ g. The M g / F e ratio is determined by the p a r a m e t e r x and the ferrous con-tent by the p a r a m e t e r s.

7

By connecting the points which represent compositions at the same tem-perature, a set of isotherms is obtained. From the phase rule it can be deduced that in the regions with two solid phases, the isotherms are straight lines (fig. 2).

MgO -X= 0-090 :-^= 0-166 -x=0-230 -X= 0-286 ^-X= 0-604-/--X^O-652 / MgfT2 0^ '-X=0-670 3 7—X= 0-683 K.—X= 0-702 —X= 0-735 \—x=0-802 -X=0-902 FeO Fe^ _F^03 2

Fig. 2 Isobaric cross-section of the phase diagram of the system MgO-FeO-Fe O at the constant oxygen partial pressure of air. The isotherms are straight lines in the regions with two solid phases and are cur-ved in the regions where only one solid phase is pre-sent.

T h e r e a r e t h r e e solid phases in the diagram: the cubic magnesio-wustite phase (MgO-FeO), the cubic spinel phase (MgFe„0 - F e F e O ) and the hexagonal hematite phase (Fe O ). The compositions in brackets a r e stoichiometric because the cations and anions occur in a ratio of simple n u m b e r s . This ratio is 1/1 for the magnesio-wustite phase, 3/4 for the spinel phase and 2/3 for the hematite phase. It follows from the phase diagram that quite l a r g e amoimts of F e _ 0 dissolve in the magnesiowustite phase and the spinel phase a s well. These nonstoichiometric s o -lid solutions have an e x c e s s of cation vacancies.

^

5/

FeO _{-^-Temperature}

Fig. 3 P h a s e diagram with the dissociation path of a composition conisisting of MgFe O + excess Fe O . The nmnbers on the dissociation path correspond with the points on the weight l o s s v s t e m p e r a t u r e curve of this composition.

Fig. 3 shows what happens to a mixture of MgFe O + Fe O upon' hea-ting in a i r . The numbers on the weight l o s s curve correspond to the numbers in the phase diagram.

The equilibrium between the cation deficient spinel and oxygen gas is r e -p r e s e n t e d by

9

F e " ' ^ + | v +\o~ ^ Fe"*""^ + 7 O 0 (reaction 3) o C ^ 4 Z

If no a s s o c i a t e s of ions and vacancies a r e formed the concentration of ions and defects can be expressed in t e r m s of x and s which a r e known from experiment. The overall composition in t e r m s of x and s is (1-x) MgO + (x-s) FeO ^ + sFeO. The number of oxygen ions p e r mole of

1. o

formula unit is equal t o ( l + - x - - s ) N and a s practically all oxygen £1 di O

s i t e s a r e occupied in the cation-deficient region of the spinel phase, this

o

is also the number of oxygen s i t e s . The number of cation s i t e s i s — of 3 3 3

this number: ( T + ' S ' X - — s ) N . The number of cations is N and

M 8 8 ' o o 3 3 1

the number on empty cation sites is consequently ( ^ x - • 5 ^ S - T ) N . The concentrations of cations and cation vacancies a r e expressed a s fractions of the available sites which they occupy. The equilibriimi con-stant of reaction 3 expressed in t e r m s of x and s thus r e a d s _{3 1}

ÏÏ 4

^^3

/ s i f6+3x-3s\° ^

When log K„ is plotted v e r s u s T^ a straight relationship i s obtained in the cation-deficient spinel region (fig. 4).

P l o t s of this kind have been made for a l a r g e nimiber of compositions and the straight lines obtained in this way practically coincide. F r o m the best fit through all the points in the log K„ vs — plots it follows that

4800

log K = 2.93 — = — from which it i s calculated that the enthalpy change for the reaction is 21. 9kcal. In the spinel lattice one third of the cations a r e on t e t r a h e d r a l or A s i t e s and two t h i r d s on octahedral or B s i t e s . The cations in the t e t r a h e d r a l sites a r e surroimded by four and those in the octahedral s i t e s by six oxygen ions. The energy of a certain cation type depends on whether it i s on an A site or a B site and consequently t h e r e is a site p r e f e r e n c e , especially at low t e m p e r a t u r e s .

In reation 3 this site preference is neglected. If we take it into account, reaction 3 must be replaced by a reaction in which the r a t i o of 2:1 for cations on B and A s i t e s respectively is p r e s e r v e d , e . g .

The concentrations of ions and vacancies on A and B sites a r e d e t e r m i -ned by t h r e e p a r a m e t e r s kj^.kg.and k „ , which a r e the equilibrium con-stants of the distribution equilibria:

F e ^ . M g ^ F e r M g

--r--B^-c--r

^ C + ^3

^ F e ^ +

V^

h

F o r a given set of k , k k values, the equilibrium constant for r e a c t i o n 4 can be expressed in t e r m s of x and s . As an example let us choose kj= 1, k 2 = 0 a n d k = 0.

This means that ferrous ions and cation vacancies a r e exclusively on B

I I ] 1 I

sites and that Mg and F e have no preference for A or B s i t e s . The value of the equilibrium constant for reaction 4 in this case becomes

3 1 „ _ / s \ /6+3x-3s\8 4 '^4"lï^,IU3x-3s-2) • POg • 1 /_1\ ^ I 8( 1-s ) \ eqn. 4 (2+x-s) ^ (6-7S-X) 8

The first p a r t of this expression is equal to K and therefore the second

o

p a r t e x p r e s s e s the effect of this specific cation distribution on the equi-librium constant. The logarithm of this second t e r m depends only slight-ly on the value of s (or temperature) and consequentslight-ly when K i s the c o r r e c t expression for a certain f e r r i t e , the log K„ v s 75; plot wUl a l s o

r

A vacancy i s surroimded by 6 or 4 negative oxygen ions which is e n e r g e -tically unfavourable. One might expect part of the negative charge to be located elsewhere in the l a t t i c e , implying that the ferrous content in-c r e a s e s : V + Fe > V + Fe .

11

Assuming that all cation vacancies have one positive charge and neglec-ting the effect of cation distribution, the equilibrium constant in t e r m s of X and s r e a d s :

5 3 1 ^ (3x+5s-2 \ ^ /6+3x-3s \ ^ . p ^ (eqn. 5) % \5x-5s+2 I • l 3 x - 3 s - 2 ) ^ 2

The log Kg vs =, plot, however, is a curved line and the experimental data, therefore, a r e in favour of the model in which the vacancies a r e uncharged (fig. 4).

F r o m the t h e r m o g r a v i m e t r i c study of the magnesio-wustite phase it h a s been found that the excess of negative charge in the vicinity of a cation vacancy is compensated by the presence of f e r r i c ions on the nearby cation s i t e s . Similar formation of a s s o c i a t e s of highly charged cations and vacancies will most probably occur in the spinel phase.

The formation of a s s o c i a t e s is e x p r e s s e d for example by

V + 2 F e ' ' " ^ . ^ fv + 2Fe"'"*^l in which c L c J

V + 2Fe J r e p r e s e n t s an associate of a cation vacancy and two f e r r i c ions.

The equilibrium constant now r e a d s

1

K6 = ( ^ T " ) - Po„ <^'l'^-^) (2+x-5)^ (3x-3s-2)8 2

By comparing the expression for K„ and Kg it is seen that they both con-tain the factors

fFe^l

3— = " g"*— . These two factors mainly d e t e r m i -(3X-3S-2) 8 [ v ] 8

+ V20" :s=!^f^ +%02

0.750 0.700 0.650

Fig. 4 P l o t s of log K VS l / T for the s a m e compositins (x=0.722). K has been calculated for t h r e e models of the defect s t r u c t u r e .

Kg: Fe 3 V 1 ^ -+ 8 <= -+ ^ :Fe

*h

No distinction is made between ions on A and B s i t e s . The vacancies a r e uncharged and t h e r e is no association of ions and defects. Same model for the defect s t r u c t u r e a s for K but with charged v a

-+ + • + + + •*

cies: Fe + V —> V + Fe . c c

Same model for the defect s t r u c t u r e a s for K„ but with a s s o c i a t e s of

o I t I ^

13

In f i g . 4 , log K , log K and log K a r e plotted v e r s u s ; j , . The e x p e r i -mental data a r e obtained from a composition with x =0, 722. Both log K„ and log K„ a r e straight lines within experimental e r r o r and we therefore cannot distinguish between models with and without a s s o c i a t e s in this experiment.

3 . The formation of f e r r i t e s from the metal oxides. ( Science of C e r a m i c s 3 , 245-261, 1967.)

A mixture of oxides usually r e a c t s upon heating to form new solid phases or solid solutions.

The reaction proceeds by diffusion of atoms or ions through the solids until the compositional gradients within each solid phase have vanished. In ionic solids the diffusional fluxes of ions a r e strictly related to each other because the total electric charge at any place at any time must r e -main z e r o . The diffusion constants of oxygen vacancies, cation vacancies and i n t e r s t a t i a l s at a given t e m p e r a t u r e a r e mainly determined by the geometry of the lattice and the charge of the ions. For example, when the oxygen ions a r e much l a r g e r than the cations and form a close-packed oxygen l a t t i c e , the diffusion constant erf oxygen vacancies is much lower than that of cation vacancies. Usually these oxides form cation-deficient solid solutions in a l a r g e compositional range whUe the extent of the anion-deficient region i s r a t h e r s m a l l . An illustration of this fact h a s been given in section 2 in the phase diagram of the system MgO-FeO-F e _ 0 : the solubility of MgO-FeO-FeO or MgO in the spinel phase or hematite phase i s low but l a r g e amounts of F e . O dissolve in the m a g n e s i o w u s -tite phase and the spinel phase to form cation-deficient solid solutions. As an example of a solid-state reaction let us consider the formation of M g F e . O in a mixture of Fe O and MgO p a r t i c l e s .

At the contact points of the MgO and Fe^O p a r t i c l e s the spinel phase is nucleated and then the reaction proceeds further by diffusion of ions through the product l a y e r . The composition of the spinel phase at the s p i n e l - F e „ 0 boundary is different from that at the spinel-MgOboudary, a s can be seen in the isothermal c r o s s - s e c t i o n of the M g O - F e O - F e „ 0 phase diagram (fig. 5 ) . '

/sobar

Isothermal cross-section of the ternary system

MgO-FeO-Fe^Os

'/2Feiq3

Charge balance:2If:^*+3If^**=2Iii^-*-Sjte balance; J ^ / + + I F ^ * +Iy^=I/^+ *

F i g . 5 Reaction schema for the solid-state reaction MgO + FCgOg- • MgFe^O^.

_..i>4,

3 /fe 7i,^Y-^Ü-^3^f ^<^yir^3

Due to the compositional gradient in the spinel layer, solid-state diffu-sion of ions through the solid takes place and the amount of spinel phase gradually i n c r e a s e s . The product layer is cation deficient with the ex-ception of a very thin region at the MgO-spinel boudary. In the broad cation-deficient region the number of oxygen vacancies is extremely low and consequently the oxygen-vacancy gradient i s z e r o . There i s , however, a l a r g e gradient in the M g ^ , F e ^ , F e " ^ and V^ concentration in the cation deficient region. The reaction thus proceeds by counter-diffusion of F e ^ ions, Fe"*^ ions and V on the one hand and Mg ions on the other, while the oxygen ions remain at their s i t e s .

The diffusional fluxes of ions and vacancies a r e related to each other in such a way a s to maintain the charge and site balance

? 15

z

- ^ ^ F e " * " ^ ^ ^ ^Fe"*^ ^ - ^ W ^ (charge balance )

< - < »•

I +++ + I ++ + L = L , ++ , .* u 1 and Fe Fe V Mg (site balance)

The diffusion of ferrous ions through the product layer means that at the spinelhematite boundary, ferrous ions a r e continuously formed by r e -duction of f e r r i c ions and that on the other side of the l a y e r , at the spinel -magnesia boundary, the ferrous ions a r e oxidized to f e r r i c ions. P a r t o f the m a t e r i a l t r a n s p o r t during the reaction thus takes place via the gas phase. The mechanism of counter-diffusing cations in s t r u c t u r e s where the anions form a close packed lattice was proposed by Koch and

4)

Wagner ' in 1936 and is usually r e f e r r e d to a s the Wagner diffusion or reaction mechanism.

4 Non-stoichiometry and sintering of ionic solids. ( - S c i e n c e of C e r a m i c s 4 , 169-188, 1968.) ( - R e a c t i v i t y of solids ^ , 99-114, 1969.)

The densification of a powder compact by sintering t a k e s place by m a t e -r i a l t -r a n s p o -r t f-rom g-rain bounda-ries to p o -r e s , the d-riving fo-rce fo-r this p r o c e s s being the d e c r e a s e of surface free energy.

P o r e s lying on grain boundaries can in principle be annihilated by g r a i n boundary and surface diffusion of atoms and ions. P o r e s which a r e c a p -tured within the g r a i n s will always meet a moving grain boundary during the sintering p r o c e s s and, consequently, a powder compact can, from a theoretical point of view, attain full density by grain boundary and s u r -face diffusion alone. T h e r e i s , however, sufficient experimental evidence that the predominant diffusion p r o c e s s during sintering is volume diffu-sion and it is therefore interesting to evaluate the dependence of volume diffusion on the defect s t r u c t u r e of the solid. In section 3 we have seen that for solid-state reactions between oxides it is sufficient that the cations in the solid a r e mobile but in the sintering p r o c e s s both cations and anions must be transported a s the p o r e s must be filled up with e l e c -trically neutral m a t e r i a l . The vacancy flux emitted by a spherical pore within a solid MO i s calculated to be

$

TT-yn/kT

D C

c c D C o o

in which "X' i s the surface free energy p e r unit a r e a , n i s the volume of a vacancy pair V + V , kT has its usual meaning, D^ and D^ a r e the diffusion coefficients of cation vacancies and anion vacancies, and C^ and C a r e the concentrations of these vacancies. The maximum flux is

c

obtained when D C = D C and whenever D > D„ it follows that C

C C o o C O o

must be l a r g e r than C for fast sintering.

The calculated vacancy flux d e c r e a s e s slowly when the anion vacancy concentration is further increased but the vacancy flux drops drastically when the composition shifts towards the cation-deficient region (fig. 6).

L ' ' 1)'^ 5.10-^ K)-^ XM (1-x)M0+xL^0 Do/Dc =10'^ A §stoich 1 1 1 «-^ 5.10-* \ —^1/3y(=-X)

X(l-Y)MO+yA;^^0

V^

Fig. 6 The vacancy flux emitted by pore in an ionic solid MO a s a function of the ai

dissolved in the host lattice.

17 .

Indeed it is found for many oxide s y s t e m s for whichD > D that the anion-defi-cient m a t e r i a l s have far better sintering p r o p e r t i e s than the cation-de-ficient m a t e r i a l s . Some sintering densities of NLAl-O containing ex-c e s s NiO or exex-cess A1„0 a r e given in fig. 7.

£t O % Porosity

I 33

I

Fig. 7 Sintering of non-stoichiometric Ni aluminates.

The solubility of NiO in the spinel phase is low, but even a small d i s s o lution i s sufficient to i n c r e a s e the oxygen vacancy concentration d r a s t i c a l -l y . The sintered samp-les containing excess NiO have porosities of 1% in this experiment . Quite l a r g e amounts of A1„0 can be dissolved in the spinel phase making this phase markedly cation-deficient. This makes volume diffusion of oxygen practically impossible, and this is reflected in the very high porosit'/^s of these s a m p l e s , which were sintered under conditions s i m i l a r to those in which the anion-deficient samples were sintered.

Oxide s y s t e m s containing cations which occur in two valencies a r e of spe-cial i n t e r e s t because the defect-structure and the sintering behaviour will depend on the partial p r e s s u r e of oxygen in this c a s e . This i s nicely d e -monstrated in the sintering behaviour of f e r r i t e s containing excess F e „ 0

When these m a t e r i a l s a r e sintered in oxygen or air the material is cation deficient during sintering and both the sintering r a t e and sintering density a r e low, b u t if the same m a t e r i a l s a r e sintered in a low partial p r e s s u r e of oxygen, e . g . in nitrogen with a small, controlled amount of oxygen, they a r e sintered practically to full density. The m i c r e s t r u c t u r e of the m a t e r i a l sintered in oxygen (fig. 8a) i s quite different from the m a t e r i a l sintered in nitrogen (fig. 8b).

Fig. 8 The m i c r o s t r u t u r e of cationdeficient f e r r i -t e s (a) (large p o r e s on -the grain boundaries and g r a i n s absolutely pore free) and the m i c r e s t r u c t u r e of anion-deficient f e r r i t e s (b) (small p o r e s within the g r a i n s and low p o r o s i t y ) .

19

In the first m i c r o s t r u c t u r e the g r a i n s a r e pore-free but quite l a r g e p o r e s a r e found on the grain boudaries while in the second m i c r o s t r u c t u r e the p o r e s a r e much s m a l l e r and a r e dispersed throughout the m a t e r i a l . This difference will now be explained. During sintering the grain bounda-r i e s move and the avebounda-rage g bounda-r a i n size i n c bounda-r e a s e s . As long a s they a bounda-r e vebounda-ry s m a l l the p o r e s a r e swept along by the moving grain boundaries, because m a t e r i a l t r a n s p o r t by volume diffusion from one side of the pore to the other is sufficiently fastfor small p o r e s . D u r i n g t h i s p r o c e s s , however, the number of p o r e s d e c r e a s e s and the average pore size i n c r e a s e s . The l a r g e r p o r e s can no longer follow the moving grain boundaries and a r e left behind within the g r a i n s . This is what normally happens with p o r e s in a sintering compact. ( fig. 8b ).

In cation-deficient f e r r i t e s the p o r e s apparently move along with the grain boundaries even when they a r e quite l a r g e , for we see that the g r a i n s a r e absolutely p o r e - f r e e and l a r g e p o r e s a r e found on the grain boimdaries. Material t r a n s p o r t from one side of the pore to the other m u s t , t h e r e f o r e , takefplace quite readily in these m a t e r i a l s . This can be satisfactorily explained by the defect s t r u c t u r e of the solid. The m a t e r i a l is cation deficient and consequently diffusion of cations readily o c c u r s . To maintain the charge balance, counter-diffusion of f e r r i c and f e r r o u s ions must take p l a c e .

1^' ' •• ••3-1- — * 2Fe + V c SFe"^ or — * • V c - * - — • « — •

[Fe^+3e

The transformatiCHi of oxygen ions to oxygen a t one side of the pore is a source for electrons and the r e v e r s e p r o c e s s at the other side an e l e c -tron sink.

The t r a n s p o r t of oxjrgen thus takes place via the gas p h a s e . The complete p r o c e s s in shown in fig. 9.

Fig. 9 A pore on a moving grain boundary in a cation-deficient f e r r i t e . The two pore surface have different radii of curvature. This gives r i se to compositional gradients and diffusional fluxes by which it i s p o s -sible that the pore is swept a l o i ^ by the moving grain boundary. The cations diffuse rapidly through the cation deficient m a t e r i a l while the oxygen is t r a n s p o r t e d from one pore surface to the other via the gas p h a s e .

L i t e r a t u r e

1) C. Wagner and W Schottky, Z physik. C h e m . , (B) 11, 163, (1931)

2) J . Frenkel, Z. P h y s . Chem. 35, 652, 1926. 3) E . J . W . Verwey, P . W . Haaijman and F . C . Romeijn.

Chem. Weekblad 44, 705, 1948.

E . J . W . Verwey, P . W . Haaijman., F . C . Romeijn and G.W. van Oosterhout, Philips R e s . R e p t s . , 5, 173, 1950. F . A . K r o g e r , H . J . Vink and J . van den Boomgaard, Z. phys. Chem 203, 1, 1954.

F . A . Kroger and H . J . Vink, Solid State P h y s . 3 , 307, 1956, Academic P r e s s New York.

F . A . K r o g e r , F . Stieltjes and H . J . Vink, PhUips R e s . R e p t s , 1 4 , 557, 1959.

21

Samenvatting

Dit proefschrift beschrijft de rol van puntdefekten in vaste-stof evenwichten, vastestof r e a k t i e s en sinteren. Een puntdefekt is een fout op a t o -m a i r e schaal in een k r i s t a l r o o s t e r . Het kan een onbezette r o o s t e r p l a a t s zijn en wordt dan een vakature genoemd ofwel het is een ion of atoom op een plaats die in het ideale r o o s t e r leeg is en heet dan een interstitieel ion of atoom. Deze puntdefekten zijn noodzakelijk voor de beweging van ionen of atomen door het r o o s t e r . Het aantal pimtdefekten in een k r i s t a l -r o o s t e -r neemt ste-rk toe met de t e m p e -r a t u u -r hetgeen een uiting is van een algemeen waar te nemen tendens in de natuur dat bij stijging van t e m p e -ratuur de wanorde van een verzameling atomen toeneemt ( v . g . l . over-gang vaste stof - vloeistof - damp). In een koperkristal is bij ~ 1000 C 1 op de 10. 000 roosterplaatsen onbezet. Zou men de atomen van de bui-tenste lagen van zo'n koperkristal kunnen m e r k e n , dan zal men na vol-doende lange tijd kunnen waarnemen dat de gemerkte Cu atomen door het gehele k r i s t a l v e r s p r e i d zijn. Dit geschiedt door vastestof diffusie w a a r van het mechanisme is dat een koperatoom in een aanliggende lege r o o s

-s t e r p l a a t -s -springt en daarbij een lege roo-sterplaat-s achter laat waarin weer een ander koper atoom springt enz. Uit dit sterk gesimplificeerde beeld is het r e e d s duidelijk dat het aantal en soort puntdefekten in een vaste stof s t e r k de reaktiviteit van de vaste stof zal bepalen.

- Vaste-stof evenwichten.

De evenwichten bij hoge t e m p e r a t u u r in het t e r n a i r e systeem MgO-FeO-Fe-O zijn bestudeerd met behulp van een thermobalans waarvan de werking v e r d e r o p zal worden uiteengezet. De M g / F e verhouding van een bepaalde samenstelling i s bekend door de inweging van de hoeveelheid MgO en Fe „O en kan door chemische analyse geverifieerd worden. Gaat men zo'n samenstelling verhitten b . v . in lucht dan zal geleidelijk de vaste stof zuurstof gaan afgeven waarbij een gedeelte van het d r i e -waardig ijzer tot twee-waardig ijzer gereduceerd wordt. De optreden-de gewichtsverliezen kan men a l s functie van optreden-de t e m p e r a t u u r nauwkeurig bepalen met een thermobalans. In dit instrument hangt het p r e -paraat met een hitte-vaste draad aan de a r m van een gevoelige balans.

Het p r e p a r a a t zelf bevindt zich in het midden van een buisoven op hoge t e m p e -r a t u u -r . Uit de aldus ve-rk-regen gegevens aan een g-rote s e -r i e samenstelling-en kan msamenstelling-en het fazediagram van het systeem afleidsamenstelling-en. Het fazediagram laat op overzichtelijke wijze zien welke vaste fazen bij een gegeven t e m -peratuur en druk met elkaar in evenwicht zijn en wat de samenstelling van die vaste fazen i s . Uit de experimentele gegevens kan men ook v e r -gaande konklusies trekken over de defekt-struktuur van de vaste stoffen. Het blijkt bijvoorbeeld dat puntdefekten in de evenwichtsbeschouwingen dienen te worden opgenomen, hetgeen een experimentele bevestiging is van wat op grond van theoretische beschouwingen kan worden verwacht. In het t e r n a i r e systeem M g O - F e O - F e „ 0 komt de samenstelling MgFe^O^ voor die met F e „ 0 . een kontinue r e e k s mengkristallen met spinelstruktuur v o r m t . In deze spinelfaze lossen grote hoeveelheden F e „ 0 op t e r

-ii o

wijl de oplosbaarheid van MgO in de spinel faze zeer gering i s . De op-lossing van F e „ 0 „ in de spinelfaze bevat een groot aantal lege

kationplaat-sen of kationvakatures.Dit z i e t m e n in door Fe_0„ te schrijven als Fe» ,„O. ofwel a l s F e » / „ V. /„ O. waarin V een lege kationplaats voorstelt. In de spinelfaze zijn namelijk voor elke 4 zuurstofplaatsen, 3 kationplaat-sen beschikbaar ( b . v . MgFe „O. ). Een gedeelte van het Fe „O dat in de spinelfaze oplost wordt bij dit oplossen gereduceerd tot Fe_0 en hierbij ontstaan geen kationvakatures. Door nu de temperatuur te verhogen of de zuurstofdruk te verlagen wordt het opgeloste F e ^ O . v e r d e r gereduceerd tot F e „ 0 in opgeloste v o r m . Dit wordt beschreven door het evenwicht.

3 ^ \ •5 F e _ 0 „ (vaste oplossing) ^ - F e „ 0 . (vaste oplossing) + ~fi^

Bedenkt men dat het oplossen van Fe O extra kationvakatures

introdu-it O

ceerd dan kan men het evenwicht beter weergeven door de vergelijking

23

In de vaste oplossing zijn de ionen en puntdefekten onafhankelijke s t r u k -tuurelementen en is de wet van de chemische massa-werking derhalve van toepassing op bovenstaande vergelijking a l s deze in ionen en puntde-fekten wordt weergegeven.

Fe + 3/8V, + 1/2 O ^ = ± Fe + 1/40

De evenwichtskonstante van deze reaktie kan men nu uitdrukken in de s a -m e n s t e l l i n s p a r a -m e t e r s x en s: (1-x) MgO + sFeO + (x-s) F e O , ^. De p a r a m e t e r x volgt uit de M g / F e verhouding die door chemische ana-lyse bepaalt kan worden en de p a r a m e t e r s , die de hoeveelheid f e r r o aangeeft, volgt uit de experimentele gegevens met de thermobalans v e r k r e -gen. De uitdrukking voor K wordt 3 -^

/ s \ /6+3X-3S \ 8 4 I x - s ) \^3x-2-3s ) ' PO2

Als log K tegen 7=, wordt uitgezet verkrijgt men een lineair verband:

log K = 2.93 —— . Hieruit volgt dat de enthalpieverandering A H voor de r e a c t i e 21.9 kcal bedraagt. Kent men aan de vakatures een positieve lading toe (Fe + V, •—> V, + Fe ) dan wordt daardoor de uit-drukking van de evenwichtskonstante K .,

/ 3 x + 5 s - 2 \ ^ / 6 + 3 x - 3 s \ 3 i l5x-5s+2y ** • l 3 x - 3 s - 2 / ^ " A

1

Het blijkt echter dat nu het verband tussen log K en t ; niet m e e r lineair is waaruit volgt dat de vakatures ongeladen zijn. Toch is het een voor de hand liggende gedachte de vakatures een positieve lading toe te kennen omdat een lege r o o s t e r p l a a t s omgeven i s door 4 of 6 zuurstof ionen z o dat t e r plaatse van de vakature een overschot aan negatieve lading a a n -wezig i s . Uit soortgelijk onderzoek v e r r i c h t aan de magnesio-wustiefaze Uijkt echter dat in deze faze de ladingskompensatie wordt verkregen door-dat de naburige roosterplaatsen van de vakature door ferri-ionen zijn bezet b . v .

waarin N. + 2 Fe een associaat voorstelt bestaande uit een kationvakature en twee naburige ferri-ionen.

Nemen we aan dat in de spinelfaze een soortgelijke associaatvorming optreedt dan wordt daardoor de waarde van de evenwichtskonstante

.4s 1/4 (2+x-s)^/^ . (3x-3s-2)^/^ ' \

Binnen de meetfout is ook voor deze uitdrukking de log K vs 7=, grafiek een rechte lijn.

Houdt men rekening met het feit dat in het spinelrooster twee typen roosterplaatsen voor kationen beschikbaar zijn, t e t r a e d e r of A plaatsen en octaeder of B plaatsen dan vindt men, afhankelijk van de verdeling van de kationen over deze plaatsen, een s e r i e uitdrukkingen voor de even-wichtskonstante die alle een lineair verband geven a l s men log K tegen ^ uitzet voor dezelfde s e r i e meetgegevens. Men kan dus niet alle finesses van de defekstructuur langs t h e r m o g r a v i m e t r i s c h e weg onderscheiden, m a a r de verwachting is wel dat n a a r m a t e betere thermobalansen beschik-baar komen, waarmee men nauwkeuriger de gewichtsverliezen kan bepa-len, m e e r gegevens over de defekstruktuur van vaste stoffen langs deze weg verkregen kunnen worden.

- Vaste-stof

reakties-De kennis van de evenwichten in het t e r n a i r e systeem MgO- FeO - Fe^Og is de basis geweest voor de bestudering van de vaste-stof reaktie tussen MgO en Fe O . In dit poedermengsel zullen op vele plaatsen de MgO en Fe O

'^ -j 2 3

deeltjes tegen elkaar liggen en op deze plaatsen zal de nieuw te vormen faze gekiemd moeten worden alvorens de reactie kan beginnen. Als dit eenmaal gebeurd is zullen de reactanten MgO en FCgO- spoedig daarna van elkaar gescheiden zijn door een dunne spinellaag en kan de reactie v e r d e r alleen verlopen door vaste-stof diffusie door de gevormde spinel-laag. Deze spinellaag zal door het oplossen van F e „ 0 een grote hoeveel-heid lege kationplaatsen bezitten waardoor de diffusie van kationen z e e r

I I I

25

F e ionen in de ene richting door de spinellaag bewegen en Mg ionen in de andere richting. De zuur stof ionen blijven op hun p l a a t s . Door de voortdurende vorming van Fe ionen aan de spinel-Fe 0 „ grens,waarbij zuurstof vrijkomt, en door oxidatie van deze Fe ionen aan de spinel-MgO grens,waarbij zuurstof wordt opgenomen,verloopt de reaktie a l s het ware gedeeltelijk via de gasfaze.

- Sinteren

De invloed van de defektstruktuur op het sintergedrag van ionogene v e r -bindingen is in het l a a t s t e deel van het proefschrift beschreven. Bij een v a s t e - stof reaktie tussen oxiden is het voldoende dat de kationen een grote beweeglijkheid hebben m a a r tijdens het sinterproces moeten zowel

kationen als anionen g e t r a n s p o r t e e r d worden omdat de poriën opgevuld m o e -ten worden met ongeladen m a t e r i e . In ionogene verbindingen waarin de g r o t e r e anionen dicht tegen elkaar liggen springen de anionen a a n m e r k e -lijk moei-lijker in een lege r o o s t e r p l a a t s dan de kationen. Uit bereke-ningen blijkt dan ook dat het s i n t e r p r o c e s sneller zal verlopen a l s het m a t e r i a a l een zodanige afwijking van de stoichiometric gegeven wordt dat extra anionvakatures aanwezig zijn. Voor een oxide MO waarin alleen kation-en anionvakatures als puntdefekten voorkomen is berekend dat de flux van zuurstofionen kationen in de richting van een porie gelijk is aan

i>

J _ + J _

D C D C o o c c

Hierin is $ d e ionenflux, r de oppervlakte energie p e r opp. eenheid van het porie oppervlak, ü het volume van een molecuuul m a t e r i e MO, k i s de konstante van Boltzmann, T i s de temperatuur in K, D en D zijn de diffusie konstanten van zuurstofvakatures en kationvakatures, C en C zijn de concentraties van deze v a k a t u r e s .

De maximale flux wordt verkregen a l s D C = D C en a l s dus, D > D

" o o c c c o moet C < C opdat het m a t e r i a a l snel kan sinteren.

c o *^

Deze theorie i s nog uitgewerkt voor m e e r complexe defektstrukturen zoals veroorzaakt door de aanwezigheid van interstitiele kationen en

aan-wezigheid van kationen m e t m e e r dan een valentie.

De juistheid van de theorie is door vele experimenten bevestigd.

Een ander aspekt van het sintergedrag is de keramische mikrostruktuur. Deze mikrostruktuur loopt voor verschillende materialen nogal uiteen. In aniondeficiente f e r r i e t e n worden alleen z e e r kleine poriën door de bewegende k o r r e l g r e n z e n meegenomen terwijl g r o t e r e poriën in de k o r -r e l s achte-rblijven. In kationdeficiente fe-r-rieten kunnen ook g-rote po-riën nog door een bewegende k o r r e l g r e n s worden meegenomen. De m i k r o -struktuur van aniondeficiente ferrieten vertoont dan ook een groot aantal kleine poriën in de k o r r e l s , terwijl in de mikrostruktuur van kationde-ficiente ferrieten een klein aantal grote poriën voorkomen en wel uit-sluitend op de k o r r e l g r e n z e n (zie fig. 8 van de engelse samenvatting). Dit uiteenlopende gedrag kan op grond van de verschillen in de defektstruk-tuur van beide m e t e r i a l e n v e r k l a a r d worden.

R660 Pliilips Res. Repts 23, 151-188, 1968

PHASE EQUILIBRIA IN THE SYSTEM MgO-FeO-Fe^Oj

by P. REIJNEN

.\bstract

The equilibria between solid phases and oxygen in the system MgO-FeO-FejOj have been studied thermogravimetrically at the constant oxygen pressure of air and temperatures up to 1400 °C. The results are condensed in a detailed phase diagram. The spinel phase in this diagram has a large extension in the cation-deficient region. The extension in the anion-deficient region is very small but detectable by thermogravimetric analysis. The isotherms in the spinel field have a point of inflexion very near to the stoichiometric line representing solid solutions of MgFe204 and FePeaO*. A large region of metastable solid solutions of Fe203 in the spinel phase has been found. The precipitation of the a-Fe203 phase from the metastable spinel phase has been studied by thermogravimetric analysis and X-ray diffraction. It is found that the magnesiowustite phase also has a large extension in the cation-deficient region. The extension in the anion-deficient region of the hematite phase is found to be a few mole % FeO in Fe203. Equilibrium equations between ions, defects and the oxygen partial pressure for idealized models of the solids are given. For some simple models the equilibrium constants have been expressed in terms of the composition parameters which are known from experiment. It appears that these log K values have linear rela-tionship with 1/7" and are therefore acceptable from a thermodynamic point of view. The influence of cation distribution on the solid-gas equi-libria has been calculated on the basis of simple models. The experimen-tal results obtained on Mg^"'^Fe^"'" ferrite, Ni^"'^Fe^* ferrite and Zn-^''^Fe^''" ferrite, which have different cation distributions, do not agree with these calculations, which indicates that the actual physical situation is not described in its very detail by the models used. Previous studies on phase diagrams and solid-gas equilibria in spinel systems are compared and discussed.

1. Introduction

Phase diagrams contain condensed experimental data on equilibria between soHds, liquids and gases. Levin, Robbins and McMurdie ') have compiled phase diagrams in which sohds and liquids are involved at high temperatures. The experimental data on which most of these diagrams are based have been obtained by quenching the samples and examining the phases micro-scopically or by X-rays. The oxygen content of an oxide depends strongly on temperature and oxygen partial pressure, when it contains ions the valency of which can be changed. Provided that the vapour pressure of the other constit-uents is vanishingly small, the overall composition of the material can most adequately be determined in that case by thermogravimetric analysis. The aim of the present investigation is to determine with great accuracy the phase dia-gram of the system MgO-FeO-FcjOs and to study the equilibria between the solid phases and the oxygen partial pressure from thermogravimetric data.

Other investigators as Paladino ^•^) and Schmaizried and Tretjakow *) have used a gas-dissociation apparatus to determine the overall composition of the solid material at a given oxygen partial pressure and temperature. This method is considered less accurate and reliable and more cumbersome than thermo-gravimetric analysis. The system MgO-FeO-Fe203 has been investigated ther-mogravimetrically by Roberts and Merwin'), by Woodhouse and White *), by Reijnen ^) and by Speidel ^). The present investigation is an extension of the preliminary work '') on this system and again most experimental data are ob-tained by thermogravimetric analysis, supported by X-ray and chemical analysis.

2, Thermogravimetry

A well mixed assembly of FcjOj and MgO particles will lose oxygen upon heating due to reduction of part of the ferric ions to ferrous ions. The corre-sponding weight losses are irreversible as long as the solid-state reaction is not completed'•^•'°) but a homogeneous and completely reacted sample shows reversible weight losses under proper experimental conditions. Only reversible weight changes ought to be considered in phase-equilibrium studies. In the present study the reversible weight changes were measured on a "Stanton" thermobalance up to 1400 °C at the constant oxygen partial pressure of air. The "Stanton" balance had been previously modified in several parts. The sensitivity of the balance was increased such that a full scale deflection corre-sponded to a weight loss of 20 mg instead of 1(X) mg, as is normal for this type of balance. In the absence of mechanical shocks the increased sensitivity gave a higher accuracy and therefore the analyses were carried out (automatically) at night. The temperature of the sample was measured by a separate Pt/Pt-Rh thermocouple of which one junction was placed just under the sample and the other one in melting ice. Thin slack wires, which did not influence the recorded weight, connected the thermocouple leads on the balance beam with the measuring system. Vertical spiralized SiC rods, which could be operated con-tinuously between 0 and 220 V a.c. and which were arranged around the central alumina tube, provided the heat source of the furnace. The air flow in the alumina tube which was open at both ends was reduced,by placing porous heat-resistent ceramic material in the upper end of the alumina tube and by setting six alumina radiation shields at both sides of the sample holder (fig. 1). The temperature in the hot zone of the furnace was found to be constant within 2 °C over a range of 3 cm at 1400 °C. The temperature of the furnace was raised in steps of 20 °C. During 20-minute periods of constant temperature the sample reached a constant weight, indicating that equilibrium had been reached. At lower temperatures this time was not always sufficient, especially when two solid phases were present. Corrections for changing buoyancy and upward force of the gas flow in the furnace were determined by separate blank

PHASE EQUIUBRIA IN THE SYSTEM MgO-FeO-FejOa 1 S 3 Porous ceramic material Heating elements Sample holder Thermocouple Radiation shields

Fig. 1. Modified furnace of thermobalance. Temperature of sample is measured by a thermo-couple, one junction of which is right under the sample holder. Constant-temperature zone is obtained by applying SiC rods, which are spiralized at the outer ends, and by applying radiation shields. Flow of air is reduced by putting a porous brick in the upper end of the furnace tube.

runs with AI2O3. The accuracy of the measured weight losses was about 0-1 mg for samples weighing about 3 g.

The samples were prepared by mixing MgO and FcjOs (Merck reagent grade) in a ball mill for 6 h and prefiring at 1000 °C for 6 h in air. Thereafter the samples were again ball-milled for 6 h and heated for 6 h at 1250 °C in air. Small portions of the samples were gently powdered in an agate mortar, heated at 700 °C in oxygen and slowly cooled in oxygen, previous to chemical and thermogravimetrical analysis. The total impurity level determined by spectro-chemical analysis was less than 0-1 weight %. The compositions of the samples as found by chemical analysis can be expressed by a composition parameter x: (1 —jc)MgO + ix Fe203. All starting compositions had a negligible amount of ferrous ions (less than 0-05 weight %) and the observed weight losses are thus proportional to the amount of ferrous ions.

3. Construction of isotherms

Instead of the complete three-dimensional phase diagram of a ternary system, it is more practical to give a number of sections at constant pressure or constant temperature. Such a section is an equilateral triangle in which the compositions are represented by triangular coordinates. In this study, one section of the diagram MgO-FeO-FcaOs at the constant partial pressure of air has been determined.

From thermogravimetric data (as an example see fig. 3) and values of the composition parameters x, the isotherms in an isobaric section of the phase diagram can be constructed directly. In the regions where two solid phases are present the isotherms are tie lines, which connect the compositions of the separate phases which are in equilibrium with each other. In the regions where only one solid phase is present the isotherms are in general curves. By these considerations it is already possible to find the single-solid-phase regions roughly from an assembly of isotherms. Consider a sample with composition parameter x. After a relative weight loss Ag/g the composition of the sample is represented in the phase diagram by a point somewhere on its dissociation path. A dissociation path in the diagram (fig. 2) is a straight line parallel to

FeO FejOg FejOj 3 2

Fig. 2. Experimental isotherms in the system MgO-FeO-Fe203. The measuring points are the intersections of isotherms and the horizontal dissociation lines. The boundaries of the spinel field and the magnesiowustite field can be roughly estimated: in the two-phase fields the isotherms ought to be straight lines.

PHASE EQUILDRIA IS THE SYSTEM MgO-FeO-FeiOs 1 55

the base of the triangle and is the locus of points representing the compositions which a sample passes through as it continually loses oxygen. If / is the length of the triangle side, M , is the weight of (1 — x) mole MgO + ^x mole FcjOj and the relative weight loss is Ag/g, then the composition is represented by a point on its dissociation path at a distance (Ag/g) {MJS)l from the side of the triangle. Further on (Ag/g) (MJB) will be indicated by s. In fact s is a second composition parameter which determines the amount of ferrous ions [Mg^ Fe^ Fe^"^s 0^~i+;j/2-s/2]- Values of J at a number of selected temperatures found from the thermogravimetric curves are plotted in the composition triangle in fij. 2. The isotherms in fig. 2 are constructed by connecting points representing compositions at the same temperature.

4. Cation-to-anion ratios in the spinel field

The cation-to-anion ratio of a sample with composition parameters .v and s is calculated to be

1 cation-to-anion ratio =

1 +ix-^s

Samples consisting of MgEcjO* + excess FCiOs (2/3 < .K < 1) have a hori-zontal part in their thermogravimetric curves (fig. 3). The cation-to-anion ratio

1200 1400

-^TCO

1600

Fig. 3. Characteristic thermogram of a composition having excess Fe203 {x > Iji). After a horizontal part in the weight-loss curve there is a rather drastic increase in weight loss at high temperatures.

of several samples has been determined in this horizontal part and found to be 0-75 within experimental error (table I). The horizontal parts thus correspond to compositions of MgFe204 and FeFe^O^. From a more detailed analysis

TABLE I

Experimental values of x and s (see text) and calculated values of the cation-to-anion ratios in the horizontal part of the thermogravimetric curves (figs 3 and 11). Within experimental error, it is found that the horizontal parts of the thermogravimetric curves correspond to stoichiometric spinel (cation-to-anion ratio = 3/4) X 0-669 0-670 0-675 0-681 0-683 0-686 0-692 0-702 0-735 0-802 j.lO^ 4-17 5-18 8-12 11-26 15-39 16-32 23-86 35-6 67-5 136-9 cation-to-anion ratio 0-750 (0-7505) 0-750 (0-7505) 0-750 (0-7500) 0-749 (0-7491) 0-750 (0-7497) 0-749 (0-7492) 0-750 (0-7496) 0-750 (0-7501) 0-750 (0-7498) 0-750 (0-7504) TABLE II

Composition parameter .v' calculated from thermogravimetric data, assuming that the cation-to-anion ratio in the horizontal part is equal to 3/4. These values deviate slightly from the values found by chemical analysis (x)

X 0-669 0-670 0-675 0-681 0-683 0-686 0-692 0-702 0-735 0-802 (Agjg). 10^ 0-50 0-62 0-97 1-34 1-83 1-94 2-82 4-18 7-80 15-20 x' 0-671 0-672 0-675 0-678 0-682 0-683 0-691 0-702 0-734 0-804

PHASE EQUIUBRIA IN THE SYSTEM Mg0-Fe0-Fe203 1 5 7

(see fig. 7) discussed in sec. 7, it follows that the compositions with the exact 3/4 ratio are lying in the single-phase-spinel field and are thus solid solutions of MgFe204 and FeFe204, which are neither oxygen- or cation-deficient. This is in strong contrast with the diagram of Paladino (ref. 2, fig. 12) where the com-position MgFe204 lies in a two-phase field. Even samples with values ofx slightly larger than 2/3 show a horizontal part in the thermogravimetric curve which proves the existence of stoichiometric MgFe204. The minor differences in the cation-to-anion ratios in table I must be attributed mainly to slight errors in the values of x, as determined by chemical analysis. By accepting a ratio of

exactly 0-75 in the horizontal parts the composition parameters x' can be

calculated independently of chemical analysis. The composition parameters x' found by thermogravimetric analysis and the values of x found by chemical analysis are compared in table II. The thermogravimetric values are considered to be more accurate and will be used in the next sections. In appendix II it is made clear theoretically that the point of inflexion in the horizontal part corre-sponds nearly to stoichiometric spinel, which is a spinel phase with a cation-to-anion ratio of exactly 3/4.

5. The point-defect concept in oxides

Point defects play an important role in solid-gas equilibria. The concept of point defects in an inorganic crystal can only be clearly understood with reference to the pure substance in internal equilibrium at 0 °K. Under these conditions, which cannot be realized experimentally, no point defects are present. By heating the substance and assuming equilibrium at each temperature, point defects are formed in order that the free-enthalpy function keeps its lowest value. The nature of a point defect can be:

(a) an ion occupying a site of another sublattice (e.g. B ion on an A site in A[B2]04),

(b) an interstitial ion, being an ion on a site which is normally empty, (c) ion vacancy, being an empty site which is normally occupied,

(d) complex defects, which are associates of interstitial ions, ion vacancies and ions.

Besides for thermal reasons, point defects are formed in solid solutions of oxides, which have different cation-to-anion ratios, according to the following scheme. Cation-to-anion ratio of host lattice S cation-to-anion ratio of oxide dissolved, where

< means material becomes anion-deficient or cation-excessive, that is anion vacancies or interstitial cations are formed;

= means no extra point defects;

> means material becomes cation-deficient or anion-excessive, that is cation vacancies or interstitial oxygen ions are formed.

• ^

All solid phases in the ternary system MgO-FeO-Fe203 belong to the class of oxides in which the cations are much smaller (r < 1 A) than the oxygen ions (r = 1-32 A). These oxides consist in general of a close- or almost close-packed oxygen lattice, cubic or hexagonal, in which the cations fill the interstitial tetra-hedral and octatetra-hedral holes in a regular way. In this class of oxides, interstitial oxygen ions are energetically highly unfavourable and it will be assumed that the defects in the cation-deficient region are predominantly cation vacancies instead of interstitial oxygen ions. Several types of defect notation are used in literature. The atomic notation of Kroger and Vink ' ' ) has been widely accepted but has some disadvantages in discussing solid-state reactions. Let us consider for the purpose of making this clear the compound AB,04, being a mixed oxide of AO and B2O3. There are two types of cation sites, B sites or octahedral sites which are occupied by the trivalent cations when the crystal has lowest internal energy, and A sites or tetrahedral sites which are occupied by faivalent ions. When a cation is on its normal site, the charge is thought to be perfectly balanced by the negative charge of the surrounding anions and no charge is attributed to this site. The notation is therefore A^ or Bg. In the ionic notation of Kroger l^nd Vink each structure element of the crystal is given its own charge assuming completely ionic behaviour. The latter system is chosen throughout in this paper. For a more complete discussion of the matter see: Kroger and V i n k " ) , Kroger '^), Rees '^), Schmaizried '*) and Gillis " ) .

atomic notation A A BB A B " B A ^ A.^-VA V A V A ^ -V a V o ' ^ A few reactions: atomic notation A A + B B ^ A e -ionic notation A A ^ " B B ^ ^ A B ^ ^ B A ^ - A,^-V A ^ ^ VA+ VA Vo^-Vo

+

cation A on its regular A site, cation B on its regular B site, cation A on the regular site of a B ion, cation B on the regular site of an A ion, cation A on interstitial site,

unoccupied A site (atom vacancy), unoccupied A site,

unoccupied A site (ion vacancy), unoccupied oxygen site (atom vacancy), unoccupied oxygen site (ion vacancy).

ionic notation A A ' ^ + B B ^ + ^ A B ^ A^+5±A,2+ + VA, VA + VQ ï± vacuum,

PHASE EQUILIBRIA IN THE SYSTEM MgO-FeO-FcjOj 1 5 9

The Fe^^/Fe-''^ equilibrium in ferrites with inverse spinel structures e.g. FeFe204, MgFe204 (see appendix I) is given in the atomic notation by

i FeB°'+ + i FCA + I V B ^ - ' - + i O ^ F e B " ' - + i O^.

The average charge of the cations on B sites is 2-5+ and thus a trivalent cation on a B site is indicated in the atomic notation by FCB"*^ etc. The same reaction

in the ionic notation is more familiar as it corresponds to ionic reactions in solutions:

i FCB^^ + i FeA^+ + I VB + i O^- ;;- Fea'^ + i O2.

6. Solid-gas equilibrium in the cation-deficient region of the spinel phase It has been found that considerable dissolution of Fe203 in the spinel phase is possible. The dissolution of FCiOj in the spinel phase introduces cation vacancies, as has already been pointed out. Moreover, part of the FcjOs is dissociated upon dissolving into Fe304 and oxygen. The spinel phase under consideration is thus in fact a sohd solution of MgFe204, FeFe204 and Fe203. The relative amounts of FeFe204 and Fe203 in the solid solution depend on temperature and oxygen partial pressure. The equilibrium between solid phase and oxygen can be described by the following equations, if oxidation and reduction are thought to take place at the outer surface of the particles:

e -f- Fe^+ ï i Fe^ +

i O ^ - ï ± i 0 2 - f c + i V o i Vo + f Vj, Ï* vacuum

Fe3+ + I Ve + i O^- 5 - Fe^^ + i O2 (6.1) The bulk of a crystallite acquires the equilibrium composition by solid-state

diffusion of the cations, in accordance with the Wagner theory on solid-state diffusion in close-packed anion lattices. The values of the equilibrium constants for the reactions i VQ + f Vc ï i vacuum and (6.1) depend somewhat on the radius of the solid particle '*). The dependence of K on r is given by

V r RT)

where y is the surface free energy, V„ is the volume of 1/8 mole ferrite and

r is the radius of the solid particle. The effect is small and can be neglected for

the present discussion:

f^ I0~ cm. RT

The above equilibrium reactions do not take into account that the cations are distributed over two types of lattice sites. The consequences of having two sublattices is discussed in appendix I.

The equilibrium constant for reaction (6.1) reads

K =

[ F e 3 - ^ ] [ V , ] 3 ' 8 [ 0 ^ - ] " 2

The concentration of the cations can be expressed in a variety of ways, but if one requires that the concentration of a specific cation is equal to unity when it occupies all available sites, then the definition for the concentrations must be chosen as follows:

number of Y ions [Y] =

[ Y A ]

[ Y B ]

number of available cation sites number of Y ions on A sites number of available A sites number of Y ions on B sites number of available B sites

The number of available cation sites in a spinel obtained from (1 — x) mole MgO + ix mole FcaOs

after a relative weight loss Ag/g is equal to 3/4 of the number of oxygen ions and oxygen vacancies. The number of ions which are involved in the equi-librium are thus found to be:

«M82+ = (1 —x)N,

/lFe3+ = (X — S)N,

"vc =(lx-l-is)N+ÓN, «Vo = :'. ON,

«02- = ( 1 +ix-^s)N,

where A^ is Avogadro's number. In the cation-deficient region, d is negligibly small and is omitted in the expression for K. The concentration of Fe-*"^ ions, [Fe^"^], is according to the definition chosen:

X — s x — s

[Fe^ + ] = «« .

PHASE EQUILIBRIA IN THE SYSTEM MgO-FeO-FczOs ₁₆₁

The other concentrations can be expressed in a similar way. The concentration of oxygen ions is taken equal to unity in all calculations. The equilibrium constant A's can now be expressed in terms of x and s:

K,= 5 (6 + 3x 2s)'" Po,"* _3/8 (x -s)(3x-2- 3s)

(A's stands for the equilibrium constant in the simple case that no distinction is made between A and B sites). Values of log A's for samples consisting of MgFe204 + excess Fe203 have been calculated using the data of thermo-gravimetric curves and the x' values of table II. In fig. 4, log Ks is plotted

,% 0 logK

t

¥•

Fig. 4. Plot of log/f versus l / T f o r a composition MgFe204 -|- excess Fe203. In the single-phase-spinel region this plot is a straight line. Upon cooling the solid solution becomes metastable (dashed straight line). The points a and b therefore indicate the extension of the stable and metastable spinel region.

versus \/T for a sample with composition parameter x = 0-802. In the single-phase-spinel region, the plot must be a straight line. In the region where two solid phases are present (spinel + a-FcjOs) the expression for A's as derived in this section is no longer valid, because the concentrations are calculated on the assumption that there is only one solid phase. Consequently the plot must show a kink (a) at the temperature where a-Fe203 is completely dissolved in the spinel phase. Upon cooling the solid solution apparently becomes super-saturated (dashed straight line in fig. 4) and the nucleation of the a-Fe203 phase is also indicated by a kink (b). In this way both the stable and the metastable spinel region are found from a series of log K^-l/T plots (fig. 5). The straight parts of all plots coincide with each other within experimental error, which

^ ^ s(6+3x-3s) _ ,^'A 0 hg K

t

Fig. 5. Plot of log AT versus 1/7 for compositions having excess Fe203. All straight lines coin-cide. The metastable regions are larger for smaller values of x.

means that the energies and entropies of the ions involved are little or not affected by the overall composition of the spinel phase. The solid solutions can therefore be regarded as ideal.

The enthalphy change over the reaction (6.1) can be calculated from the slope of the straight line in fig. 5 and is found to be: zl ƒ/ = 21-9 kcal per for-mula unit. The phase boundary of the stable spinel field and also of the meta-stable spinel region at the Fe203 side of the diagram has been constructed from the kinks in the log A:-l/r plots (fig. 6). By interpolation the points on the boundary corresponding to 1000, 1100, 1150, 1200, 1250, 1300 and 1350 °C are also found. The experimental isotherms of fig. 2 have been superimposed on the phase diagram in fig. 6 and an excellent fit is observed. This means that the experimental data are consistent with each other. Note that the phase boundary of the stable spinel field in the FezOj-FcjO* system (fig. 6) corre-sponds to the Darken and Gurry diagram ^^) of that system.

7. Discussion on stoichiometric and anion-deficient Mg ferrite

The dissolution of MgO in the spinel phase introduces interstitial cations or oxygen vacancies. The sharp increase in weight loss in the thermogravimetric curve (fig. 3) when the horizontal part is passed, has previously '') led us to the conclusion that the spinel phase does not exist with a cation-to-anion ratio larger than 3/4. The thermogravimetric analysis of a large number of samples near the stoichiometric MgFe204 point has now made it possible to construct the isotherms in this region with great accuracy (fig. 7). The diagram shows

PHASE EQUILIBRIA I N THE SYSTEM MgO-FeO-FejOs 163 -x= 0-60i-•X=0.652__MgR20^ X= 0-670 ^ X=0-6S3 X=0-702 X=0-735 X=0-e02 -X=0-902

Fig. 6. The spinel region of the phase diagram. The stable and the metastable phase boundaries are found from the kinks in the log K-\IT plots (see fig. 5). The open circles on the stable boundary correspond to a set of selected temperatures (ICXK), 1100, 1150, 1200, 1250, 1300, 1350 ^C). The experimental isotherms of fig. 2 are superimposed on the diagram. The excellent fit proves that the experimental data are consistent with each other.

that the isotherms are slightly curved in a region where the cation-to-anion ratio is larger than 3/4. This means that the spinel phase can exist with a very small oxygen deficiency, because in a two-phase field the isotherms should be straight lines. The change-overs of the curved isotherms to the straight isotherms are so smooth that the phase boundary cannot be determined with good accuracy. Saturation-magnetization values of samples near the stoichiometric point (fig. 8), that were heat-treated in one furnace (fired at 1300°C in air and slowly cooled 100 "C/h), show that the saturation-magnetization values depend on com-position in a region having excess MgO. This also indicates that MgO has been dissolved and is not or incompletely precipitated upon cooling, which result has previously been found by Blackman '"'). The magnetization values increase when more MgO is dissolved in the spinel phase while a decrease should be expected on first sight irrespective of the nature of the defect structure (oxygen vacancies or interstitial cations). An explanation for this unexpected behaviour is based on the fact that the material is not in internal equilibrium at room

-X= 0-638 ^ c \ % -X=0-6^7 (Paladino) ' -x= 0-650 .-X= 0-663 _Mi^O^ ,-X=0-669 ^ -X= 0-675 -X= 0-681 -X= 0-686 'X=0-6S2

Fig. 7. Detail of the phase diagram in the region of MgFe204. The isotherms are slightly curved in a region with a cation-to-anlon ratio larger than 3'4, indicating a very small solu-bility of Mgt) in the spinel phase. According to a previous phase diagram by Paladino -), the single-phase region has its limit at the MgO side of the diagram at x = 0-647.

(gauss cm ) I 26 25 24 23 22 21 20 MgFe^ 0^+xMgO •* 0-20 0-10 Mgre^O^+yi^zOj 0-10 0-20

Fig. 8. Values of saturated magnetization of Mg ferrite as a function of composition. The change of magnetization in the region with excess MgO indicates the formation of solid solutions In this region.