JOURNAL

Morphology

of the AMERICAN CERAMIC SOCIETY

Volume 64, Number 5 May 1981

and Stability of the Glass Phase in Glass-Ceramic Systems RISHI RAJ*

Bard Hall, Cornell University, Ithaca, New York 14853

An explanation is given for the apparent stability of a minor phase in a two-phase system, such as a glass in a glass-ceramic, even though normally the glass would be expected to dissolve completely into the crystalline phase. It is shown that the glass would be stable only if it is segregated to the triple-junction nodes and only if the dihedral angle is <a/3. The size and morphology of the glass pocket are obtained in terms of the interface energies, the undercooling below the solidus temperature, and the dihedral angle.

I. Introduction

LASS can be present in an otherwise crystalline ceramic either G because of the use of a “liquid phase” during hot consolidation of the powder’ or because the ceramic phase is obtained through devitrification of a glass.2 Hot-pressed silicon nitride is an example of the first case and P-spodumene lithia-aluminosilicate glass-ce- ramic is an example of the second. Technically, both are glass-

Presented at the 81st Annual Meeting, The American Ceramic Society, Cin- cinnati, Ohio, May I , 1979 (Basic Science Division No. 57-B-79). Received March 17, 1980; revised copy received July 11, 1980.

Supported in part by the Department of Energy, Division of Basic Sciences, under Contract No. EG-77-S-02-4386 and in part by Rockwell International Science Center.

‘This paper was written while Rishi Raj was on leave at Rockwell International Science Center, Thousand Oaks, California 9 1360.

Fig. 1. Geometry of a pocket of glass phase segregated at a triple junction node when O<a/6.

ceramic materials. The glass is usually segregated to the triple- junction nodes at the grain bo~ndar i e s ,~ although it may also pen- etrate the two-grain junctions4 when the glass wets the grain bound- aries completely, i.e. the dihedral angle is nearly zero.

Some interesting questions arise regarding the morphology and stability of the glass phase. For example, why does most of the glass segregate to the triple junctions? Why is a residual glass observed in p-spodumene glass-ceramic even when it is annealed below the solidus temperature in a region in which the glass forms a solid solution with the crystalline phase?5 It is shown in this paper that these observations have a thermodynamic basis.

11. Morphology of the Triple-Junction Node

An idealized picture of a glass pocket at a triple junction is sketched in Fig. 1 for a case when the dihedral angle 20<a/3. As we shall see later, the glass is stable only for this condition. In the third dimension, normal to the plane of the paper, the cross section remains the same until a grain corner is reached, i.e. the glass pocket is continuous between two adjacent four-grain junctions.

If we assume that the glass-crystal interface energy is isotropic, then surface diffusion will ensure that the interfaces have a uniform radius of curvature; let this be equal to r. It can then be shown that the size of the glass pocket, u, is related to r through the equation:

u=r[sec(a/6) cos0- I ] (1)

The area occupied by the glass in the two-dimensional picture is given by

whereas the volume fraction of the glass phase, assuming that most of it exists a t the triple junctions, is given by:

vg=8Ag/nd2 (3)

where d is the equiaxed grain size.6 Equations (1) through (3) permit vg to be calculated in terms of u, the pocket size, or r, the radius of curvature. When 8+0:

vg=0.41 (r2/d2) or 17.1 3(u2/dz) (4)

The total area of the glass-crystal interface, per unit length of the triple-junction pocket, will also depend on r and 0

S= r(a - 68) ( 5 )

The angle 0 is related to the glass-ceramic interface energy, y ~ , and the grain-boundary energy, yb, through the following equation:

COS8 = yb/2y, (6)

24’5

246 Journal of the American Ceramic Society-Raj

TEMP (“C)

P- spodumene

1 I I 1 I

1450

J-

800

75 70 65 60 siop ( w t “lo)

Fig. 2. ponent system Li,O. AI2O3 .4SiOz-Si0,.

A portion of the phase diagram for the two-com-

The relative magnitudes of yb and yl depend on the composition of the segregated phase. The angle 8 can have values ranging from near zero to those approaching 7~/2.’ For a more general description of the geometry of the second phase, even when 8>7r/6, the reader is referred to the work of Clemm and Fisher6 and Wray.9

111. Stability of the Glass Pocket

As an example, we consider a two-component system consisting of 6-spodumene (LizO. AI2O3. 4s io2 ) and S O 2 . The phase diagram for this system shows a large single-phase region in which S i 0 2 and P-spodumene form a solid solution.I0 A portion of the phase diagram is included in Fig. 2. The interesting point is that when a material of composition xo is equilibrated at a temperature To which lies below the solidus temperature, the glass persists at the triple-junc- tion nodes, although from the phase diagram one would expect it to dissolve completely into the crystalline phase.* We will show that because of the peculiar morphology of the glass in the triple junc- tion, the glass will be thermodynamically stable even in the solid- solution region.

We consider the change in free energy for a change of state from configuration I to I1 as shown in Fig. 3, the average composition, xo, and the temperature, To, remaining constant. The change in free energy, AG=G, , -GI , will consist of three terms: ( I ) an increase in free energy because a nonequilibrium glass phase exists in state 11, (2) an increase in G because of the presence of the glass-crystal interface in state 11, and (3) a decrease in G because of the crystal- crystal grain boundary which exists in state I but not in state 11. Let 1 AGJ be the change in the free energy when one unit volume of the liquid of composition xl* is dissolved as a solid solution into the crystalline phase of composition x,, at a temperature To. Then the first term will be given by I AG,IrzFv(8) where rZFV(8) is the volume of the glass pocket (per unit length normal to the plane of the paper); the second term would be given by ylrF,(B) and the third term by ybrFb(@), where the F(8)’s are a means of relating particular geometrical properties of the pocket to r and 8 as de- scribed in Section 11. We then obtain the following equation for AG:

A G = G I I - G I = ~ ~ F , ( ~ ) I AG, 1 +rFS(8)y/-rFb(8)yb (7 )

where

*It has been correctly pointed out by the reviewers that xf may vary with the radius of curvature, r, and, therefore, AGv could also change with r. It is shown in the Appendix that in a binary system in which the components form an ideal solution, AG” would be independent of r, and that xf will be determined by the temperature and the composition of the crytalline phase. It should be realized. though, that the essential result in this paper, that the minimum in the free energy occurs at a finite volume of the glass phase, arises merely from the geometry of the problem.

Vol. 64, No. 5

State I State II

AG

Fig. 3. The difference in free energy between state I 1 and state I at a temperature and composition at which the glass is completely soluble in the crystalline phase; 8 < ~ / 6 .

FV(8)=2 c0sz8 s in7~/3-~/2+3(8-% sin28)

FJ8) = K - 68

and

Fb(8) = 3 cos8 (tana/6 - t a d ) (8)

Note that equations for F,, F,, and Fb are derived from geo- metrical considerations only. For example, r2Fv is the volume of the pipe per unit length, rF, is the surface area of the solid-liquid interface in the pipe, and rFb is the crystal-crystal grain boundary area which would form if the glass were to be replaced by crystal.

As shown in Fig. 3, Eq. (7) will have a minimum at r=r , if 8<a/6. Equating the first derivative of Eq. (7) to zero and sub- stituting for y / / y b from Eq. (6 ) and for F(8)’s from Eq. (8) we obtain the simple result that:

r,=yrllAG”I (9)

AG,= - (y?/ I )Fd@ (10)

when 8<7r/6. Substituting for r=r , into Eq. ( 7 ) we find that

A plot of F,(8) is given in Fig. 4. We note that the “depth” of the minimum shown in Fig. 3 will decrese as f? increases until the min- imum disappears when 8-a/6.

IV. Discussion

( I ) The results contained in Eqs. (9) and (10) and the plot in Fig.

4 essentially summarize our findings. As long as AG, remains neg- ative, i.e. 8<a/6, a minimum in the free-energy curve (Fig. 3) would exist and the glass pocket would be stable even though the

Volume Fraction of the Glass Phase

May 1981

8" Fv(8) 0 0.16125

Morphology and Stability of the Glass Phase in Glass-Ceramic Systems 247

bulk thermodynamics would require complete dissolution of the glass into the crystal as a solid solution. The radius of curvature at the glass-crystal interface is also explicitly defined in terms of the bulk thermodynamic properties y I and AC,. It is even possible to calculate the volume fraction of the glass phase which will be retained in equilibrium below the solidus temperature (Fig. 2) from Eqs. (21, (3). (81, and (9):

As B+n/6, F,(#)-+O and, therefore, v,-+O. On the other hand, vg is a maximum when #=O, i.e. when Fv(B)=0.16.

(2) Distribution of the Class when the Dihedral Angle #=O or when 2y,5yb

When the dihedral angle is equal to zero, the question can be posed as to why the glass segregates to the triple junctions instead of spreading evenly across all grain boundaries. It follows from the definitions of F,(#) and Fb(#) that this woyld occur only if r(F,y, - FbYb)>O, or on substituting from Eqs. (8) and ( 6 ) if -2ryf,(0)>0. Since F,(B) is positive when B=O, the inequality is not satisfied and, therefore, the glass would segregate to the triple junctions. This result, in fact, becomes immediately obvious on comparison of states I and I1 (Fig. 5) , in both of which #=0 but where in state I the glass is distributed evenly, whereas in state I1 it is segregated at the triple junctions. If Yb=27/, the energy of the grain interfaces is not affected by the presence of the glass but at the triple junction, ABC, the configuration in state I1 his a lower energy than that in state I because the interface area associated with the region ABC is smaller in state I1 than in state I. It follows that state I1 is a lower-energy configuration than state I.

Another interesting question is how much glass, if any, would be present in the two-grain junctions. Energetically, if Yb=2y, exactly, then the crystal-crystal grain boundary is exactly equivalent to a crystal-glass-crystal grain boundary and, as explained in the pre- ceding paragraph, one would expect all glass to segregate to the triple junctions. But if 2y/<Yb* then the two-grain junctions will contain a glass layer of a finite thickness. If y, is independent of the boundary thickness, i.e. the energy of the crystal-glass interface on one side of the boundary is not influenced by the proximity of a similar interface on the other side, then the glass thickness will be the minimum possible, i.e. about a monolayer. This, however, is not likely to be the case. yI is a measure of the strain energy in the bonds at and near the interface as one moves from a well- ordered structure in the crystal to the disordered structure in the liquid. If the thickness of the glass layer is larger than the distance to which this strain field extends, then yI would be independent of

*Physically, the meaning of this condition is that two crystal-glass interfaces repel each other when brought to close proximity. Although this would not occur in ionic or metallic systems where bonding is nondirectional (Ref. 1 1). in covalently bonded materials it is at least conceivble that the interfaces would repel rather than attract each other.

8 (radians)

0- 0 10 2 0 30

8 (degrees)

Fig. 4. A plot of the function F,(O)

the glass thickness. If not, then the effective y, would increase because of the proximity of another glass-crystal interface. At some point, therefore, a balance would be reached when twice the ef- fective yl becomes equal to yb and the boundary width would sta- bilize. A larger width would not be permissible because then the liquid would want to escape into the triple junctions, whereas a smaller width would cause twice the effective yI to become larger than Yb. If this reasoning is correct, then it is possible that the actual width of the two-grain junctions is about twice the size of the strain field of an unconstrained crystal-liquid interface. In MgO- fluxed hot-pressed silicon nitride, the width of the two-grain junc- tions is = 1.2 nm334 which would mean that for a single crystal-glass interface the strain field would extend to -0.6 nm into the liquid.

(3) The temperature dependence of v8 in Eq. (1 1) is contained pri-

marily in the term AC, since it would depend on the undercooling below the solidus temperature. In a real situation, however, the time-temperature history of the sample will determine the amount of retained glass. Equation (1 1) should provide reasonable values for vg when the sample is equilibrated just below the solidus tem- perature where the kinetics is reasonably fast. However, at still lower temperatures, the actual vg may be controlled more by kinetic than equilibrium considerations. Therefore, to minimize the actual

Importance of Time- Temperature History

Fig. 5. A comparison of sur- face energy in two possible con- figurations; Yb=2y,. State I State II

248

(A)

T

t

Vol. 64, No. 5 Journal of the American Ceramic Society-Raj

LIQUID

I I I I I I

I I

SOLID I xo

A + x B

LIQUID i

I

I

P i f

Fig. A l . ( A ) Phase diagram for ideal solid-solution binary system. ( B ) Relative free energy curves for the liquid and solid phase for composition xo and temperature To; AG,,, is the free energy of mixing using solid A and solid B as the standard states.

amount of vR. the sample should be annealed well enough below the solidus temperature so that AG, is large enough while the kinetics of glass dissolution is also fast enough. At the higher or lower temperatures, vR would be greater because of smaller AG, at the higher temperatures and slower kinetics at the lower temperatures.

V. Conclusions

The stability of the glass phase in those glass-ceramics systems i n which the glass forms a solid solution in the crystalline phase was considered. It was found that:

If the dihedral angle is <r/3, glass would be retained at triple-junction nodes in equilibrium even though the bulk glass may be soluble in the crystalline phase.

The radius of curvature of the glass-crystal interface and the volume fraction of the glass are explicitly given by Eqs. (9) and ( 1 1) in terms of the interface energies, the bulk thermodynamic properties, and the grain size.

(3) The actual volume fraction of the glass would be determined by equilibrium thermodynamics, as derived here, and by the time- temperature history of the specimen, which will influence the ki- netics of the glass dissolution into the crystalline phase. I n general, the retained glass fraction would be larger than anticipated from equilibrium considerations,

( 1 )

(2)

APPENDIX

It is shown here that AG, will not be a function of the pocket size, u, at least for the case of an ideal solid-solution binary system. Let us calculate AG, for an average composition xo and a uniform temperature To as shown in the phase diagram in Fig. Al(A). We assume that the volume fraction of the crystal phase is larger than that of the glass, so that the composition of the crystal phase is equal to the average composition xo and does not vary when the pocket size of the glass changes.

Let (FA#, gpg~) and (pAr, pBe) be the chemical potentials of the two components in the glass and crystal phases, respectively. Fur- ther, let VA and VB be the partial molar volume of A and B in the glass phase. If wAd and fiBd are the chemical potential in “free” glass of the same composition, i.e. glass with a bulk composition the same as the segregated glass but which is not segregated in the grain boundaries, then it follows thati2

y, and r have the same meaning as given in the text. The second term arises from the mechanical pressure (-y,/r) generated by the curvature of the interface at the glass pocket.

Now in the state of equilibrium we must have that:

PA’=@A‘

and

PBR=PB‘

Therefore from Eqs. (A-1) and (A-2) we obtain

and

PB*- PB‘ = (7//4 FB (A-3)

All chemical potentials on the left side of Eq. (A-3) can be deter- mined from the free energy curves foj the liquid and crystal phases at To. If we assume that VA=vB=V, which would be the case if the solution were ideal, then we obtain the condition that

@Ad- P A c = PEP‘- PBc (A-4)

which depends on the bulk thermodynamic properties of the ma- terial only. The graphic construction to satisfy this condition is illustrated in Fig. Al(B). It defines the glass composition x,, which depends only on x,,. AG, can now be calculated in terms of the thermodynamic parameters. By definition:

(A-5)

All parameters in Eq. (A-5) are obtained from the graphic con-

The author thanks F F. Lange for comments and sug- struction shown in Fig. AI(B). Acknowledgmenk gestions.

References

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