glassformation and crystallization, compendium from the 1996sestak/yyx/glass1996.pdf ·...

Glassformation and crystallization, compendium from the 1996 Larry L. Hench LIFE AND DEATH: THE ULTIMATE PHASE TRANSFORMATION Thermochimica Acta 280/281 (1996) 1-13 Edgar Dutra Zanotto THE APPLICABILITY OF THE GENERAL THEORY OF PHASE TRANSFORMATIONS TO GLASS CRYSTALLIZATION Thermochimica Acta 280/281 (1996) 73 82 Michael C. Weinberg GLASS-FORMATION AND CRYSTALLIZATION KINETICS Thermochimica Acta 280/281 (1996) 63-71 C. T. Moynihan, S.-K. Lee, M. Tatsumisago, T. Minami ESTIMATION OF ACTIVATION ENERGIES FOR STRUCTURAL RELAXATION AND VISCOUS FLOW FROM DTA AND DSC EXPERIMENTS Thermochimica Acta 280/281 (1996) 153 162 C.S. Ray, D.E. Day IDENTIFYING INTERNAL AND SURFACE CRYSTALLIZATION BY DIFFERENTIAL THERMAL ANALYSIS FOR THE GLASS-TO-CRYSTAL TRANSFORMATIONS Thermochimica Acta 280/281 (1996) 163 174 Jaroslav Sestak USE OF PHENOMENOLOGICAL KINETICS AND THE ENTHALPY VERSUS TEMPERATURE DIAGRAM (AND ITS DERIVATIVE - DTA) FOR A BETTER UNDERSTANDING OF TRANSITION (CRYSTALLIZATION) PROCESSES IN GLASSES Thermochimica Acta 280/281 (1996) 511 521

thermochimica acta

E L S E V I E R Thermochimica Acta 280/281 (1996) 1-13

Life and death: the ultimate phase transformation 1

L a r r y L. H e n c h *

Department of Materials Science and Engineerin 9, University of Florida, Gainsville, FL32611, USA

Abstract

The order ~ disorder phase transformation involved in life and death is discussed. Five bioactive substrate ordering factors are proposed to provide the mechanisms for overcoming the entropic barrier to forming life from a disordered molecular assemblage. Molecular orbital models (AM1 method) are used to analyze the ordering factors. The results are consistent with the genetic activation characteristics and tissue bonding of bioactive silicate glasses.

Keywords: Chiral; Death; Entropic barrier; Life; Molecular orbital calculations; Order factors; Phase transformation

I. Overview

Disorder ~ order reactions are one of the most important classes of phase transformation; i.e. gas ~ liquid; liquid ~ solid; glass ~ crystal. The free energy is lowered in each case at the expense of entropy, or vice versa, depending upon the direction of the transformation. An enormous number of such reactions have been studied and detailed physical chemical theories exist, as demonstrated by the many papers in this volume.

Biological systems also fit into the category of order ~ disorder reactions. The biological phase transformation has profound consequences; with order comes life, with disorder, death. Living organisms transform a random mixture of organic and inorganic molecules into highly organized macromolecular structures.

Life transforms disorder into order. Death is the onset of the reverse reaction; i.e.

Life (Order) ~ Death (Disorder) (1)

* Fax: + 1 904 392 8664. 1 Dedicated to Professor Hiroshi Suga.

0040-6031/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0040-6031(95)02632-0

2 L.L. Hench/Thermochimica Acta 280/281 (1996) 1-13

As required by the third law of thermodynamics, life is achieved and maintained at the expense of the entropy of its environment. The mystery of life is the origin of this antientropic transformation; the mystery of death is what stops it.

For nonliving systems the physics of a phase transformation is straightforward; changes in pressure, temperature, or volume modify the interaction of electron wave functions of neighboring atoms which leads to a lower, or higher, total energy for the assembly of atoms. A significant feature of this physical process is its reversibility. As the term implies, invariant points on T-P-C diagrams are independent of the direction of heating or cooling. Although kinetic barriers to order ~ disorder transformations exist, thereby permitting, for example, glass formation, equilibrium requires that all supercooled or superheated systems will eventually and reversibly transform. A crystal can be cycled through Tm innumerable times and the same space group will always reform upon crystallization.

The phase transformation of a living system is remarkably different. The death of a cell is irreversible. Chemical analysis will show that the same molecules are present in the dead cell; even the supramolecular architecture of the cell wall, nucleus, and organelles is still present. But entropy irreversibly increases following death (Fig. 1). There is no change in temperature, pressure, or chemical environment that will resurrect vitality in a dead cell. The biological transformation from order ~ disorder is one way.

u)

n 0 n- l-- Z i i i

O L D C E L L

LIFE

D E A T H

CHEMICAL DISSOLUTION

TIME, t

B I R T H

HETEROGENEOUS NUCLEATION BY

DNA

MITOSIS

Fig. 1. Schematic of entropy changes involved in life and death transformations.

N E W C E L L

LIFE

L.L. Hench/Thermochimica Acta 280/281 (1996) 1 13 3

Restoration of life requires total chemical dissolution of the organism and regrowth, by mitosis (cell division) of a new cell (Fig. 1). This disorder --. order transformation to form a new cell is not self nucleating, the concept of spontaneous or "homogeneous" nucleation of life is called "a miracle," and is in the domain of metaphysics rather than physics. As is well known, all aspects of cell formation are directed by the base pair sequences in DNA. No laboratory simulations have been able to produce even a small segment of a DNA molecule or even the simplest proteins without first seeding the experiment with DNA [1].

Thus, it is accepted that the disorder ---, order phase transformation characteristic of life involves the equivalent of"heterogeneous" nucleation by DNA [2].

It is equally well accepted that all DNA is synthesized within a cell, directed by the DNA already present [-2]. No DNA is synthesized de novo outside a cell and transported into a cell to serve its replicative function. Even viruses, which can be considered as mostly "naked" segments of DNA, are replicated within a host cell using the biosynthesis pathways of the host.

2. The paradox

The problem of understanding the origin of the disorder ~ order transformation of biological systems takes the form of a paradox:

(1) DNA synthesis requires DNA. How did DNA biosynthesis begin without DNA? (2) Likewise, protein synthesis requires enzymes. Enzymes are proteins. Thus, how

did protein biosynthesis begin without enzymes?

In other words, how could the reverse transformation reaction in Eq. (1) (Dis- order ~ Order) occur; i.e., "homogeneous" nucleation?

3. Alternative hypotheses

An extensive literature review [1, 3 9] describes four general scientific hypotheses for the origin of order and life.

(1) Single biolo(tical origin: replication and metabolic processes appear simultaneously within a membrane.

(2) Double bioloyical oriyin: metabolic processes appear first in a pre-biotic organism followed by symbiosis of a replication structure or vice versa.

(3) Clay-based origin: metabolic processes evolve on clay-based substrates that provide order and self-replication of proteins; eventually replaced by DNA.

(4) Outer space origin: alien life forms adherent to cosmic dust particles enter the earth's atmosphere via comets to seed nutrient pools.

None of these hypotheses, or their subsets, provides a mechanistic solution to the entropic barrier, illustrated in Fig. 1, to forming an ordered assemblage from a disordered one. The "life from space" conjecture [9] simply shifts the question to an

4 L.L. Hench/Thermochimica Acta 280/281 (1996) 1 13

extraterrestrial origin without providing a means of proof; i.e. is not science. The other three hypotheses also offer no means of proof. Stanley Miller's classic experiment in 1955 demonstrated that simple amino acids and sugars can be synthesized from a random mixture of gases in a reducing atmosphere exposed to electrical discharges [5]. However, the macromolecular structures characteristic of life cannot be produced by such nondirected synthesis.

Several years ago, I presented a hypothesis for the origin of ordered biotic structures from a random assemblage of disordered organic molecules [3].

(5) Bioactive substrates origin: ordered protein and DNA structures are formed on bioactive inorganic substrates.

The bioactive substrate hypothesis suggests that selective adsorption of bio- monomers contributes five ordering factors that eliminate randomness, reduce entropy, and ensure repeatability of the biopolymers formed [3]. The ordering factors are:

(a) substrate steric factors which impose repeatable spatial requirements; (b) monomeric optic axis orientation which imposes chiral growth; (c) substrate optic axis orientation which imposes match of monomeric chirality; (d) irreversible condensation reactions which result in a stable biopolymer; and (e) polymeric steric factors which limit the selection of additional monomeric units.

4. Evidence for bioactive substrates

The original evidence in support of a bioactive substrate theory for the disorder --* order transformation was the epitaxial binding of the levorotatory form of the amino acid alanine (poly-L-alanine) to alpha quartz crystals [10]. Highly specific orientations of poly-L-alanine deposits were formed on (1010) or (101 l) crystal planes where repeating silanol groups matched alanine binding sites (Fig. 2a). The epitaxially oriented amino acids were tightly bonded and resisted mechanical abrasion or chemical attack, unlike poly-D-alanine or substrates where spatial distances between bonding sites were not matched. These experiments provided evidence for ordering factors (a), (b), and (c) above and perhaps (d) because the adherent agglomerates could not be removed from the substrate without substantial mechanical force.

5. An inorganic route to biosynthesis

Based upon these empirical findings, Jon West and I have recently completed several calculational studies that indicate a possible mechanistic solution to the order ~ disorder paradox of life [11-14]. We have used semi-empirical molecular orbital (MO) calculations (AM 1 method) to study the energetics of interaction of hydroxylated SiO 2 clusters and amino acids. Our rationale is that silica, silicates, and water have always been the most abundant compounds of the lithosphere [15] and provide the most probable interfaces for interactions with pre-biotic amino acids.

The MO calculations show that hydrated three-membered silica rings are easily formed during the fracture of silica and silicates [ 16]. The three-membered silica rings

L.L. HenchlThermochimica Acta 2801281 (I 996) 1 -I 3

POLY-L-ALANINE BONDS TO AM1 MO MODELS OF RE_PEAT SITES ON L-QUARTZ SILICA-ALANINE INTERACTIONS

(1010) SURFACE IN THE [Olll] PREDICTS BONDING TO OCCUR DIRECTION WITH THE LOWEST BARRIER FOR

THE CARBOXY GROUP

/ PREFERRED DIRECTION FOR EPITAXIAL BONDING

L-QUARTZ SURFACE

AC- DIPEPTIDE FORMATION WITHOUT EMZYMES or BIOACTIVE SUBSTRATI

ALANINE AMINE GROUP BONDING TO SILICA

DIPEPTIDE FORMATION USING A BIOACTIVE SUBSTRATE

Ai$zJijf TO SILICA

I

REACTION COORDINATE

5

HYDROGEN

[B] TRANSITION STATE DURING [D] TRANSITION STATE DURING SILICA-ALANINE BONDING VIA SILICA-ALANINE BONDING VIA

AMINE GROUP CARBOXY GROUP

Fig. 2. Ordering and biosynthesis interactions of amino acids with hydrated silica substrates.

are energetically metastable due to quantum-mechanical strain of the bridging Si-O- Si bonds (Figs. 3 and 4). The strained three-membered silica rings provide a pathway for binding of alanine [CH,CH(NH,)COOH] with an energy barrier as low as + 2.2 kcal mol- ’ (Figs. 2b-2d). The low energy barrier is easily reversible at 25550°C and is relatively insensitive to hydrolysis conditions of the molecules. The low energy barrier is due to the formation of a pentacoordinate Si atom in a metastable transition state (Fig. 5). The pentacoordinate silicon state occurs when the -COOH group of an amino acid interacts with a trisiloxane ring. The carboxyl bond polarizes the Si-O-Si bond in the three-membered ring and opens it into a three-membered chain [ 123. Water attacks

L.L. Hench/Thermochimica Acta 280/281 (1996) 1 13

DISTRIBUTION OF ENERGIES FOR AM1 and PM3 SILICA RING MODELS

- 2 2 0

o

v ' lo v t -

0 1

Z I.I. 0 I - < I.IJ "1-

-230

-240

-250

-260

e~

-270

- 2 8 0

5 K c a l / M o l ,

t ~- Q_ 4

q

\ \,

\,

~, "~ t PM3 2 K c a i / M ;b [ - - - . _~ ~ ' ~ ' ~

-q3 D-_

f -'---~-o

i i I i

1 2 3 4 5 6

NUMBER OF TETRAHEDRA

Fig. 3. Effect o f n u m b e r o f s i l i c a t e t r a h e d r a i n r i n g s o n rela t iveheatsofformation.(Noteenerget icdifferences between 3-membered trisiloxane rings and the more stable 4- and 5-membered rings.)

three-membered rings in a similar manner, as described experimentally and theoretically using hydrated silica gel systems [17, 18]. (This equivalence of the polar behavior of carboxyl groups of amino acids to H20 is vital when we examine the bonding of bioactive glasses to living tissues in the next section.)

We have recently completed MO calculations which show that the metastable pentacoordinate Si-OH complex acts like as inorganic enzyme by providing a favorable reaction pathway for polypeptide synthesis [12]. The easily reversible opening and closing of the hydrated silica rings provides a pentacoordinate Si transition state which

L.L. Hench/Thermochimica Acta 280/281 (1996) 1-13 7

BOND ANGLES AND DISTANCES

IN SILICA RING MODELS

co ~J

Otu Zn- 0 ~ 111 w

co. . . . -

_.1

2 0 0 -

1 5 0

1 0 0 '

5 0

2 °

,.,, ,a, ll,.. ~ . . . . ]-,-'m"- ~ - - . .o AM1 ANGLES

" / f - ..3 1 °

~'.-. ,~ ~ l l . 74 "" " '¢. , 1 . 7 2

"-- 0.02A ~ AM1 , 1 . 7 0

~I' ,1.68 BONDS

0.02A .M3 ,1 . 6 6 BONDS

~ 7 , 1 . 6 4

o<~

co IJJ O z I- co

Z O

om co d

s

co

NUMBER OF TETRAHEDRA

Fig. 4. Effect of number of silica tetrahedra in rings on angles and bond distances. (Note ditational and rotational strains in 3-membered rings.)

3-MEMBERED HYDROLYSIS VIA 3-MEMBERED SILICA R I N G PENTACOORDINATE SILICA CHAIN

SILICON

Fig. 5. Molecular reaction in hydrolysis opening of a trisiloxane ring. (Note pentacoordinate silicon.)


serves an enzymatic function providing a low energy pathway to create the dipeptide bond. The MO calculations show that addition of a second amino acid (glycine) to a trisiloxane + alanine cluster results in formation of an alanine-glycine dipeptide and release ofa trisiloxane chain. The energy barrier of the saddle point for the biosynthesis reaction of alanine + glycine + trisiloxane is + 17.9 kcal mol 1. This barrier for polypeptide formation via inorganic biosynthesis is greater than enzyme-catalyzed peptide synthesis. However, this value is substantially less than the + 70.6 kcal tool 1 barrier to the formation of peptide bonds without a common intermediate. Fig. 2b shows that the energy barrier for formation ofa dipeptide bond without the presence of an enzyme-like structure is very high, + 70.6 kcal mol - 1 calculated with the same MO method.

These calculations demonstrate a possible, energetically feasible pathway for achieving the fourth ordering factor (d); i.e, the irreversible condensation reactions involved in protein synthesis, prior to the existence of enzymes.

The MO calculations also show that the metastable silica amino transition states have an absorption band in the 220 nm region of the ultraviolet [19]. This is a significant finding because one of the most likely explanations for the chirality of life forms involves circularly polarized sunlight in this region of the spectrum [4]. Living organisms contain only levorotatory amino acids in proteins. When a racemic mixture of amino acids is exposed to right circularly polarized UV photons in this range of energy, the preponderance of amino acids that survive is levorotatory [4]. The L amino acids exposed to the metastable surface transition states of silica and silicates would be preferentially chemisorbed on the substrates due to ordering factor (b) listed above. The UV absorption at 220 nm would provide photocatalysis of peptides on the bioactive substrates thereby stabilizing the L amino acids in the proteins. The chirality of the photocatalyzed reaction would be maintained by selective polymerization of only the monomers which satisfied polymeric steric factors, ordering factor (e).

Thus, before the existence of enzymes or DNA the entropic barrier for disorder order (non-living ~ living) illustrated in Fig. 1, would be overcome by the five

ordering factors catalyzed by the bioactive substrate, as indicated in Fig. 6. Lowering the entropic energy barrier (Fig. 6) combined with a low enthalpy of

reaction of amino acids with metastable surface states (Fig. 2), and photo catalysis, leads to a plausible low free energy of reaction for protein synthesis and the disorder

order reaction required to initiate self propagating life forms from a nonliving mixture of amino acids.

6. Present day implications

The combination of specificity and variability of inorganic bioactive substrates [3] provides the foundation for what have become highly specific and enormously variable organic structures capable of being replicated over and over again. We can conclude that the inorganic origin of biopolymers is irreversibly and immutably locked into the very beginning of the genetic code.

L.L. Hench/Thermochimica Acta 280/281 (1996) 1-13

NON-LIVING ~ LIVING

03

>.- o. O n- I - Z uJ

O L D C E L L

LIFE

D E A T H

B i o a c t i v e :

Substrat~ O r d e r i n g ' ,

F a c t o r s

B I R T H

l i . . . .

(c) i . . . . . . . . . . .

(d) L . . . . . . . . . .

(e)

CHEMICAL DISSOLUTION

TIME, t

HETEROGENEOUS NUCLEATION BY

DNA

MITOSIS

N E W C E L L

LIFE

Fig. 6. Hypothesized effect of bioactive ordering factors (a e) on overcoming entropic barrier in creating life.

Hildebrand et al. [20] have used modern genetic engineering techniques to demonstrate that certain genes are activated by hydrated silicon. Volcani reports that more than 60 genes are Si-sensitive [21]. Keeting et al. [22, 23] have shown that hydrated soluble silicon will enhance the proliferation of bone cells (osteoblasts) and active cellular production of transforming growth factors. These findings help explain the unique characteristics of certain compositional ranges of silicate glasses (termed bioactive glasses) which form a bond with living tissues [24 30].

Bioactive glasses have very narrow compositional boundaries to their ability to form bonds between living and non-living matter. Fig. 7 shows a ternary compositional cross section of the Na20 CaO-S iOz-PzO 5 glass system. There is a constant 6 wt% P20 s in all the compositions. The limit of the bone-bonding boundary is at 60% SiO 2. Glasses with more than 60% SiO 2 are biologically inert. Glasses with compositions from 42 to 52% bond to both soft connective tissues as well as to bone [30]. These compositions are designated as having Class A bioactivity [26]. An important aspect of this compositional range is the osteogenic properties of the glasses. Class A bioactive glasses release concentrations of soluble hydrated silicon which activate bone cells to produce growth factors. The hydrated silica gel layer that forms on the glass surface [28] adsorbs and desorbs the cell growth factors which enhance bone growth many times faster than nonosteogenic Class B bioactive materials, and even more rapidly


S!O 2

FIBROUS TISSUE ENCAPSULATION

BONE BONDING

(CLASS B)

, ' " BONE and "', ,' SOFT TISSUE ~' ', BONDING ) ' ' ~ (CLASS A) ~ "

D I S S O L U T I O N

N O N - G L A S S FORMING

CaO Na20

Fig. 7. Bioactive glass bonding boundaries for bone and soft tissues. All compositions contain 6 wt% PzOs.

than autogenous bone by itself [-26, 31 ]. Details of the consequences of the sequence of the inorganic-organic reactions involved in growth of living matter to the nonliving substrate are given elsewhere [26].

A plot of the rate of surface reactions of silicate glasses as a function of S i O 2 c o n t e n t is shown in Fig. 8. The ratio of Na20 /CaO in the glasses is maintained constant for the compositions plotted, thus Fig. 8 is a cross section of Fig. 7 which passes through the three regions of no bioactivity, Class B bioactivity, and Class A bioactivity. Fig. 8 looks remarkably similar to compositional plots for other types of phase transformation; i.e., paraelectric --* ferroelectric; paramagnetic ~ ferromagnetic, etc, where small changes in composition affect the sensitivity of a structure to a change in symmetry and physical properties.

In the case of bioactive glasses the boundaries shown reflect the sensitivity of the nonliving material to inducing, or perhaps even controlling, the disorder ~ order transformations involved in forming the macromolecular structures of living tissues.

7. Implications for the future

The discovery that man-made materials may activate genes that produce growth factors has profound implications. Class A bioactive glasses are already being used

L.L. Hench/Thermochimica Acta 280/281 (1996) I 13 11

O3 >. <~ c~ v

UJ

I--

Z 0 I - -

¢¢ LU r r

2

0

CLASS A BIOACTIVITY

B O N E G R O W T H

--- R E L E A S E OF G R O W T H F A C T O R S

A C T I V A T I O N OF B O N E C E L L S

F O R M A T I O N OF --- H Y D R O X Y C A R B O N A T E A P A T I T E

( H C A ) L A Y E R

--- R E A C T I V E S I L I C A - G E L L A Y E R

--- R E L E A S E OF S O L U B L E S I L I C A

I I I I I I

4 2 4 4 4 6 4 8 5 0 5 2

_ H C A L A Y E R

CLASS B BIOACTIVITY

, i , 5 4 5 6 5 8

,o] BIOACTIV

SILICA CONTENT (MOLE % S i O 2 )

F'ig. 8. Biological phase boundaries for NazO CaO P2Os SiO 2 bioactive glasses as a function of SiO 2 content. (Note large differences in rate of biological reactions as the phase boundaries are crossed.)

clinically for replacement of the ossicles of the middle ear [32], as tooth root implants to maintain the stability of the alveolar ridge for edentulous patients [33], as a particulate to stimulate bone repair around teeth with previous periodontal disease [34], and as a particulate to augment bone repair around failed orthopedic devices. It should be possible in the future to use this concept of bioactive substrate activation of cells to reverse such devastating diseases as osteoporosis which is caused by a "slowing down" of the growth of osteoblasts. The understanding of inorganic activation of organic, macro-molecular ordering reactions may also lead to the design of therapeutic treatments for diseases of the skeletal system or perhaps even dietary supplements which will inhibit onset of the diseases. Thus, understanding the mechanisms underly- ing the mystery of the disorder --, order transformation from nonliving to living offers hope for prolonging the quality of life, one of the most important goals of science today.

12 L.L. Hench/Thermochimica Acta 280/281 (1996) 1 - 13

Acknowledgment

T h e a u t h o r g r a t e fu l l y a c k n o w l e d g e s f i n a n c i a l s u p p o r t of A i r F o r c e Off ice of Sc ien-

tific R e s e a r c h G r a n t N o . F 4 9 6 2 0 - 9 2 - J - 0 3 5 1 .

References

I-1] R. Shapiro, Origins: A Skeptics Guide to the Creation of Life on Earth, Penguin Books, 1988. [-2] A.L Lehninger, Biochemistry, Worth Publishers, Inc., New York, 1982. [3] L.L. Hench, J. Biomed. Mater. Res., 23 (1989) 685 703. [4] S.F. Mason, Chemical Evolution, Oxford University Press, 1992, Chapter 12. [-5] S.L Miller and L.E. Orgel, The Origins of Life on the Earth, Prentice-Hall, Englewood Cliffs, NY, 1974. [-6] A.G. Cairns-Smith and H. Hartman, Clay Minerals and the Origin of Life, Cambridge University

Press, 1986. [-7] F. Dyson, Origins of Life, Cambridge University Press, New York, 1985. [8] W. Schwemmler, Reconstruction of Cell Evolution: A Periodic System, CRC Press, Boca Raton, FL

1984. [-9] F. Hoyle and N.C. Wickramasinghe, Cosmic Origins of Life, J.N. Dent and Sons, London, 1981.

[-10] B. Hartwig and L.L. Hench, J. Biomed. Mater. Res., 6(5) (1972) 413-424. I-11] L.L. Hench and J.K. West, in Paul Ducheyne and David Christiansen (Eds.), Bioceramics 6, Butter-

worth-Heinemann Ltd., Oxford, UK, 1993, pp. 35 40. [12] L.L Hench and J.K. West, J. Vasc. Soc. Technol. A 1215] (1994) 536 543. [13] J.K. West and L.L. Hench, J. Biomed. Maters. Res., 28 (1994) 625-633. [14] J.K. West and L.L. Hench, in T. Yamamuro, T. Kokubo and T. Nakamura (Eds.), Bioceramics 5,

Kobonshi Kankokai, Inc., Kyoto, Japan, 1992, 75 86. [15] D. Evered and M. O'Connor (Eds.), Silicon Biochemistry CIBA Foundation Symposium, Wiley, New

York, 1986, p. 121. [16] J.K. West and L.L. Hench, J. Mater. Sci., 29 (1994) 3601 3606. [17] S. Wallace, J.K. West and L.L. Hench, J. Non-Cryst. Solids, 152 (1993) 101-108. [18] J.K. West and S. Wallace, J. Non-Cryst. Solids, 152 (1993) 109 117. [19] L.L Hench and J.K. West, Life Chemistry Reports, (1995), 1 55. [20] M. Hildebrand, D.R. Higgins, K. Busser and B.E. Volcani, Gene, 132 (1993) 213. [21] B.E. Volcani, personal communication. [22] P.E. Keeting, M.J. Oursler, K.E. Wiegand, S.K. Bonds, T.C. Spelsberg and B.L. Riggs, J. Bone Miner.

Res., 7111] (1992) 1281-1289. [23] P.E. Keeting, K.E. Wiegand, T.C. Spelsberg and B.L. Riggs, A Novel Silicon-containing osteotropic

agent, zeolite A, induces proliferation and differentiation of normal human osteoblast-like cells. American Soc. Bone and Mineral Res. Annual Meeting, Atlanta, G.A. USA, Abst. # 184.

[24] L.L. Hench, R.J. Splinter, W.C. Allen and T.K. Greenlee, Jr., J. Biomed. Mater. Res., 2(1) (1972) 117 141.

[25] L.L. Hench, in J.E. Davies (Ed.), The Bone Biomaterial Interface, University of Toronto Press, 1991, pp. 33 44.

[26] L.L. Hench, in O.H. Andersson and A. Yli-Urpo (Eds.), Bioceramics 7, Butterworth-Heinemann Ltd., Oxford, England, 1994, pp. 3 14.

[27] U. Gross, R. Kinne, H.J. Schmitz and V. Strunz, in D.L. Williams (Ed.), CRC Critical Reviews in Biocompatibility, Vol. 4, Issue 2, CRC Press, Boca Raton, FL, 1988, p. 155.

[28] L.L. Hench, J. Am. Ceram. Soc., 74(8)(1991) 1487-1510. [29] L.L. Hench and June Wilson (Eds.), Introduction to Bioceramics, World Scientific Publishers, London

and Singapore, 1993. [30] J. Wilson, G H. Pigott, F.J. Schoen and L.L. Hench, J. Biomed. Mater. Res., 15 (1981)

805-817.

L.L. Hench/Thermochimica Acta 280/281 (1996) 1-13 13

[31 ] J. Wilson, L.T. Yu and B.S. Beale, in T. Yamamuro, T. K okubo and T. Nakamura (Eds.), Bioceramics 5, Kobonshi Kankokai, lnc., Kyoto, Japan, 1992, 139 146.

[32] G.E. Merwin, in T. Yamamuro, L.L. Hench and J. Wilson (Eds.), Handbook of Bioactive Ceramics, Vol 1, CRC Press, Boca Raton, EL, 1990, pp. 323-328.

[33] J. Wilson, A.E. Clark, E. Douek, J. Krieger, W. King Smith and J. Saville, in O.H. Anderssen and A. Yli-Urpo (Eds.), Bioceramics 7, Butterworth-Heinemann Ltd., Oxford, England, 1994, pp. 415 422.

[34] J. Wilson and S.B. Low, J. Appl. Biomater., 3 (1992) 123 I29.

ELSEVIER Thermochimica Acta 280/281 (1996) 73 82

therm0chimica acta

The applicability of the general theory of phase transformations to glass crystallization 1

Edgar Dutra Zanotto

Department ~f Materials En(tineerin 9- DEMa, Federal University ~/'S£to Carlos- U FSCar. 13565-905, Sgzo Carlos-SP, Brazil

Abstract

The general theory of phase transformation kinetics, derived independently by Kolmogorov, Johnson and Mehl, and Avrami (KJMA), has been used intensively by materials scientists to study various mechanisms of phase transformations in metals, polymers and glasses, especially in kinetic calculations of glass formation. The theory is certainly valid within the framework of its assumptions, and thus its violation in certain cases can be used to infer hidden details of the crystallization mechanisms of supercooled liquids. In this paper the applicability of the KJMA theory to glass crystallization is discussed for three transformation mechanisms: (i) volume crystallization of single crystals, (ii) volume crystallization of branched crystals, and (iii) surface crystallization, in the following silicate glasses: Na20.2CaO.3SiO2; Li20.2SiO z and CaO.MgO.2SiO 2.

Keywords: Crystallization; General theory; Glass; Phase transformation

1. Introduction--the importance of crystallization and the theory of phase transformations

The most obvious crystallization process is that frequently employed by chemists for the synthesis of new compounds , i.e. the precipitation of powder particles from super- saturated solutions. Geologists rely on the "pos t -mor tem" study of crystallization to understand the formation of minerals and solidified magmas. Many solid-state physi-

Dedicated to Professor Hiroshi Suga.

/ 0040-6031,96/$15.00 .~ ~ 1996 Elsevier Science B.V. All rights reserved SSDI 0040-6031(95)02636-3

74 E.D. Zanotto/Thermochimica Acta 280/28,1 (1996) 73 82

cists depend on crystal growth from seeded melts to obtain single-crystal specimens as well as commercially important materials such as silicon and lithium niobate. Cerami- cists and materials scientists dedicate a lot of effort to the synthesis of novel ceramics and glasses employing sol-gel technology. In this case the avoidance of crystal nucleation and growth in the gel, during the sintering step, can lead to a glass. The final microstructure and properties of metals obtained by solidification depend on the crystallization kinetics of the cooling path.

In the glass field, the catalyzed crystallization of vitreous materials can lead to a wide range of pore-free glass-ceramics, with unusual microstructures and properties, such as

transparency, machinability and excellent dielectric, chemical, mechanical and thermal-shock behavior. Finally, the glassy state is only attainable when crystallization (the thermodynamically favorable path) can be avoided during fabrication! Thus the scientific and technological importance of understanding and controlling the crystallization kinetics of glasses is clear. Hence, the use of phase transformation theories can be of substantial help. In addition, this theory is frequently employed to infer phase transformation mechanisms from both isothermal and non-isothermal (DTA or DSC) experiments [la, lb].

2. Nucleation, growth and overall crystallization

When a liquid is cooled below its melting point, crystal nucleation can occur homogeneously or heterogeneously in the volume or on the surface. The classical nucleation theory (CNT) was derived in the late 50's by Turnbull and Fisher [2], and in spite of its limitations it has been used intensively. The subsequent step, crystal growth of the nuclei, leads to a polycrystalline material.

The overall crystallization of a liquid occurs by a combination of nucleation and growth. The kinetics of such a process is usually described by a theory derived independently, in the late 30's, by Kolmogorov [3], Johnson and Mehl [4] and Avrami [5 7], best known as the Kolmogorov Avrami or Johnson Mehl-Avrami theory, hereafter called the KJMA theory. This theory has been used intensively by materials scientists to study various mechanisms of phase transformations in metals, polymers and glasses. Examples of technological importance include the study of the stability of glassy metals, the curing of odontological plasters, the devitrification time of radwast glasses, the development of glass-ceramics, and kinetics calculations of glass formation [8].

Avrami [5 7] assumed that: (i) nucleation is random, i.e. the probability of forming a nucleus in unit time is the same for all infinitesimal volume elements of the assembly; (ii) nucleation occurs from a certain number of embryos (N) which are gradually exhausted; the number of embryos decreases in two ways: by growing to critical sizes (becoming critical nuclei) at a rate v per embryo, and by absorption by the growing phase; (iii) the growth rate (U) is constant until the growing regions impinge on each other and growth ceases at the common interface, although it continues normally elsewhere.

E.D. Zanotto/Thermochimica Acta 280/281 (1996) 73 82 75

Under these conditions, Avrami [5 7] has shown that the volume fraction transformed c~ in isothermal conditions is given by

c~= 1 - e x p v3 e x p ( - v t ) - 1 + v t - ~ - + (1)

where O is a shape factor, equal to 4~/3 for spherical grains, and t is the time period. There are two limiting forms of this equation, corresponding to very small or very

large values of vt. Small values imply that the nucleation rate, I = Nv exp ( - vt), is constant. Expanding exp ( - v t) in Eq. (1) and dropping fifth- and higher-order terms gives

= 1 - exp ( - 9I o Uat4/4) (2)

where I o = .gv. This is the special case treated by Johnson and Mehl [4] and is valid for very large N, when the number of embryos is not exhausted until the end of the transformation (homogeneous nucleation).

Large values of vt, in contrast, mean that all nucleation centers are exhausted at an early stage in the reaction. The limiting value of Eq. (1) is then

= 1 - e x p ( - q N U 3 t 3) (3)

Eq. (3) applies for small IV (fast heterogeneous nucleation). Avrami has proposed that for a three-dimensional nucleation and growth process,

the following general relation should be used

= 1 - exp ( - K t") (4)

where 3 < m < 4. This expression covers all cases where I is some decreasing function of time, up to the limit when I is constant. Eq. (4) also covers the case of heterogeneous nucleation from a constant number of sites, which are activated at a constant rate until becoming depleted at some intermediate stage of the transformation. Table 1 shows values ofm for different transformation mechanisms. Thus, if spherical particles grow in the internal volume of the sample then m should vary from 1.5 to 4. If growth proceeds from the external surfaces towards the center (columnar shape) then m will be different.

In the more general case, where I and U are time-dependent, one can write

:~= 1 e x p ( - 47r 3

\

Table 1 Avrami parameters m for several mechanisms (spherical growth)

Interface-controlled growth Diffusion-controlled growth

Constant I 4 2.5 Decreasing I 3 4 1.5-2.5 Constant number of sites 3 1.5

76 E.D. Zanotto/Thermochimica Acta 280/281 (1996) 73 82

where r is the time of birth of the new phase particles. The above treatment, whilst including the effects of impingement, neglects the effect of free surfaces (thin specimens). This problem was treated by Weinberg [9]

Eq. (4) is usually written as

lnln(1 - ~) l = l n ( K ) + m l n t (6)

This expression is used intensively by materials scientists to infer the mechanisms of several classes of phase transformations from the experimental values of m, i.e. the slope of In in (1 -c~)- 1 versus In t plots. The linearity of such plots is taken as an indication of the validity of the KJMA equation. It should be emphasized, however, that ln-ln plots are insensitive to variations of c~ and t, and that the value of the intercept K is seldom compared to the theoretical value. This is mainly due to the great difficulty in measuring the high nucleation and growth rates in metallic and ceramic (low viscosity) systems.

3. Application to glass crystallization

The KJMA theory can be shown to be exact within the framework of its assumptions. Hence, any violation must be a result of applying it to situations where its assumptions are violated, which may be the case in many crystallization situations.

In an extensive number of studies, the KJMA theory has been employed to analyze experimental data for crystallinity versus time in both isothermal and non-isothermal heat treatments of glass systems. Emphasis was usually given to values of m obtained from the slopes of experimental In In (1 - ~ ) 1 versus In t plots. In Refs. [10-14] for instance, m ranged from 1 for surface nucleation to 3 for internal nucleation. In no case has the intercept been compared with the theoretical value.

Recently, Zanotto and co-authors [15 17] carried out a series of experiments to test the applicability of the Kolmogorov Johnson-Mehl Avrami theory to several cases of glass crystallization. These inlcuded volume nucleation of single crystals and branched crystals as well as surface nucleation of single crystals. Some of these studies will be summarized below.

3.1. Volume nucleation in Na20.2CaO.3SiO 2 glass

In 1988, Zanotto and Galhardi [15] determined the isothermal crystallization kinetics of a nearly stoichiometric Na20.2CaO.3SiO 2 glass at 627 and 629°C (Tg ~ 570°C) by optical microscopy, density measurements and X-ray diffraction. Both nucleation (Io) and growth (U) rates were measured by single- and double-stage heat treatments, up to high volume fractions transformed (e ~ 0.5).

Fig. 1 shows the experimental data for crystallinity, measured up to 0.97, compared with the calculated values, using the independently measured values of I o and U, and m = 4, in Eq. (2). The experimental points initially (up to 15-20% crystallinity) coincide with the theoretical curve but drop for longer times (higher crystallinities). A possible explanation for this behavior is as follows. The large majority of crystalline phases


0 .9

0.7

I-- Z ._1 --J 0 .5

.0.0 / / o x rd ~27°C

eO o m i~

629°C

+t ) - ° I~

ca 0.3

o

0.1

4 8 12 16 T I M E ( h o u r s )

Fig. I. Volume fraction crystallized as a function of time for specimens heated at 627 and 629C: optical microscopy (O, ©), X-ray diffraction (Z]) and density (A) measurements, respectively.

growing in viscous liquids are spherulitic (only partially crystalline). In such cases the "crystallinity" values measured depend on the experimental technique employed. The good agreement between the values determined by the three techniques used in Ref. [15] (XRD, density and optical microscopy) indicates that the NC2S 3 particles are single crystals, in agreement with the electron diffraction determinations of Gonzalez- Oliver [18]. Thus, the decrease in the overall crystallinity with time (when compared with the predicted values) could be due to rejection of impurities at the growth fronts. Indeed, a semi-quantitative EDS analysis (Fig. 2) indicated an increase in Si and a decrease in Na and Ca in the glassy matrix with respect to the crystalline particles. Therefore, it is probable that this would locally increase the viscosity, lowering the crystallization rate. If the particles were spherulites this effect probably would not be so drastic since the rejected impurities could be engulfed between the crystalline arms of the spherulites.

In summary, the early crystallization stages were well described by theory for the limiting case of homogeneous nucleation and interface-controlled growth. For higher degrees of crystallinity, both growth and overall crystallization rate decreased due to compositional changes of the glassy matrix, and the experimental kinetics could be described by theory only if diffusion-controlled growth was assumed. It was also demonstrated that the sole use of numerical fittings to analyze phase transformation

78 E.D. Zanotto/Thermochimica Acta 280./281 (1996) 73 82

Si I ;~ Ca

(a) Na ~l ~Ca I I ' , /

~ i C R Y S T A L L I N E P H A S E

si I

A Ii I t

I, i I f t C a I t

( b ) No I t I Ca

~ G L A S S Y P H A S E

Fig. 2. EDS scan of both glassy and crystalline phases in a sample heated at 627 'C for 14.3 h (c~ ~ 0.7).

kinetics via Eq. (6), as very often repor ted in the literature, can give misleading interpretat ions [15]. It was concluded that if p roper precaut ions are taken, the theory predicts well the initial stages of the glass-crysta l t ransformat ion for the case of homogeneous nucleat ion of single-crystal particles.

3.2. Surface nucleation in CaO.MgO.2SiO 2 (diopside) 91ass

In 1991, the present au thor [16] tested the applicabili ty of the K J M A theory to the case of nucleat ion of single crystals on free glass surfaces. A diopside glass containing 5% A1203, with polished surfaces heat- t reated at 820°C (Tg ~ 720°C), was used. Figs. 3a, b, c show the t ime dependence of the average number of crystals per unit area of glass N~, the m a x i m u m crystal dimension, Rrn , and the fractional area crystallized, respectively.

Fig. 3a shows a large statistical scatter a round an average value (N~ ~ 79 000 crystals per m m 2) which is independent of heat t rea tment time. This behavior is typical for surface nucleation and has been recently repor ted for a number of glasses [19]. Therefore, surface nucleat ion saturates in the early crystallization stages. Fig. 3b shows that the growth rate (U = 0.001 m m h 1) is constant up to 4 h at 820°C (c~ ~ 0.65).


'E E

'O

i x m

E 3 E

O E 1

1.0

uJ

1,4

.J - I

u) >- r r

~ 0 . 8 Z

I-- Call n- Il. ~C tal 0¢ ~t

+ T 0 . . . . . . . .

I °

(a)

s s

s s ~ s

s O S

(b)

O O M

(c)

0 i I

I f

J

t

- - ~ I I I I 2 3

TIME/hours

Fig. 3. Time dependence of(a) the average number of crystals per unit area, N,; (b) m a x i m u m edge size; and (c) fract ional area crystal l ized de te rmined by opt ical microscopy (©) and by scanning electron microscopy ([]). The dot ted line is the theoret ical curve ca lcula ted by Eq. (6).

For the special case under study, i.e. fast nucleation of square-shaped crystals from a constant number of sites on a glass surface,

~+=1 e x p ( - N + U 2t 2) (6a)

where ~+ is the fractional area crystallized. Fig. 3c shows that, despite the statistical scatter, and taking into account the extreme

sensitivity of the exponential expression to small errors in N+ and, especially in U, the evolution of the fractional area crystallized is reasonably well described by Eq. (6a) up to 65% crystallization, the limit of experimental evaluation. In summary, the KJMA describes well the case of rapid heterogeneous nucleation from a fixed number of sites at the glass surface. Thus, in the case of surface crystallization, the possible rejection of

80 E.D. Zanotto/Thermochimica Acta 280/281 (1996) 73-82

impurities at the crystal/glass fronts did not occur or did not affect significantly the crystallization rate up to 65% crystallinity.

3.3. Volume nucleation of branched crystals in Li20.2SiO 2 glass

In a recent study [17], the volume crystallization of Li20.2SiO 2 (kS2) branched crystals in a n L S 2 glass was followed at 500°C (Tg ~ 450°C). This composition is of special interest because it is the canonical system for crystallization studies and a controversy concerning its nucleation behavior (homogeneous versus heterogeneous) was recently resuscitated by Deubener et al. [20]. As the L S 2 spherulites are prolate ellipsoids, the general equation reduces to [17]

= 1 - exp ( - ~I o U a U 2 t4/3) (7)

where U a and U b are the growth rate of the largest and smallest dimensions of the crystals, respectively, as shown by Fig. 4. In this case it is assumed that U b = U c.

Table 2 and Fig. 5 show the experimental volume fraction crystallized, determined by optical microscopy, and the values calculated by Eq. (7), using the independently determined values of crystal nucleation (Io) and growth (U) rates and induction time (r) by both reflected light (RLM) and transmitted light (TLM) microscopy.

In this case, an induction period (r) of approximately 3 h was observed in the nucleation (N V versus time) and growth (length vs. time) curves, and thus t was substituted by (t - r) in the calculations of Eq. (7). Taking into account the difficulty in precisely defining and measuring z, and the exponential sensitivity of Eq. (7) to ~, U b, U a

J

Fig. 4. C r o s s sect ion of a n LS 2 crys ta l s h o w i n g the p r inc ipa l axes.

Tab le 2

Crys t a l l i za t ion p a r a m e t e r s for LS 2 glass a t 5 0 0 ' C [17]

l o /m m 3 h 1 U a / m m h i U b / m m h i z/h

T L M 784 3 8 3 x 10 ~' 231 x 10 6 3 R L M 794 3 7 8 x 10 6 2 0 6 x 10 6 3

E.D. Zanotto/Thermochimica Acta 280/281 (1996) 73-82 81

1.O

? // t,/" /

, / 0.6 / /

~_ TLM ii / o / RLM

g 0 4 - . ,~

¢..)

L L /

0.2 /~ f / / /

t. j

J I I I

20 4 0 60 80 100 120 140

Treatment Time / hours

Fig. 5. Experimental(@) and calculated values ofcrystallinity of Li20.2SiO 2 at 500 C: TLM, values of I and U obtained by transmitted light: RLM, values ofl and U obtained by reflected light.

and Io, it can be concluded that the agreement between the calculated and experimental crystallinity is remarkably good up to 25% crystallinity. Further experiments are being carried out in our laboratory for longer treatment times to reach more advanced crystallinity degrees. Fig. 5 also shows that the predicted crystallinity using the RLM data is in better agreement with experiment than that using the TLM data.

4. Concluding remarks

The general theory of phase transformation kinetics was tested for three crystallization cases without any adjustable parameter. The theory was shown to describe remarkably well the evolution of crystallinity up to ,-~ 20 25% transformation, for volume nucleation in two glasses which nucleate homogeneously, Na20.2CaO.3SiO 2 and Li20.2SiO 2. In the first glass, whose grains are single crystals, the predicted crystallinity is underestimated for more advanced stages due to impurity rejection at the growth fronts. For the other glass, whose grains are branched crystals, no data is yet available for the latter crystallization stages. For a CaO.MgO.2SiO 2 glass, which nucleates heterogeneously on the external surfaces, the theory predicted well the time

82 E.D. Zanotto/Thermochimica Acta 280/281 (1996) 73 82

evo lu t i on of c rys ta l l in i ty up to the highest c rys ta l l in i ty m e a s u r e d ( ~ 65%). In summary , the K J M A theory can be used conf iden t ly to infer the c rys ta l l i za t ion m e c h a n i s m of glasses, bu t careful ly d e t e r m i n e d da t a are needed, at least for the ini t ia l crystal l iza- t ion stages (up to 2 0 % crystal l ini ty) . The m o r e a d v a n c e d stages can often be in f luenced by c o m p o s i t i o n a l changes in the ma t r ix phase. In add i t ion , it can def ini t ively be c o n c l u d e d tha t homogeneous nucleation is the p r e d o m i n a n t m e c h a n i s m for the overal l c rys ta l l i za t ion of LS 2 glass at 500°C.

Acknowledgements

The a u t h o r t h a n k s his fo rmer s t uden t s A.C. G a l h a r d i , R. Basso a n d M.L.G. Leite for o b t a i n i n g mos t of the expe r imen ta l d a t a p resen ted in this paper , T h a n k s are also due to P A D C T (New Mater ials) , con t r ac t no. 620058/91-9 , a n d C N P q , con t r ac t no. 301549 /85-0 for f inanc ia l suppor t .

References

[la] J. Sestak, Thermal Analysis, Elsevier, Amsterdam, 1984, Chapt. 10. [lb] N. Koga, J. Sestak and Z. Strnad, Thermochim. Acta, 203 (1992) 361. [2] D. Turnbull and J.C. Fisher, J. Chem. Phys., 17 (1949) 71. [3] A.N. Kolmogorov, lzv. Akad. Nauk. SSSR, 3 (1937) 355. [4] W.A. Johnson and R. Mehl, Trans. AIME, 135 (1939) 416. [5] M. Avrami, J. Chem. Phys., 7 (1939) 1103. [6] M. Avrami, J. Chem. Phys., 8 (1940) 212. [7] M. Avrami, J. Chem. Phys., 8 (1941) 177. [8] D.R. Uhlmann, J. Non-Cryst. Solids, 7 (1972) 337: J. Am. Ceram. Soc., 66 (1983) 95. [9] M.C. Weinberg, J. Non-Cryst~ Solids, 72 (1985) 301.

[10] S.W. Freiman and L.L. Hench, J. Am. Ceram. Soc., 51 (1968) 382. [11] M. Dusil and L. Cervinka, Glass Technol., 17 (1976) 106. [-12] A. Marona, A. Buri and G.L. Valenti, J. Mater. Sci., 13 (1978) 2493. [13] P. Hautojarvi, A. Vehanen, V. Komppa and E.J. Pajanne, Non-Cryst. Solids, 29 (1978) 365. [14] N.J. Francillon, F. Pacadd and P. Querille, Proc. Int. Symp. Radwaste Management, Berlin, RFA,

1982. [15] ED. Zanotto and A.C. Galhardi, J. Non-Cryst. Solids, 104 (1988) 73. [16] E.D. Zanotto, J. Non-Cryst. Solids, 130 (1991) 217. [17] M.L.G. Leite and E.D. Zanotto, Defended in February 1995, MSc dissertation, DEMa/UFSCar,

Brazil. [18] C.J.R. Gonzalez-Oliver, PhD Thesis, Sheffield University, UK, 1980. [19] E.D. Zanotto, in Nucleation and Crystallization in Glasses and Liquids, American Ceramic Society,

Columbus, OH, 1993, p. 65. [201 J. Deubener; R. Bruckner and M. Sternitzke, J. Non-Cryst. Solids, 163 (1993) 1.

E L S E V I E R Thermochimica Acta 280/281 (1996) 63-71

thermochimica acta

Glass-formation and crystallization kinetics 1

Michael C. Weinberg

Arizona Materials Laboratory, University of Arizona, Tucson, AZ 85712, USA

Abstract

A review is presented of several aspects of crystallization processes in glass and their impact on glass-forming ability. Herein, both isothermal and non-isothermal processes are considered. For isothermal crystallization processes, generalizations of the Johnson Mehl Avrami Kol- mogorov (JMAK) theory are described and the status of certain features of nucleation rate and crystal growth calculations in glasses are given. For non-isothermal crystallization, some features of DTA/DSC analyses and critical cooling rate calculations for glass-formation are discussed, and generalizations of the standard theory for computation of the fraction crystallized are presented.

Keywords: Crystallization; Glass; Isothermal; Kinetics; Non-isothermal

1. Introduction

The topic of crystallization kinetics is of essential importance for glass formation and glass-ceramic synthesis. According to the kinetic viewpoint of glass formation [1, 2], the glass-forming ability of a composition may be assessed by its reluctance to undergo crystallization. Hence, glass formation may be considered in terms of a competition between crystallization and cooling. These notions have been formulated in a quantitative manner, and critical cooling rates for glass formation have been calculated for simple systems [3-5].

The existing descriptions of isothermal and non-isothermal crystallization processes in glasses are based upon two fundamental theories: classical nucleation theory (CNT) [6-8], and the Johnson Mehl-Avrami-Kolmogorov (JMAK) theory [9 12]. The former provides a means for the computation of crystal cluster size distributions and

1 Dedicated to Professor Hiroshi Suga.

0040-6031/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0040-6031(95)02635-5

64 M.C. Weinberg/Thermochimica Acta 280/281 (1996) 63 71

cluster growth rates, while the latter allows one to calculate the volume fraction crystallized in terms of crystal nucleation and growth rates. Thus, CNT provides the input parameters for the formal theory of transformation kinetics expressed by JMAK theory. However, within the past couple of decades, it has been recognized that both CNT and JMAK formalisms have certain limitations and must be modified or generalized in order to be applicable to crystallization processes in glasses [13 15]. Herein, some of these generalizations will be discussed.

Crystallization processes can be studied under isothermal or non-isothermal conditions. Although isothermal crystallization is easier to describe, most recent studies have been performed non-isothermally due to the widespread availability of DTA/DSC instruments. Often the analyses of the latter class of experiments have not been executed critically, and hence many of the DSC/DTA studies which have been published provide little insight into the crystallization mechanisms which are operative in various compositions. Although Yinnon and Uhlmann [16] have given a critical evaluation of the analysis of non-isothermal DTA/DSC experiments in the not too distant past, some discussion of this topic will be presented in this paper.

In the next section, the isothermal crystallization of glass will be considered. In particular, the calculation of the volume fraction crystallized as a function of time, X ( t ) ,

will be reviewed and discussed for several different cases. In addition, theoretical expressions commonly used for crystal nucleation and growth rates will be assessed. In the section thereafter, non-isothermal crystallization processes will be analyzed. Sev- eral issues relating to calculation of critical cooling rates and analysis of DTA/DSC experiments will be discussed. The final section will present some concluding remarks.

2. Isothermal crystallization

For isothermal crystallization, X ( t ) is usually computed from the JMAK equation

ln(1 - X ( t ) ) 1 = C i U , t , + : / ( n + 1) (1)

where U is the crystal growth rate, I is the steady-state crystal nucleation rate, C is a geometric factor dependent upon crystal shape, and n is an integer and corresponds to the dimensionality of the crystallization process. Although it is not always explicitly stated, Eq. (1) is limited by the following conditions: (a) I is time-independent, (b) U is size-independent, (c) nucleation occurs randomly and uniformly, and (d) growth is interface-controlled.

Zanotto and coworkers [17, 18] and Weinberg [19] have evaluated the accuracy of the JMAK equation. However, from the work of Bradley [20] it is clear that the JMAK equation is precise in the limit of large nucleation statistics. In other words, if boundaries can be ignored and nucleation densities are sufficiently large, then the JMAK equation is virtually exact. However, Eq. (1) is not applicable if conditions (a)-(d) are not satisfied, and in such cases other transformation equations must be employed. Below, two examples are considered where generalizations of Eq. (1) are required.

M.C. Weinbery/Thermochimica Acta 280/281 (1996) 63 71 65

Weinberg and Kapral [21] derived expressions for X(t) for systems in which finite size as well as inhomogeneous nucleation effects were present. The theory was developed for both the cases of initial nucleation and continuous nucleation. A simple discrete space and time model was used to compute the kinetics, and analytical results were obtained for X(t). The validity of the analytical results were tested by comparisons with numerical simulations of the nucleation and growth processes for a two-dimensional version of the model. The numerical and analytical results were in precise agreement, confirming the theoretical formulae given for X(t).

Weinberg [14] employed a continuum version of the approach given in Ref. [21] to derive the transformation laws for 2- and 3-dimensional surface-nucleated processes and for the crystallization kinetics of spherical particles where combined surface and bulk nucleation occur. For the analyses of the purely surface-nucleated transformations, initial seeding (nucleation) was e, ssumed. Analytical expressions were given for X(t) and the influence of sample geometry on transformation rate was explored. It was found that 2-D samples of equal area but different shape transformed at nearly equal rates if their perimeter lengths were similar. Standard JMAK theory, e.g. Eq. (1), indicates that a plot of l n ( - l n (1 -X(t))) vs. log t should be linear. However, it was demonstrated that such plots need not be linear for surface-nucleated crystallization. In considering the transformation kinetics of particles with surface and bulk nucleation, a model system was used which satisfied the following conditions: (a) particles are spherical, (b) surface nuclei are present at initial times (with seeding density/unit area = Pr), but nucleation occurs continuously and at a constant rate I in the bulk, (c) crystal growth rate U is constant and follows a spherical growth law for crystals nucleated on the surface or in the bulk. Explicit expressions were derived for X(t), and the relative importance of surface and bulk crystallization kinetics was assessed. For small surface-seeding probability, it was found that the crystallization kinetics were nearly those of a bulk-nucleated sample. However, a plot of l n ( - l n ( 1 -X(t ) ) ) vs. log t did not have a slope of 4 due to finite particle size effects. For comparable values ofl/U and Pr, the crystallization kinetics were found to be quite similar to those from a purely surface-nucleated crystallization process. This feature followed from the fact that at the initial time all surface nuclei were present while the bulk was free of seeds. Although for many conditions linear l n ( - l n ( 1 -X( t ) ) ) vs. log t plots were obtained (with integer values of n), it was shown that such plots did not provide information regarding crystallization mechanism.

The second generalization of JMAK theory deals with the influence of transient effects upon the crystallization rate. For 3-D spherical growth, Eq. (1) may be rewritten a s

l n ( 1 - X( t ) ) l= (41 t /3 ) f to l ( t ' )R3( t ; t ' ) d f (2)

in order to account for time-dependent effects. In Eq. (2), I is the nucleation rate, and R(t; t') is the radius of a particle at time t which nucleated at a time t'. For interface- controlled crystal growth, most previous investigators have treated the time dependence of the nucleation rate, while assuming U constant, e.g. see Ref. [22] and references cited therein. However, Shneidman and Weinberg [-14] have shown that it is an


inconsistent procedure to allow the nucleation rate to be time-dependent without taking into account the size dependence of the growth rate. When this time-dependent problem is treated in a consistent manner, one finds that the corrections to the JMAK equation (Eq. (1)) are larger than anticipated [23, 24]. Shneidman and Weinberg found that corrections to the JMAK equation are particularly significant for moderate to small barrier heights (W*/k T < 30, where W* is the free energy barrier to form a critical nucleus) where nucleation rates are large. For large nucleation barriers, corrections to the usual t 4 law occur only in the time regime where the total volume-fraction crystallized is very small, and thus such corrections are unimportant. Furthermore, it was shown that at long times the corrections to the t 4 law cannot be expressed solely in terms of powers of v/t (where v is the relaxation time) since the correction contains a term dependent upon the logarithm of this ratio.

Hence, for isothermal crystallization processes there are at least two situations where the standard JMAK equation must be generalized. For the case of small barriers to nucleation, transient effects will be important, and time-dependent nucleation and radius-dependent growth must be taken into consideration. In this situation, Eq. (1) will give the leading order contribution for the volume fraction crystallized. If nucleation occurs non-uniformly throughout the sample (such as in surface crystallization), then other expressions are needed for X(t). Such expressions can be derived using the same statistical arguments needed for the derivation of the JMAK equation. Finally, it should be noted that even if nucleation is uniform and transient effects are unimportant, corrections to Eq. (1) can arise from finite sample size effects. However, except for special cases where the surface area is unusually large, these corrections are quite small.

In order to compute X(t), one must be able to calculate the steady-state nucleation rate I. About 15 years ago, comparisons were made between calculated values of I, using CNT, and experimental measurements for crystal nucleation in lithium disilicate (LS2) glass [25, 26], and a huge discrepancy was discovered. Although the temperature dependences of the nucleation rate was described reasonably well by CNT, the predicted magnitude of I was many orders of magnitude too small. Subsequently, James and coworkers (see Ref. [13] and references cited therein) found that the temperature dependences of the crystal nucleation rates in several additional silicate glasses were described quite precisely by CNT. Also, James demonstrated that if the liquid-crystal surface tension was allowed to have a weak temperature dependence, then calculated values of the nucleation rate (using CNT) could be made to agree quite well with experimental data. However, Weinberg and coworkers have found certain difficulties with the application of CNT to problems of nucleation in glass [27 29]. For example, Smith and Weinberg [27] found that CNT could not fit the magnitude or the temperature dependence of the measured nucleation rates in lithium diborate glass. Weinberg and Zanotto [28] indicated that the kinetic prefactor in CNT could be obtained without the use of the Stokes Einstein assumption if transient nucleation data were employed. They used this procedure to re-examine the temperature dependence of the crystal nucleation rate in LS2 glass and several other silicate glasses. They discovered that in all cases differences between CNT and experimental data appeared in the region of large undercoolings. Thus, they concluded that the temperature

M.C. Weinbery/Thermochimica Acta 280/281 (1996) 63 71 67

dependence of the classical nucleation rate was still a matter of controversy. Shneidman and Weinberg [-29] used an asymptotic solution of the transient nucleation equation to compute the barrier to nucleation in LS2 glass. In this manner they were able to asses various kinetic models of nucleation. They found systematic deviations between experimental data and the predictions of all models, which they interpreted as either indicating the presence of pre-existing nuclei or the failure of CNT.

The crystal growth rate U must be known as a function of temperature, too, in order to calculate the fraction crystallized at various temperatures. Unlike metallic systems where crystal growth is governed by thermal diffusion, the kinetics of crystal growth in glasses is most often interface- or diffusion-controlled. Three models have been used to describe interface growth in glasses [30]: normal growth, screw dislocation growth, and surface nucleation controlled growth. For normal growth, the growth rate may be expressed as

U = va [1 - e x p ( - ASfA T/T) (3)

where a is the unit distance advanced by the interface, v is the jump frequency at the interface, and ASf is the molar entropy of fusion in units of the gas constant. In normal growth, the interface is rough on an atomic scale and the crystal liquid interface is non-faceted. According to the work of Jackson [31,32], normal growth will occur for small entropy of fusion materials. The growth rate for screw dislocation growth is

U = f v a [- 1 - exp ( - A Sf A T/T) (4)

where f is the fraction of preferred growth sites. In screw dislocation growth, the interface is smooth and faceted, and growth occurs preferentially on the steps of the dislocation defect. Screw dislocation growth is expected in high entropy of fusion materials. Finally, for surface nucleation growth, the growth rate is given by

U = C v e x p ( - B / T A T ) (5)

where B and C are constants. Surface nucleation growth refers to a growth mechanism in which two-dimensional nuclei form on a crystal surface and grow along the surface. Surface nucleation growth is anticipated in high entropy fusion materials, and it is characterized by a smooth faceted interface.

The status of crystal growth in glasses has been reviewed recently by Uhlmann and Uhlmann [33]. From Ref. [33], one observes that a limited number of quantitative comparisons have been made between the experimentally determined and theoretically predicted crystal growth rates as a function of temperature, and that for only two systems which are predicted to exhibit normal growth (GeO 2 and SiO2) do the measured temperature dependences of the growth rates agree well with theory. However, the experimental and theoretical magnitudes of the growth rates in these systems differ by about an order of magnitude. For systems with high entropies of fusion, there is little agreement between theory and experiment. For example, for systems in which growth occurs by a 2-D nucleation mechanism, a plot of the log of the growth rate times the viscosity vs. 1/(T(T m - T) should be linear (where T m is the melting temperature). This relationship has been experimentally tested for several glass-forming compositions which are believed to exhibit 2-D nucleation crystal


growth [30, 33], and in nearly all cases the observed temperature dependence of the growth rate does not conform to the above-mentioned functional form although the magnitudes of the predicted growth rates are within an order of magnitude of those measured. A notable exception to these observations is the case of crystal growth in lithium diborate where the temperature dependence of the experimental growth rates at large undercoolings is described well by the 2-D nucleation crystal growth model, but the computed values of the growth rates are found to be many orders of magnitude too small [34].

In summary, although the basic theory required for the computation of X(t) for isothermal transformations is in place, a number of "loose ends" exist which make crystallization calculations for glasses somewhat suspect.

3. Non-isothermal crystallization

There have been numerous studies of non-isothermal crystallization kinetics, many of which are related to interpretation of non-isothermal DTA/DSC experiments [35-44]. For non-isothermal transformations, the fraction transformed as a function of time can be written in the form of Eq. (2), where now the time dependence arises from the temperature dependence of the nucleation and growth rates [44]. Additional time-dependent effects may arise if transient nucleation effects are important [46], but herein we focus on situations where steady-state nucleation and size-independent growth rates can be utilized. For such circumstances, and for constant cooling rates, one may transform Eq. (2) into the following expression which involves temperature integrals

X(T)=(4~S 4/3) I (r ' )dT ' U(T")dT" (6) Ti Ti

In Eq. (6), S is the magnitude of the cooling rate, I(T') is the steady-state nucleation rate, U(T") is the crystal growth rate, and T~, Tf are the initial and final temperatures.

If one chooses T~ as the melting temperature and Tf as room temperature (or some sufficiently low temperature where crystallization does not occur) and selects X < 10 o6 as a criterion for glass formation [2], then S c, the critical cooling rate for glass formation, can be obtained by using these parameters in Eq. (6). Although this equation provides a precise expression for S c (within the framework of the quasi-steady scheme), several approximate methods have been utilized for critical cooling rate calculations. Most notably, it has been shown that reasonably good estimates ofS c may be obtained by the "nose method", which relies upon computing the cooling curve which just passes through the nose of a T - T - T (time-temperature-transformation) diagram [47,48]. If T n, t n designate the temperature and time, respectively, at the nose of a T - T - T diagram corresponding to X = 10-o6, then the critical cooling rate is given by

S c = (T m - Tn)/t n (7)

where T m is the melting temperature. It has been observed that critical cooling rates computed by the "nose method" were generally greater than those calculated by other

M.C. Weinber,q/Therrnochimica Acta 280/281 (1996) 63 71 69

methods. Weinberg and coworkers [49] have provided an explanation for this finding. Also, an assessment of the various factors involved in critical rate calculations has been made [48]. In this analysis the following three ingredients were considered: (1) the selection of a kinetic model, e.g. use of Eq. (6) or Eq. (7), etc., (2) the choice of the nucleation and growth expressions, and (3) the selection of the values of several parameters which enter into the nucleation and growth equations. Calculations were performed using parameters appropriate for SiO 2 and GeO 2 glasses. It was shown that S~ is fairly insensitive to the kinetic model, but is quite dependent upon the material parameters which enter into the crystal nucleation and growth expressions. In particular, the critical cooling rate is a strong function of the value of the liquid-crystal surface tension, or the Turnbull parameter [50] which approximately characterizes the latter. Also, S c was shown to be quite sensitive to the value of ACp/ASf (where ACp is the difference between crystal and liquid specific heats).

The critical cooling rate calculations for glass formation described above assume that crystallization initiates homogeneously. However, it is known that most compositions devitrify by a surface crystallization mechanism, and thus traditional critical cooling rate calculations are limited to a particular class of compositions. Recently, a description of surface-initiated non-isothermal crystallization kinetics was given and applied to SiO 2- or GeO2-type glasses [51 ]. Calculations were performed to elucidate the effects of surface nuclei density and cooling rates on the crystallization kinetics. Also, the cooling rates required for the avoidance of a specific value crystallized were computed. It was demonstrated that, unlike the situation for bulk nucleation, critical cooling rates for glass formation in surface-initiated transformations are size-dependent. Thus, glass-formation depends not only on the cooling rate, but also upon the sample dimensions.

As mentioned, about a decade ago Yinnon and Uhlmann [-16] provided a detailed analysis of the myriad of mathematical methods proposed for the analysis of DTA/DSC experiments. In the abstract of Ref. [16] it is stated, "All these methods are based on the Avrami treatment of transformation kinetics and define an effective crystallization rate coefficient having an Arrhenian temperature dependence. Several different ways of mathematically treating the data have been proposed. Most are shown to be based on an incorrect neglect of the temperature dependence of the rate coefficient." The authors go on to say, "It is further argued that in general the overall crystallization rate coefficient is not Arrhenian in character. Thus, in general, non-isothermal transformation cannot be treated analytically." In view of these comments, Weinberg [-52] undertook an effort to delineate those cases where an Arrhenian assumption might be justified. The condition of site saturation was imposed, and the three standard models of crystal growth were utilized. It was shown that for the case of normal growth, the requirement of an Arrhenian form for K, the reaction rate coefficient, could be satisfied by imposing a minimum value to a parameter which is proportional to the number of particles nucleated and inversely proportional to the heating rate. For surface nucleated or screw dislocation crystal growth, though, it was demonstrated that an Arrhenian form for K is not valid. However, even for the latter growth models if a similar parameter exceeds a critical value, then it was shown that the equations which describe non-isothermal DTA/DSC


experiments are quite close in form to those derived using an Arrhenian assumption for K.

The latter analysis was limited in two respects. First, site saturation was assumed without discussing the anticipated domain of validity of this approximation. Also, only the Kissinger [53] method was considered. Thus, in a more recent work [54], the generality of the site saturation assumption was evaluated, and the validity of an Arrhenian assumption for K using other mathematical models for evaluation of experimental data was tested. The site saturation approximation is accurate if homogeneous nucleation and growth curves are reasonably well separated. It was demonstrated that this is the case for most simple inorganic oxide glass-forming systems, and hence site saturation is a good approximation if one restricts attention to sufficiently high temperatures in a DTA/DSC heating experiment. Also, it was concluded that although plots of (l/3)ln(ln(1 - X ) 1) vs. 1 /Tare often found to be linear, only for certain values of the parameters which control crystallization may the slope be associated with an activation energy.

4. Conclusions

Although much information has been compiled regarding crystallization of glasses and supercooled liquids, difficulties still exist in producing reliable crystallization calculations for isothermal or non-isothermal transformations. In the case of isothermal transformations, the basic theory is "in place", but problems arise due to the inadequacy of the nucleation and crystal growth expressions. For non-isothermal transformations, there is a larger gap in knowledge since no theoretical framework exists for the treatment of problems where transient effects are of importance.

Acknowledgements

The author wishes to extend his appreciation to the Office of Basic Energy Sciences of the U.S. Department of Energy, and to the Division of Microgravity Science and Applications of NASA for partial support of this work.

References

[1] D. Turnbull, Contemp. Phys., 10 (1976) 159. [2] D.R. Uhlmann, J. Non-Cryst. Solids, 7 (1972) 337. [3] M.C. Weinberg, B.J. Zelinski, D.R. Uhlmann and E.D. Zanotto, J. Non. Cryst. Solids, 123 (1990) 90. [4] K.F. Kelton and A.L. Greer, J. Non-Cryst. Solids, 79 (1986) 295. I-5] C. Fang, H. Yinnon, and D.R. Uhlmann, J. Non-Cryst. Solids, 57 (1983) 465. 1-6] M. Volmer and A. Weber, Z. Phys. Chem., 119 (1926) 227. [7] R. Becket and W. Doring, Ann. Phys., 24 (1935) 719. 1-8] D. Turnbull apd J.C. Fisher, J. Chem. Phys., 17 (1949) 71. 1-9] W.A. Johnson and R. Mehl, AIME, 135 (1939) 416.

M.C. Weinberg/Thermochimica Acta 280/281 (1996) 63 71 71

[10] M. Avrami, J. Chem. Phys., 7 (1939) 1103. [l 1] M. Avrami, J. Chem. Phys., 8 (1940) 212. [12] A.N. Kolmogorov, Isz. Akad. Nauk SSR, Ser. Fiz. 3, (1937) 355. [13] P.F. James, J. Non-Cryst. Solids, 73 (1985) 517. [14] V.A. Shneidman and M.C. Weinberg, J. Non-Cryst. Solids, 160 (1993) 89. [15] M.C. Weinberg, J. Non-Cryst. Solids, 134 (1991) 116; 142 (1992) 126. El6] H. Yinnon and D.R. Uhlmann, J. Non-Cryst. Solids, 54 (1983) 253. [17] E.D. Zanotto and P.F. James, J. Non-Cryst. Solids, 104 (1988) 73. [18] E.D. Zanotto and A.C. Galhardi, J. Non-Cryst. Solids, 104 (1988) 70. [19] M.C. Weinberg, J. Cryst. Growth, 82 (1987) 779. [20] R.M. Bradley, J. Chem. Phys., 86 (1987) 7245. [21] M. Weinberg and R. Kapral, J. Chem. Phys., 91 (1989) 7146. [22] I. Gutzow, D. Kashchiev and I. Avramov, J. Non-Cryst. Solids, 73 (1985) 477. [23] D. Kashchiev, in J. Nyvlt and S. Zacek (Eds.), Proc. 10th Symp. on Industrial Crystallization, Beckyne,

Czechoslovakia, Sept. 1987, Elsevier, Amsterdam, 1989, p. 3. [24] G. Shi and J.H. Seinfeld, J. Mater. Res., 6 (1991) 2097. [25] G.F. Neilson and M.C. Weinberg, J. Non-Cryst. Solids, 34 (1979) 137. [26] E.G. Rowlands and P.F. James, Phys. Chem. Glasses, 20 (1979) 1. [27] G.L. Smith and M.C. Weinberg, Phys. Chem. Glasses, 32 (1991) 37. [28] M.C. Weinberg and E.D. Zanotto, J. Non-Cryst. Solids, 108 (1989) 99. [29] V.A. Shneidman abd M.C. Weinberg, in M.C. Weinberg (Ed.), Ceramic Transactions, Vol. 30,

Nucleation and Crystallization in Liquids and Glasses, American Ceramic Society, Westerville, Ohio, 1993, p. 375.

[30] D.R. Uhlmann, in J.H. Simmons, D.R. Uhlmann and G.H. Beall (Eds.), Nucleation and Crystallization in Glasses, Advances in Ceramics, Vol. 4, American Ceramic Society, Columbus, Ohio, 1982, p. 80.

[31] K.A. Jackson, Crystal Growth, Pergamon Press, New York, N.Y., 1967, p. 17. [32] K.A. Jackson, in R.H. Doremus et al. (Eds.), Growth and Perfection of Crystals, Wiley, New York, 1958,

p. 319. [33] D.R. Uhlmann and E.V. Uhlmann, in M.C. Weinberg (Ed.), Nucleation and Crystallization in Liquids

and Glasses, Ceramic Transitions, Vol. 30, American Ceramic Society, Westerville, Ohio, 1993, p. 109. [34] G.L. Smith and M.C. Weinberg, Phys. Chem. Glasses, 35, (1994) 6. [35] T. Kemeny and J. Sestak, Thermochim. Acta, 110 (1987) 113. [36] D.T.J. Hurle, Handbook of Crystal Growth, Vol. 1, North Holland, Amsterdam, 1993. [37] J. Sestak, Phys. Chem. Glasses, 15 (1974) 137. [38] A.W. Coates and J.P. Redfern, Nature, 201 (1964) 68. [39] T. Ozawa, Polymer, 12 (1971)150. [40] D.W. Henderson, J. Therm. Anal., 15 (19791 325. [41] H.J. Borchardt and F. Daniels, J. Am. Chem. Soc., 79 (1957) 41. [42] H.S. Chen, J. Non-Cryst. Solids, 27 (1978) 257. [43] J.A. Angis and J.E. Bennet, J. Therm. Anal., 13 (1978) 283. [44] T.J.W. De Bruijn, W.A. De Jong and P.J. Van Den Berg, Thermochim. Acta, 45 (198l) 315. [45] R.W. Hopper, G. Scherer and D.R. Uhlmann, J. Non-Cryst. Solids, 15 (1974) 45. [46] K.F. Kelton, J. Non-Crysl. Solids, 163 (1993) 283. [47] D.R Uhlmann, L. Klein, P.I.K. Onorato and R.W. Hopper, Proc. 1 lth Lunar Sci. Conf., 1980, 197. [48] D.R. Uhlmann, B.J. Zelinski, E.D. Zanotto and M.C. Weinberg, Proc. 15th International Congress on

Glass, Leningrad, USSR, Vol. la, 1989, 156. [49] M.C. Weinberg, D.R. Uhlmann and E.D. Zanotto, J. Am. Ceram. Soc., 72 (1989) 2054. [50] D. Turnbull. J. Appl. Phys., 21 (1950) 1022. [51] M.C. Weinberg, J. Non-Cryst. Solids, 151 (i992)81. [52] M.C. Weinberg, J. Non-Cryst. Solids, 127 (1991) 151. [53] H.E. Kissinger, J. Res. Natl. Bur. Stand., 57 (1956) 217. [54] M.C. Weinberg, Thermochim. Acta, 194 (19921 93.

thermochimica acta

E L S E V I E R Thermochimica Acta 280/281 (1996) 153 162

Estimation of activation energies for structural relaxation and viscous flow from DTA and DSC experiments 1

C. T. M o y n i h a n a,,, S.-K. Lee b, M. Ta t sumisago b, T. Minami b

Materials Science and Engineering Department, Rensselaer Polytechnic' Institute, Troy, NY12180-3590, USA

b Department of Applied Materials Science, Osaka Prefecture University, Sakai, Osaka 593, Japan

Abstract

The activation energy AH* for structural relaxation m the glass transition region can be determined from the heating rate dependence of the glass transition temperature Tg or the cooling rate dependence of the limiting fictive temperature T} measured using DSC or DTA. AH* values determined this way are in good agreement with the shear viscosity activation energies AH*. AH* for high Tg inorganic glasses can also be estimated from the width A(1/Tg) of the glass transition region measured by DSC or DTA using an empirical constant C = (AH*/R)A(1/Tg). New data for sodium borate glasses yield values of the constant C in agreement with those from earlier studies.

Keywords: Activation energy; Differential scanning calorimetry; Differential thermal analysis; Glass transition; Viscosity

1. Introduction

Differential scanning calorimetry (DSC) or differential thermal analysis (DTA) monitors the heat absorbed or evolved by a material as a function of time, most commonly during rate heating or cooling. If a kinetically impeded process such as crystallization or a chemical reaction which is accompanied by absorption or evolution of heat occurs within the sample, DSC and DTA can be used to characterize the kinetics of the process. In this paper we discuss DSC and DTA determinations of the temperature dependence of the relaxation times controlling the structural relaxation and viscous flow processes associated with the glass transition.

* Corresponding author. Dedicated to the Professor Hiroshi suga.

0040-6031/96/$15.00 ;¢) 1996 Elsevier Science B.V. All rights reserved SSDI 0 0 4 0 - 6 0 3 1 ( 9 5 ) 0 2 7 8 1 - 5

154 C.T. Moynihan el al./Thermochimica Acta 280/281 (1996) 153 I62

2. Structural relaxation and the glass transition

Structural relaxation in a glass-forming liquid refers to the kinetically impeded rearrangement of the liquid structure in response to changes in external variables such as temperature T (the variable relevant to DSC and DTA studies), pressure or electric field. Good reviews of the methodology discussed below and used to analyze structural relaxation experiments are given in Refs. [1-8]. Hodge's recent and extensive review [8] is particularly recommended.

Shown schematically in Fig. 1 is the response of the enthalpy H of a glass-forming melt initially in equilibrium to a step change in temperature from T t to T 2 imposed at time t = 0. The melt initially exhibits a "fast" glass-like change in H associated primarily with the vibrational degrees of freedom. This is followed by a "slow" or kinetically impeded further change in H associated with liquid-like degrees of freedom and generally thought to involve rearrangement of the liquid structure. This structural relaxation progresses until equilibrium is reached at the new temperature T 2. For computational convenience, results of experiments of this sort are frequently described in terms of the fictive temperature Tf, which is the relaxational part of the measured property (enthalpy H, here) expressed in equivalent temperature units. Fictive temperature is defined so that at equilibrium Tf = T. As shown in Fig. 1, during the slow structural relaxation following a step change in temperature from T 1 to T 2, Tf varies from T I to T 2 in parallel with the change in H.

Structural Relaxation in Liquids

>

1-1

T2

H slow,

0 time

11

h

Fig. 1. Schematic plot of enthalpy H and fictive temperature Tf versus time during isothermal structural relaxation following a step change in temperature.

C.T Moynihan et al./Thermochimica Acta 280/281 (1996) 153 162 155

The time dependence of the fictive temperature for the isothermal relaxation in Fig. 1 can be described quite accurately for moderate departures from equilibrium by the relaxation function

(1)

The relaxation function ~b(t) is, first, nonexponential and requires a distribution of relaxation times ri with corresponding weighting factors gi (~ggg = 1). Second, the relaxation function is nonlinear, so that the relaxation times ~ depend both on temperature T and on the instantaneous structure, often parameterized by Tf. Since Tf varies with time, this imparts an implicit time dependence to the % One expression for the ~ which works well for moderate departures from equilibrium and over a moderate temperature range in the glass transition region is a modified form of the Arrhenius equation often called the Tool Narayanaswamy equation

xAH* - x)AH* lnri =lnTi° + ~ +(1 RTf (2)

The ~0 are pre-exponential constants, R the ideal gas constant, AH* the activation energy (more properly, the activation enthalpy) for structural relaxation, and x (0 < x < 1) a nonlinearity parameter which partitions the dependence of the vg on T and Tf. Note that both for relaxation close to equilibrium (1/Tf ~ l/T) and for relaxation a moderate but roughly constant "distance from equilibrium (1/Tf = 1/T + const.), the temperature dependence of ~i is given by

dlnr#d(1/T) = AH*/R (3)

Cooling or heating a liquid or glass at a rate q = dT/dt can be thought of as a series of small temperature steps ATfollowed by isothermal holds of duration At = AT/q. This is illustrated schematically in Fig. 2a [6] for the variation in the experimentally measured enthalpy H and in the equilibrium enthalpy H e and temperature T(equilibrium fictive temperature) during stepwise cooling and subsequent reheating through the glass transition region. At high temperatures, the r~ are short enough that the system relaxes completely and hence exhibits liquid-like behavior during the characteristic time interval At. At low temperatures the r~ are very long compared to At, so that no relaxation occurs and glass-like behavior is observed in the time interval At. The temperature range in between these two extremes is the glass transition region and is delineated in Fig. 2b [6], which shows the variation in H during cooling and reheating and the characteristic hysteresis between the cooling and reheating curves.

A decrease in cooling or heating rate q will increase the characteristic time scale At. Since the relaxation times z~ increase with decreasing temperature, this will shift the glass transition region to lower temperatures. This is shown schematically in Fig. 3 [6, 9] both for the H vs. Tcooling and reheating curves and for the corresponding plots of heat capacity Cp( = dH/dT) vs. T. The location of the glass transition is commonly demarcated by some characteristic temperature (the glass transition temperature) on the H or C, vs. Tcurves. This might be taken as the temperature Tg for the extrapolated

156 C.T. Moynihan et al./Thermochimica Acta 280/281 (1996) 153 162

o~ .1-

L_ ,

L _ _

Q I I

I L - - - 1

I I I I I - - - J

×

F - - - ~1

t I

I

i I'---

' H I

- - - T,H e

H

t ime

glass

I 1 I I I

T

Fig. 2. Schematic plots of(a) variation with time of temperature T, equilibrium enthalpy H e and experimental enthalpy H, and (b) H versus Tduring stepwise cooling and reheating in the glass transition region.

onset of rise of the Cp vs. T reheating curve, as shown in the lower part of Fig. 3, or as the extrapolated temperature T'f of intersection of the equilibrium liquid and glass H vs. Tcool ing or reheating curves, as shown in the upper part of Fig. 3. T'f is often referred to as the limiting fictive temperature attained by the glass on cooling. T'f can be determined by suitable integration of the Cp or heat flow reheating curve as measured by DSC or D T A and illustrated in Fig. 4.

3. Activation energies from the cooling or heating rate dependence of T~ or T'f

The output o f a DSC or DT A during heating is propor t ional to the heat capacity Cp of the specimen and consequently provides a straightforward way of detecting the glass transition. Moreover , since the characteristic temperatures Tg or T'f demarking the

C.T Moynihan et al./Thermochimica Acta 280/281 (1996) 153 162 157

H

Cpe

Cp

Cpg

glass[ transition region I liquid /

(A) fast cool and . ~ heat at rate IqAI

(B) slow cool and / 'f

t

Temperature

Fig. 3. Schematic plot of enthalpy H and heat capacity Cp during cooling and reheating through the glass transition region at two different rates. From Ref. [9].

glass transition depend on the cooling or reheating rate q, and since this dependence arises from the temperature dependence of the relaxation times ri, DSC or DTA heating data can be used to determine the activation energy AH* of Eq. (3) for the structural relaxation times. Two general techniques for this have gained acceptance over the past twenty years [8].

The first technique involves measurement of the glass transition temperature Tg as a function of heating rate qh. A necessary constraint is that prior to reheating the glass must be cooled from above to well below the glass transition region at a rate qc which is either equal to or proportional to the reheating rate qh. Subject to this constraint, it may be shown [9] that

d In q h / d ( 1 / T g ) = - A H * / R (4)

The second technique involves measurement of the limiting fictive temperature T'f as a function of cooling rate q~. T'f may be determined (as in Fig. 4) during subsequent


0,5

0.3

4 6 0 ' 4 ' ' 80 T(K) 560 6~,0 720

Fig. 4. Heat capacity versus temperature measured during reheating of B203 glass at I 0K m i n 1 after cooling through the glass transition region at - 10 K min - 1. T'f is the limiting fictive temperature attained by the glass on cooling to well below Tg at this rate. From Ref. [10].

reheating at any rate (typically ~ 10 K min- 1) which gives good precision on the DSC or DTA. The only experimental constraint is that the cooling must commence above and end well below the glass transition region. In this case one obtains [10]

d In Iq¢l/d(1/T'f) = - AH*/R (5)

As noted by Hodge [8], this second technique is considerably more practicable than the first technique. Fig. 5 shows a plot ofloglqcl vs. 103/T'f for AszSe 3 glass Ell] from which AH* may be obtained via Eq. (5).

Shown in Table 1 for a number of glass-forming liquids are values of the activation energy AH* for structural relaxation in the glass transition region determined via Eqs. (4) or (5), along with values of Tg measured during heating at a rate between 5 and 20Kmin 1 following cooling at a comparable rate. Also shown are experimental

y o

~'(K) 4 5 5 4 5 0 4 4 5 4 4 0 435

2.18 ' 2.~'2 ' 2.26' ' 2.~( 103/Tf'(K)

Fig. 5. Logarithm of cooling rate Iqcl versus reciprocal of limiting fictive temperature T'f for As2Se 3 glass. From Ref. [11].

C.T. Moynihan et al./Thermochimica Acta 280/281 (1996) 153 162 159

Table 1 Glass transition temperatures Tg measured at heating rates of 5 20 K min 1, activation energies AH* for structural relaxation determined via Eq. (4) or (5), and activation energies AH* for shear viscosity in the glass transition region for various glass-forming liquids

Glass AH*/ AH*/ Refs. Tg/(K) (kJmol 1) (kJmol 1)

5P4E(5-phenyl-4-ether) 244 322 294 [1] CK N (0.4Ca(NO3) 2 339 590 590 [ 10,12]

0.6KNO 3) AszSe 3 454 342 322 [11, 13] B20 3 557 385 385 [10,13] ZBLA(58ZrF 4 33BaF 2 587 1400 1140 [5, 14]

- 5LaF34AIF3) NBS 711 (lead silicate) 722 374 411 [13, 15] NBS 710 (alkali lime silicate) 839 612 612 1-13, 16] BSC (alkali borosilicate) 843 615 615 [10, 13]

values of the activation energy AH* for the shear viscosity q in the glass transition region. It is generally thought that the atomic or molecular rearrangements taking place during structural relaxation are similar to those occurring during viscous flow in response to a shear stress. In support of this, one finds that the activation energies AH* and AH* are equal within experimental error (about 10%) for seven of the eight glass-forming liquids in Table 1. The one apparent exception is the heavy metal fluoride glass ZBLA, where the AH*/AH* ratio appears to be roughly 1.2. This AH*/AH* ratio for ZBLA may still be unity within experimental error, given the extreme steepness of the In [qcl vs. 1/r'f and In q vs. 1/Tplots which correspond to these very large AH* and AH* values. Angell [17] has noted that some discrepancy between AH* and AH* might be expected for very fragile liquids with large AH*/Tg values and may be indicative of decoupling near Tg of the faster viscous flow process from the slower structural relaxation process. Nonetheless, as illustrated by the data in Table 1, it appears that AH* values determined via Eq. (4) or (5) give an accurate estimate of the shear viscosity activation energy AH*.

4. Activation energies from the width of the glass transition region on DSC or DTA heating curves

The breadth or width of the glass transition region as measured during heating by DSC or DTA may be parameterized as the extrapolated temperatures Tg for onset of rapid rise of the Cp or heat flow vs. T curve and T'g for completion of the Cp overshoot [13]. These are illustrated in Fig. 6 for a 40Na20 60B203 glass [18]. The shapes of Cp or heat flow reheating curves and hence the relative values of Tg and T'g are sensitive to the thermal history (cooling rate, time and temperature of sub-Tg anneals) prior to reheating. However, it is found experimentally [13] for a given reheating rate

160

0 B

O ) ¢ -

C.T. Moynihan et al.i'Thermochimica Acta 280/281 (1996) 153 162

40Na20 - 60B203

Tg

I I I I

300 350 400 450 500

T(°C)

Fig. 6. Heat flow measured by DSC during heating of 40Na20 60B203 glass at l0 K rain i after cooling through the glass transition region at a slow rate. Tg and T'g mark respectively the extrapolated onset and finish of the glass transition region during rate heating.

that the values of Tg and T'g are unaffected within experimental error if the glass is reheated immediately after cooling through the glass transition region and if the ratio of the cooling to reheating rate is kept in the range roughly

0.2 < Iqjqhl < 5 (6)

As shown in Fig. 2, the low temperature boundary of the glass transition region, Tg, is the temperature at which the structural relaxation times r i become on average short enough that the glass can start to exhibit detectable relaxation on the characteristic time scale At = AT/q h. The high temperature boundary, T'~, is the temperature at which the ri become extremely short, so that the glass can almost completely equilibrate in time At. In terms of the parameters of Eqs. (1) and (2), the distance between Tg and T'g will become larger the smaller the temperature dependence of the relaxation times (the smaller AH*) and the broader the distribution of relaxation times, but will be relatively insensitive to the value of the nonlinearity parameter x [13]. If the experimental constraint of Eq.(6) is met, one thus predicts for glasses with identical distributions of relaxation times on a logarithmic scale that

( A H * ~ ( 1 ~)~(AH*~(1 _~)_(AH*']A(l~=constant=C ~ J \ T = \ R ] \T= \ R J \T=] (7)

C.T. Moynihan et al./Thermochimica Acta 280/281 (1996) 153 162 161

In Ref. [13], Eq. (7) was tested and found to be valid to a high degree of approximation for 17 inorganic glasses with Tg well above ambient temperature. These glasses included the last six entries in Table 1 (AszSe 3 through BSC), as well as other silicates, chalcogenides and heavy metal fluorides. The observed range in the C values was 4.3-5.6, with a mean of

C = 4.8 + 0.4 (8)

Recently, shear viscosity and DSC heating data have been acquired in the glass transition region for a series of sodium borate glasses with compositions in the range 15 40 mol% Na20 [18]. The DSC heating curves were measured at a heating rate of 10 K min 1 after prior cooling through the glass transition region at a slow rate, so that the constraint of Eq. (6) is approximately satisfied. The DSC heat flow curve is shown in

! Fig. 6 for one of these glasses. In Table 2 are listed Tg, Tg and AH* values along with the values of (AH*/R)A(1/Tg) = C calculated from Eq. (7) for these five sodium borate glasses. For completeness, data for pure B z O 3 glass taken from Ref. [13] are also included in Table 2. The mean value of C for the six glasses in Table 2 is

C = 5.5 + 0.4

which is at the high end of, but overlaps, the range of C values determined for the 17 glasses in Ref. [13].

An updated generic value of C may be obtained by averaging the C values of the five sodium borate glasses of Table 2 and the C values of the 17 glasses in Ref. [13]. This gives

C = 5.0 _+ 0.5 (9)

As detailed in Ref. [ 13], this mean value of C may be used in Eq. (7) to make a very good estimate for high Tg inorganic glasses of the shear viscosity activation energy AH* in the glass transition region using only values of Tg and T'g obtained from a single DSC or DTA heating curve. As also detailed in Ref. [13], however, it should be remembered that the constancy of C is a strictly empirical result, since it depends on the assumption that the distribution of relaxation times is roughly the same for all these glasses. There

Table 2 Lower and upper limits, Tg and T'g, of the glass transition region during heating at 10 K min 1, shear viscosity activation energy AH* in the glass transition region and (AH*/R)A(1/Tg) calculated from Eq. (7) for B20 3 glass [13] and five sodium borate glasses [18]

Composition AH*/ (mol%) Tg/(K) T'g(K) (kJ tool ~) (AH*/R)A(1/Tg)

B20 3 557 591 385 4.8 15Na20 85B103 696 731 682 5.6 20Na20 80B203 743 774 847 5.5 25Na20 75B203 755 785 988 6.0 30Na20 70BzO 3 756 781 1137 5.8 40Na20 60B20 3 719 741 1035 5.1


exist low-Tg inorganic glasses, such as the CKN glass in Table 1, for which the experimental value of C is nearly twice that given in Eq. (9).

5. Conclusions

From a pragmatic standpoint, the main virtue of using DSC or DTA data to obtain viscosity activation energies is probably convenience. One can obtain DSC or DTA data on very small samples, and there is no need to fabricate larger samples of precise dimensions, as in the preparation of beam-bending or fiber-elongation viscosity specimens. With some further assumptions about the constancy of the viscosity t/at Tg, one can use the AH* values from DSC or DTA to make estimates of the melt viscosities up through the working range [13]. The validity of A H * ( ~ AH*) determinations via Eq. (4) or (5) seems now to be well established, but these determinations require measurement of a number of DSC or DTA heating scans for different prior cooling rates. Evaluation of AH* via Eqs. (7) and (9) is much easier, since only one scan is required. However, the implied constancy of C in Eq. (9) needs to be checked out for a much larger number of inorganic glass-forming systems to obtain a better feel for just what classes of glass conform to this empirical correlation.

References

[1] C.T. Moynihan et al., Ann. NY Acad. Sci., 279 (1976) 15. [2] G.W. Scherer, J. Am. Ceram. Soc., 67 (1984) 504. [3] G.W. Scherer, Relaxation in Glass and Composites, Wiley-lnterscience, New York, 1986. [4] G.W. Scherer, J. Non-Cryst. Solids, 123 (1990) 75. [5] C.T. Moynihan, S.N. Crichton and S.M. Opalka, J. Non-Cryst. Solids, 131 133 (1991) 420. [6] C.T. Moynihan, in R.J. Seyler (Ed.), Assignment of the Glass Transition, ASTM STP 1249, Philadel-

phia, 1994, p. 32. [7] J.-P. Ducroux, S.M. Rekhson and F.L. Merat, J. Non-Cryst. Solids, 172 174 (1994) 541. [8] I.M. Hodge, J. Non-Cryst. Solids, 169 (1994) 211. [9] C.T. Moynihan, A.J. Easteal, J. Wilder and J. Tucker, J. Phys. Chem., 78 (1974) 2673.

[10] C.T. Moynihan, A.J. Easteal, M.A. DeBolt and J. Tucker, J. Am. Ceram. Soc., 59 (1976) 12. [11] A.J. Easteal, J.A. Wilder, R.K. Mohr and C.T. Moynihan, J. Am. Ceram. Soc., 60 (1977) 134. [12] H. Tweer, N. Laberge and P.B. Macedo, J. Am. Ceram. Soc., 54 (1971) 121. [13] C.T. Moynihan, J. Am. Ceram. Soc., 76 (1993) 1081. [14] S.N. Crichton, M.S. Thesis, Rensselaer Polytechnic Institute, 1986. [15] S.N. Crichton and C.T. Moynihan, J. Non-Cryst. Solids, 102 (1988) 222. [16] H. Sasabe, M.A. DeBolt, P.B. Macedo and C.T. Moynihan, Proc. l lth int. Congress on Glass, Prague,

1977, p. 339. [17] C.A. Angell, J. Non-Cryst. Solids, 102 (1988) 205. [18] S.-K. Lee, M. Tatsumisago and T. Minami, J. Ceram. Soc. Jpn., 103 (1995) 398.

ELSEVIER Thermochimica Acta 280/281 (1996) 163 174

thermochimica acta

Identifying internal and surface crystallization by differential thermal analysis for the

glass-to-crystal transformations1

C.S. Ray*, D.E. Day Ceramic En(jineeriny Department and Graduate Center Jor Materials Research,

University of Missouri-Rolla, Rolla, MO 65409, USA

Abstract

A differential thermal analysis (DTA) method has been developed that identifies and distinguishes surface and internal (bulk) crystallization that occurs during the crystallization of a glass. This method is rapid, convenient and requires only a few (about 6 8) DTA experiments to identify the dominant crystallization mechanism (bulk vs. surface) in the glass. In this method, either the maximum height of the DTA crystallization peak, (6 T)p, or the ratio Tp 2/(A T)p where Vp is the temperature at (~T)p and (z~ T)p is the peak half-width, is determined as a function of size of the glass particles used for the DTA measurements.

When analyzed by this technique, an as-quenched lithium disilicate (LS2) glass was found to crystallize predominantly by surface crystallization. The tendency for surface crystallization was enhanced when the glass particles were exposed to moisture prior to DTA. Internal or bulk crystallization dominated over surface crystallization when this LS 2 glass was doped with small amounts of platinum.

The DTA curves in the literature for several soda lime silica glasses as a function of particle size were analyzed by the present method. The analysis showed that Na20.CaO.2SiO 2 and Na20.2CaO.3SiO 2 glasses crystallized by internal crystallization, but surface crystallization was the dominant crystallization mechanism for an Na20.CaO.3SiO 2 glass. These results agree with those obtained from an analysis of the apparent activation energy for crystallization as a function of particle size for these glasses.

Keywords: Crystallization; DTA; Glass; Mechanism; Transformation

*Corresponding author; Tel: (314)341-6432; Fax:(314)341-2071. 1 Dedicated to Professor Hiroshi Suga.

0040-6031/96/'$15.00 (~_~ 1996 Elsevier Science B.V. All rights reserved SSD! 0040-6031(95)02640- 1

164 C.S. Ray, D.E. Day/Thermochimica Acta 280/281 (1996) 163 174

1. Introduction

The overall crystallization process that occurs in glasses during reheating is a complex structural reorganization that is hard to classify uniquely into the so-called internal (bulk) or surface mechanisms. In most glasses, crystallization by internal and surface mechanisms proceeds simultaneously and competitively. The kinetic and thermodynamic properties, such as the diffusion coefficient, difference in free energy (per unit volume) and interracial energy between the crystalline and glassy phases, molar volume, and the entropy of fusion, all of which depend upon the glass composition, generally determine which specific mechanism would dominate the crystallization of a particular glass [1 3]. Knowledge of the dominant crystallization mechanism for the glass-to-crystal transformation has both scientific and technological value. For example, in developing glass-ceramics with desirable microstructure and properties, prior knowledge of the crystallization mechanism is useful in selecting the appropriate temperature time processing conditions. For glasses having a strong tendency to crystallize internally, the controlled nucleation and crystallization of the glasses in bulk form is the desirable route to prepare glass-ceramics [4]. Glasses where surface crystallization is dominant are cerammed by sintering glass particles at a suitable temperature and pressure [4]. To prepare tape-cast, screen-printed, piezoelectric or pyroelectric glass-ceramics, a base glass which crystallizes by a surface mechanism is highly desirable [5].

This paper describes an experimental method that can rapidly and conveniently determine whether a glass crystallizes primarily by surface or bulk crystallization. In the present technique, which uses differential thermal analysis (DTA), the maximum height or half-width of the DTA crystallization peak is measured as a function of the particle size of the glass sample, while using a constant DTA heating rate for all the measurements. An Li20.2SiO2(LS2) glass was used primarily in the present investigation. The tendency for internal crystallization was increased and its effect investigated using glasses containing small amounts of platinum which functioned as a nucleating agent. Likewise, the surface nucleation sites were increased by deliberately exposing the glass particles to a moist atmosphere prior to the DTA measurements. The applicability of the present technique was verified using the DTA curves reported [6 8] previously for the N a 2 0 - C a O SiO 2 glasses. The theoretical justification of this experimental method for differentiating internal and surface crystallization is currently under study.

2. Experimental procedure

A well-mixed, 50g batch of the Li20.2SiO 2 (LS2) composition was melted in a platinum crucible at 1475°C for 3 h (air atmosphere) and then cast between two steel plates. Before casting, the melt was stirred periodically (30-40min intervals) with a silica rod to ensure homogenization of the glass. X-ray diffraction (XRD) and examination by scanning electron microscopy (SEM) showed no evidence of unmelted or crystalline particles in the as-quenched glass. The silica content of the glass was

C.S. Ray, D.E. Day/Thermochimica Acta 280/281 (1996) 163 174 165

determined by measuring the intensity (counts per s) of the Si- K~ peak for the glass by energy-dispersive X-ray analysis (EDAX) and comparing it with that of a silica standard for an identical beam current and scan area. Based on 12 different locations, each --~0.1 x 0.1 mm 2 area, the average silica content of the glass was within + 1.5 wt% of that in the batch. The quenched glass was ground and screened to five different particle sizes: 0.025 0.045, 0.075-0.106, 0.180-0.300, 0.425 0.500, and 0.850- 1.190 ram, which were stored in a vacuum desiccator until used for DTA measurements. For analyzing the experimental results as a function of glass particle size, each of the five different ranges of particle size was represented by the numbers 0.035, 0.090, 0.240, 0.462, and 1.020 mm, which are close to the average for each size range.

The effect of particle size on the DTA peak profile for the L S 2 glass was investigated by heating glass particles of different size in the DTA apparatus at 15°C min- 1 from room temperature until crystallization was complete. Platinum containers were used for both the glass and reference (alumina) samples and a flowing nitrogen atmosphere of --~ 50 cm 3 min- 1 was maintained during the DTA measurements. The DTA machine was calibrated periodically using In, Zn, and A1 standards at the same conditions used for the sample measurements. The data were recorded and analyzed in a computer interfaced with the DTA machine. Except for the platinum-containing glasses, the weight of the glass samples was held constant at 40 mg. The samples used for DTA were as-quenched glass and received no pre-nucleation treatment.

To enhance the effect of surface crystallization, surface nucleation sites were deliberately created by exposing the glass particles to moist air at 100% relative humidity for 80 h before they were used for the DTA measurements (wet samples). Three different particle sizes, namely 0.035, 0.090, 0.462mm, were used for this investigation. The same-sized glass particles were also kept in an oven at 120°C for the same 80 h time (dry samples) and then crystallized in the DTA in order to compare the DTA results for dry and wet glass particles.

Since the tendency for internal crystallization is expected to increase when heterogeneous nuclei are uniformly distributed throughout the glass, DTA measurements were repeated for L S 2 glasses containing 0.001 and 0.005 wt% platinum. The results were compared with those of the L S 2 base glass (free of platinum). The platinum- containing glasses were prepared by adding a PtC14 solution to the batch, melting the batch and quenching the melt in the same way as was done for the undoped LS 2 glass.

3. Results and discussion

3.1. Li20.2SiO 2 ( LS2) glass (undoped )

The decrease in the maximum height of the DTA crystallization peak (6T)p measured at 15°Cmin - 1 for 40mg of as-quenched LS 2 glass, which occurred with increasing glass particle size, is shown in Fig. 1. The ratio of the total effective surface area to the total volume for a fixed weight of glass decreases with increasing particle size. Consequently, the number of active surface nuclei and, hence, the heat generated from the surface crystallization decrease as the particle size of the glass increases. The

166 C.S. Ray, D.E. Day/Thermochimica Acta 280,/281 (1996) 163 174

L)

,z

¢h

,< [.,

9 . 0

8 .0

7.0

6.0

5 .0

4.0

3.0 0.00 0.20 0 .40 0 .60 0 .80 1.00 1 .20

Particle Size, m m

Fig. 1. The maximum height of the DTA crystallization peak (6T)p for lithium disilicate glass as a function of particle size. Sample weight: 40rag. DTA heating rate: 15 "C rain ~.

decrease in (c]T)p observed with inc reas ing par t ic le size, therefore, ind ica tes tha t an a s - q u e n c h e d LS 2 glass crystal l izes p r imar i l y by surface crys ta l l iza t ion . H a d b u l k ( internal) c rys ta l l i za t ion been the d o m i n a n t m e c h a n i s m for the c rys ta l l i za t ion of this glass, (6T)p wou ld have increased r a the r t h a n decreased wi th inc reas ing par t ic le size.

The (~T)p for 40 m g of the dry a n d wet LS 2 glasses is c o m p a r e d in Fig. 2 as a func t ion of par t ic le size. Clearly, the effect of surface c rys ta l l i za t ion on (c~ T)p is larger in the wet

12.0

Q'k • Dry \ 0 Wet

~ 10.0 • Fig. I

8 .0

6.0 <

e~

4.0 • 0 .00 0 .10 0 .20 0 .30 0 .40 0 .50

Particle Size, m m

Fig. 2. Dependence of maximum height of the DTA crystallization peak (~T)p on particle size for lithium disilicate glass under different conditions of storing the glasses prior to DTA. The glass particles were stored in a vacuum desiccator (desiccated, shown in Fig. 1), in an oven at 120 °C for 80 h (dry), or in moist air at 100% relative humidity for 80 h (wet). Sample weight: 40 rag. DTA heating rate: 15 °C rain- 1.

C.S. Ray. D.E. Day/Thermochimica Acta 280/281 (1996) 163 174 167

samples than that in the dry samples. The difference in (6T)p between the wet and dry samples is more pronounced for the smaller particles, where the total effective surface area is larger, than for the larger particles. The decrease in (6 T)p with increasing particle size even for the dry samples confirms that surface crystallization is the dominant mechanism in the crystallization process of this lithium disilicate glass. Also compared in Fig. 2 (solid triangles) is the dependence of (OT)p on particle size shown in Fig. 1 for the glass stored in a vacuum desiccator. This comparison shows that the curve for the glass stored in the vacuum desiccator is very close to that for the dry glass, which suggests that storing in a vacuum desiccator keeps this glass reasonably dry. However, the slightly larger (6T)p values for the samples stored in the vacuum desiccator, compared to the samples dried at 120°C for 80h, suggest a slight moisture attack occurred on the particles in the vacuum desiccator.

The temperature corresponding to the maximum of the DTA crystallization peak, Tp, increased with increasing particle size, Fig. 3, in an identical fashion for the wet, dry, or desiccated glass. This increase in Tp indicates that crystallization becomes increas- ingly difficult with increasing particle size. Except for the smallest 0.035 mm particles of the wet glass, the Tp for the dry, wet, and desiccated glasses was almost indistinguish- able at any particle size. The Tp for the 0.035 mm wet particles was about 8°C lower than that of the dry and desiccated particles. As indicated by the peak height plot in Fig. 2, the wet glass contains a higher number of surface nuclei, especially for the smaller particle sizes, which makes these particles easier to crystallize. So, the observed lower value of Tp for the 0.035 mm wet particles compared to that for the dry and desiccated particles is expected.

The crystal growth dimension n (also known as Avrami parameter), given by [9]

n = [2.5/(A T)p]/(E/R T 2) (1)

670

~ 660-

~. 650-

6 4 0

~" 6 3 0

~ 6 2 0

~" 6 1 0 <

~ 61111

• Dry 0 Wet.

0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0

Partic le Size, m m

Fig. 3. The temperature at the maximum height of the DTA crystallization peak Tp for lithium disilicate glass particles which have been stored at different conditions prior to DTA. For conditions of experiments and of storing the glass particles, see Fig. 2.

168 c.s. Ray. D.E. Day/Thermochimica Acta 280/281 (1996) 163 174

is frequently used to describe surface or internal crystallization. A value of n close to 3 signifies internal or three-dimensional crystallization and a value close to 1 indicates surface crystallization. In termedia te values of n between 1 and 3 are indicative of both surface and internal crystallization. In Eq. (1), (AT)p is the width of the D T A peak at ha l f -maxima (half-width), R is the gas constant , and E is the act ivat ion energy for crystal growth. If E is assumed to be independent of particle size, [Tp 2/(AT)p] would be propor t iona l to n and plott ing it as a function of particle size could give a quali tat ive idea for the dominan t crystall ization mechanism in the glass.

The plot of [T2/(AT)p] as a function of particle size is shown in Fig. 4 for the dry, wet, and desiccated LS 2 glass particles. These curves show a dependence on the particle size similar to that in Fig. 2 for (6T)p, i.e. [T2/(AT)p] decreases with increasing particle size for all the glasses. If internal crystall ization was the dominan t mechanism, [ Tp 2/(A T)p ] would increase with particle size ensuring, according to Eq. (1), an increase in the value for n. However , a decrease in [T2p/(AT)p] with incl:easing particle size is also not quite reasonable even if surface crystall ization was the dominan t mechanism. For predominate ly surface crystallization, the values of [T2/(AT)p] for the dry, wet, and desiccated samples should ideally be the same, part icularly for the smallest 0.035 m m particles, thereby, yielding a value of n close to 1 for all the samples at this particle size. If surface crystall ization continues to be the p r imary mechanism and E does not change appreciably with particle size, [TZ/(AT)p] for all the particle size samples should be nearly the same. In other words, the plot of [ T 2/(A T)p] as a function of particle size should be a straight line nearly parallel to the horizontal axis. The reason for the decrease in [TZ/(AT)p] with increasing particle size is not clear at this time. Despite this unexplained feature the curves in Fig. 4 demons t ra te that it is not internal but rather surface crystall ization which dominates the crystall ization process of the lithium disilicate glass.

4 .0"

• Dry 0 Wet

3.0

,~ 2.0

1.0 0 . 0 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0

Partic le Size, mm

Fig. 4. The ratio of the square of the peak temperature Tp to peak half-width (ATlp as a function of particle size for lithium disilicate glass particles stored under different conditions prior to DTA. For conditions of experiments and of storing the glass particles, see Fig. 2.


3.2. L S 2 glass doped with platinum

To further verify whether surface crystallization is the dominant mechanism for the crystallization of undoped LS 2 glass, identical DTA measurements were repeated for LS 2 glasses containing 0.001 and 0.005 wt% platinum as a heterogeneous nucleating agent. The results were compared with those for the undoped LS 2 base glass. Platinum-nucleated glasses are expected to crystallize primarily by bulk or internal crystallization. Since the platinum-nucleated glasses yield such a large DTA peak height that the temperature of the DTA furnace becomes unstable, 40 mg of sample, as was used for the undoped base glass, could not be used for these measurements. A 40mg sample of LS 2 glass composed of 0.462mm average size particles and containing 0.005 wt% Pt yields a peak whose height is 40°C larger than that of the base glass, at a scanning rate of 15°C min 1. The weight of the platinum-containing glasses was adjusted according to their platinum content so that the DTA peak height did not exceed 8 10C when 0.462mm glass particles was scanned at 15°C rain 1. This weight of glass was then held constant for the measurements on other particle sizes (about 15 mg for the 0.005 wt% and 20 mg for the 0.001 wt% Pt- containing glasses).

Since the weight of glass was different for different samples, a direct comparison of (6T)p for these glasses was difficult. A new term, "reduced peak height", which maintains the same functional trend as the original peak height with particle size but makes a direct comparison possible, was defined as

Reduced (6T)p = [(6T)p for a particular particle size]/[(6T)p for the smallest particle size, 0.035 mm in this case]

The reduced (•T)p for the undoped LS 2 glass and those containing 0.001 and 0.005 wt% Pt are compared in Fig. 5 as a function of particle size. The shape of the curves in Fig. 5 clearly demonstrates that internal or bulk crystallization dominates in the platinum-containing glasses, especially, the glass containing 0.005 wt% Pt where the reduced (6T)p increases with increasing particle size as opposed to the decrease observed for the undoped base glass. For the glass containing 0.001 wt% Pt, the reduced (ST)p decreases just slightly with increasing particle size. In this case, as the particle size increases the decrease in (6T)p caused by the decrease in surface crystallization (less surface area) is nearly counterbalanced by the increase in (~T)p caused by increasing internal crystallization (more heterogeneous nuclei).

As was observed for dry and wet undoped glasses, Tp for the platinum-containing glasses also increased with increasing particle size, compare Fig. 3 with Fig. 6. It is also clear from Fig. 6 that Tp decreases with increasing concentration of platinum in the glass for any particle size. This reduction is expected since it suggests that crystallization becomes more favorable with increasing platinum content in the glass. However, unlike (6T)p, Tp for a glass always increases with increasing particle size, no matter whether the glass is doped, undoped, wet, or dry. When compared with a glass of identical composition and particle size, the measured Tp can only predict which glass contains a higher concentration of nuclei. The dependence of Tp on particle size,


3 . 0 " .It

2.5 ~ • Undoped

~ O 0.00l wt% Pt

2.0 ~ • 0.005 wt% Pt

~- 1.5

~ 1.0

0.5] ~

I 0.0 I . . . . . . . . . .

0.035 0 .20 0 .40 0 .60 0 .80 1.00 1.20

Particle Size, mm

Fig. 5. Reduced maximum DTA peak height (6T)p as a function of particle size for undoped and platinum- containing lithium disilicate glasses. See text for the definition of reduced peak height. Sample weight: 40 mg (undoped glass), 20 mg (glass containing 0.00i wt% Pt), 15 mg (glass containing 0.005 wt% Pt). DTA heating rate: 15°Cmin 1.

o

,z

L. .= 1.1

.<

700 t 680 ]

660 t o

640

6 2 0

6 0 0

580

.-------'0

O

• Undoped

O 0.001 wt% Pt

• 0.005 wt% Pt

0.00 0 .20 0 .40 0.60 0 .80 1.00 1.20 Particle Size, mm

Fig. 6. Temperature of the DTA peak maximum Tp as a function of particle size for undoped and platinum-containing lithium disilicate glasses. Sample weight and DTA heating rate are the same as those given in Fig. 5.

therefore, cannot p rovide informat ion for the crys ta l l iza t ion mechanism (surface and /o r internal) that occurs when these glasses crystallize.

As ment ioned previously, the dependence of [ Tp 2/(A T)p ] on par t ic le size may closely resemble the dependence of the crystal g rowth d imens ion n of the glass. Values of [TZ/(AT)p] for the p l a t inum-con ta in ing glasses were ca lcula ted from the measured Tp


and (AT)p and reduced, for a bet ter compar ison, in the same way as was done for (6 T)p, i.e.

Reduced [T~/(AT)p] = [T~/(AT)p] for any particle size/[TZ /(AT)p] for the smallest particle size, 0.035 m m in this case

Fig. 7 shows that the plots of reduced [TZ/(AT)p] as a function of particle size for the doped and undoped LS 2 glasses closely resemble the plots in Fig. 5 of the reduced (6T)p as a function of particle size for these glasses. Thus, like reduced (c]T)p, reduced [T2/(AT)p] also provides the same information, namely, surface crystall ization dominates in the undoped LS 2 glass. When doped with plat inum, bulk crystall ization becomes the dominan t mechanism in this glass, as indicated by an increase in [TZ/(AT)p] or the growth dimension n, see Eq. (1), with increasing particle size.

3.3. Na20.CaO.2SiO 2 (NCS 2, Na20.CaO.3SiO 2 (NCS3), and Na20.2CaO.3SiO 2 ( NC2S ~ ) glasses

The reduced (•T)p and reduced [TZ/(AT)p] curves for the NCS2, NCS3, and NCzS 3 glasses, calculated f rom the D T A the rmograms repor ted previously by others [6,7] and ourselves [8], are shown in Figs. 8 and 9 as a function of particle size. As was observed for the LS 2 glasses, the plots of ( ~ T ) p for these three glasses as a function of particle size resemble the plots of [TZ/(AT)p], compare Figs. 8 and 9. Both (6T)p and [TZ/(AT)p] decrease for the NCS 3 glass and increase for the NC2S 3 glass as the particle size increases. For the NCzS 3 glass, only two data points were

oo

2 . 5

2.0

1 . 5

1.01

0 . 5

• Undoped

0 0.001 wt% Pt

• 0.005 wt% Pt

~O

0.0 0 . 0 3 5 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0 1 . 2 0

Partic le Size, m m

Fig. 7. Reduced ratio of the square of the DTA peak temperature Tp To peak half-width (A T)p as a function of particle size for undoped and platinum-containing lithium disilicate glasses. Sample weight and DTA heating rate are the same as those given in Fig. 5.

3.0

et e~

[...

2.0

1.0

0.0

e~

~o

O NC2S3

• NCS 2

• NCS3

J

0 . 0 5 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2

Part ic le Size, m m

Fig. 8. Reduced maximum DTA peak height (ST)p as a function of particle size for Na20.2CaO.3SiO 2 (NC283) , NazO.CaO.2SiO 2 (NCS2) and NazO.CaO.3SiO 2 (NCS3) glasses. For these plots, the DTA curves o fNCS 2 and NCS 3 glasses in Ref. [6], and NC2S 3 glass in Ref. [8] were used. Sample weight: 200mg for NCS 2 and NCS 3 glasses, 30mg for NCzS 3 glass. DTA heating rate: 10 "C min 1.

3.0

2.0 ¸

O NCzS3

• NCS2

• NCS3

1.0

0.tl

172 C.S. Ray, D.E. Day/Thermochimica Acta 280//281 (1996) 163 174

0 . 0 5 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0 1 . 2 0

Particle Size, m m

Fig. 9. Reduced ratio of the square of the DTA peak temperature Tp to the peak half-width (AT)p as a function of particle size for NC/S 3, NCS 2 , and NCS 3 glasses. The DTA curves of NCS 2 and NCS 3 glasses in Ref. [6], and NCzS 3 glass in Ref. [8] were used to generate these plots. For sample weight and DTA heating rate, see Fig. 8.

a v a i l a b l e f o r a n a l y s i s , b u t t h e i n c r e a s e in e i t h e r ( 6 T ) p o r [TZ/(AT)p] w i t h i n c r e a s i n g

p a r t i c l e s ize is c l e a r l y e v i d e n t . F o r t h e N C S 2 g las s , ( 6 T ) p o r ETZ/(AT)p] d e c r e a s e s

i n i t i a l l y w i t h i n c r e a s i n g p a r t i c l e size, a n d t h e n i n c r e a s e s as t h e p a r t i c l e s ize e x c e e d s

a b o u t 0.3 m m .

c.s. Ray, D.E. Day/Thermochimica Acta 280/281 (1996) 163 174 173

Based on our interpretation for the LS 2 glasses, the curves in Figs. 8 and 9 indicate that the NCS 3 and NC2S a glasses crystallize primarily by surface and internal crystallization, respectively. The NCS 2 glass crystallizes primarily by internal crystallization, although surface crystallization appears to be dominant for smaller particle sizes. The conclusion for the crystallization processes of NCS 2, NCS 3, and NCzS 3 glasses made from the DTA data analysis techniques used in the present investigation agrees with that reported previously [-6-8] from an analysis of apparent activation energy for crystallization of these glasses as a function of particle size.

4. Conclusions

A DTA method has been developed that identifies and distinguishes surface from internal crystallization processes in glasses. This technique which requires only a small quantity of glass and a few DTA scans using glass particles of different size, is a rapid and convenient method of determining whether a glass crystallizes primarily by surface or internal crystallization. No special sample preparation technique involving cutting, grinding, polishing, or etching is required. There is no other convenient and less time-consuming method known to date, which can provide information for the dominant crystallization mechanism in a glass.

It has been shown in the present investigation that a lithium disilicate (LS2) glass crystallizes primarily by surface crystallization as opposed to the general belief [ 10, 11 ] that homogeneous nucleation leading to internal (or bulk) crystallization occurs in an undoped LS 2 glass. The theoretical justification of the present technique in identifying surface from internal crystallization is currently under investigation. Preliminary results [-12] show that the DTA curves for the LS 2 glass generated by computer modeling represent the experimental curves more closely when surface crystallization is assumed to be dominant over internal crystallization. The present method successfully demonstrates the enhancement of surface crystallization when the LS/glass particles are exposed to moisture prior to crystallization. Likewise, the internal crystallization becomes clearly evident when this glass is doped with small amounts of platinum that provide heterogeneous nucleation sites inside the glass.

When DTA curves in the literature for NazO.CaO.2SiO 2 (NCS 2), NazO.CaO.3SiO 2 (NCS3), and NazO.2CaO.3SiO 2 (NC2S3) glasses were analyzed by the present technique, the NCS 2 and NCzS 3 glasses were found to crystallize primarily by internal crystallization, whereas surface crystallization was dominant in the NCS 3 glass. These results agreed with those determined from an analysis of the apparent activation energy for crystallization of these glasses as a function of particle size.

Acknowledgement

This work was supported by the National Aeronautics and Space Administration through contracts NAG8-898 and NAGW-2846.


References

[ l ] D.R. Uhlmann, Materials Science Research, Vol. 4, New York, Plenum Press, 1969, p. 172; Advances in Nucleation and Crystallization in Glasses, Columbus, OH, American Ceramic Society, 1972, p. 91.

[_2] G.W. Scherer, Materials Science and Technology, Vol. 9, VCH Publishers Inc., New York, 1991, p. 119. [3] J.MA. Rincon, Polym.-Plast. Technol. Eng., 31 (1992) 309. I-4] G. Partridge, C.A. Elyard and M.I. Budd, Glasses and Glass-Ceramics, Chapman and Hall, New York,

1989, p. 226. [_5] A. Halliyal, A.S. Bhalta, R.E. Newnham and L.E. Cross, Glasses and Glass-Ceramics, Chapman and

Hall, New York, 1989, p. 272. [6] N. Koga, J. Sestak and Z. Strand, Thermochim. Acta, 203 (1992) 361. [7] N. Koga and J. Sestak, Bol. Soc. Esp. Ceram. Vidr., 31 (1992) 185. [_8] X.J. Xu, C.S. Ray and D.E. Day, J. Am. Ceram. Soc., 74 (1991) 909. [9] J.A. Augis and J. E. Bennett, J. Therm. Anal., 13 (1978) 283.

[10] P.F. James, Glasses and Glass-Ceramics, Chapman and Hall, New York, 1989, p. 59. 1-11] K.F. Kelton, J. Non-Cryst. Solids, 163 (1993) 283; J. Am. Ceram. Soc., 75 (1992) 2449. [12] K.F. Kelton, K.L. Narayan, T.S. Cull, and C.S. Ray, J. Non-Cryst. Solids, in press.

therm0chimica acta

E L S E V I E R Thermochimica Acta 280/281 (1996) 175 190

Use of phenomenological kinetics and the enthalpy versus temperature diagram (and its derivative - - DTA)

for a better understanding of transition processes in glasses 1

J a r o s l a v Sestfik

Institute o[' Physies Division o/'Solid-State Physics, Academy of Sciences of the Czech Republic, Cukrot, arnickd 10, 16200 Praha 6, Czech Republic

Abstract

Thermophysical bases of vitrification and crystallization are discussed in terms of the enthalpy versus temperature diagram and its derivative (DTA) illustrating relaxation and glass transformation as well as crystallization of glassy and amorphous states to form stable and metastable phases. Non-isothermal kinetics are thoroughly discussed and the use of one- (JMAEK) and two- (SB) parameter equations is described. The practical case of the complementary 70SiO 2

10A1203 20ZnO glass treatment is demonstrated for both standard nucleation-growth curves and corresponding DTA recordings; these show a good coincidence of the activation energies obtained.

Keywords: Crystallization; DSC; DTA; Enthalpy temperature diagram; Glass transitions; Non-isothermal kinetics; SiO 2 A120 3 Z n O ; Vitrification

1. Introduction

Second phase nucleation and its consequent growth can be unders tood as a general process of new phase formation. It touches almost all aspects of our universe account- ing for the formation of smog as well as the embryology of viruses [1]. On the other hand the supression of nucleation is impor tant for the general process of vitrification, itself impor tant in such diverse fields as metaglasses or cryobiology trying to achieve

1 Dedicated to Professor Hiroshi Suga.

0040-6031,/96,,'$15.00 i; 1996 Elsevier Science B.V. All rights reserved SSDI 0 0 4 0 - 6 0 3 1 (95)02641-X

176 d. fesRkk/Thermochimica A eta 280/281 ( 1996 ) 175 190

non-crystallinity for highly non-glass-forming alloys or intra- and extracellular solutions needed for cryopreservation.

A characteristic process worthy of specific note is the sequence of relaxation nucleation growth processes associated with the transition of the non-crystalline state to the crystalline one. Such a process is always accompanied by a change of enthalpy content which, in all cases, is detectable by thermometric (differential thermal analysis, DTA) and/or calorimetric (differential scanning calorimetry, DSC) measurements [2, 3]. Some essential aspects of such types of investigation will be reviewed and attempts will be made to explain them more simply.

The article has two parts with different information levels. The aim of Section 3 is to show an easy way of graphically illustrating processes associated with devitrification. The diagram of enthalpy versus temperature is employed, including its relationship to conventional DTA DSC measurements; this is easily understandable in terms of the temperature derivatives [3]. Inherent information, however, is very complex and its successful utilization, particularly in determining the reaction kinetics of crystallization, is not easy. Therefore a more specialised section 4 is added in order to review some current practices of non-isothermal kinetics evaluations.

2. Theoretical basis of D T A - D S C

Most DTA instruments can be described in terms of a double non-stationary calorimeter [3] in which the thermal behaviour of the sample (S) is compared with that of an inert reference (R) material. The resulting effects produced by the change of the sample enthalpy content can be analysed at four different levels, namely:

(1) identity (fingerprinting of sequences of individual thermal effects); (2) quality (determination of points characteristic of individual effects such as begin- nings, onsets, outsets, inflections and apexes); (3) quantity (areas, etc.); (4) kinetics (dynamics of heat sink and/or generation responsible for the peak shapes).

The identity level plays an important role in deriving the enthalpy against temperature plots while study of kinetics pays attention to profile of their derivatives.

From the balance of thermal fluxes the DTA equation can be established [3,4] relating the measured quantity, i.e., the temperature d(~'erence between the sample and reference, A TDTA = Ts - TR, against the investigated reaction rate, d~/dt. The value of the extent of reaction, :¢, is then evaluated by simple peak area integration assuming the averaged values of extensive (~) and intensive (T) properties obtained from the sample specimen. It may be complicated by the inherent effect of heat inertia, Cp(dA TDTA/dT), when the DTA peak background is not simply given by a linear interpolation [2, 3].

A similarly derived DSC equation shows the direct relationship for the measured compensation thermalflux, AQ, supplied to both specimens under the conditions where the specimen difference A TDTA serves only as a regulated quantity kept as close to zero as possible. Therefore during any DTA experiments the actual heating rate of the

J. S'estfik/Thermochimica Acta 280/281 (1996) 175 190 177

sample is in reality changed [4] because of the DTA deflection due to heat release and/or absorption, which is also a direct measure of the deviation between the true and programmed (predetermined) temperatures (maintained by the reference). At the moment when a completely controlled thermal condition of the specimens is attained [-3] the DTA peak would be diminished while that of DSC becomes more accurate. Both types of measurement provide a unique source of input data which is usually undervalued and underestimated for more sophisticated kinetic analysis.

Initially we should make clear that ordinary DTA can never satisfy all the demands arising from specific characteristics of the glassy state and the non-equilibrium nature of its investigation. Micro-DTA, where possible temperature gradients of microgram samples are maximally decreased, is an useful tool for a precise detection of characteristic temperatures while macro-DTA enables more effective measurement of integral changes needed in calorimetry. However, glassy samples of shapes like fibres or very thin metallic ribbons or even fine powders must be treated with increased attention as the process of sample pretreatment as well as filling a DTA cell can introduce not only additional interfaces and defects but also unpredictable mechanical tension and interactions with the sample holder surface all of which are sensitively detected and exhibited on thermometric recordings. Under limiting conditions, and particularly for a very small sample, it is then difficult to distinguish between effects caused by bulk and surface properties of a sample as well as by a self-catalyzing generation of heat. Consequently heat transfer from the reacting zone may become a rate-controlling process [-3].

3. Thermophysics of vitrification and crystallization

Glasses (obtained by a suitable rapid cooling of melts) or amorphous solids (reached by an effective disordering process) are in a constrained (unstable) thermodynamic state [5] which tends to be transformed to the nearest, more stable state on reheating [2, 6]. The well-known dependence of a sample's extensive property on temperature can best illustrate the possible processes which can take place during changes of sample temperature [-3]. A most normal plot was found in the form of H versus T which can be easily derived using an ordinary lever rule from a concentration section of a conventional phase diagram [7]. The temperature derivatives of this course (dAH/dT) resembles the DTA DSC traces each peak and/or step corresponding to the individual transformation exhibited by the step and/or break in the related H vs T plot.

3.1. Fundamentals of glass-jormation processes

Fig. 1 illustrates possible ways of achieving a non-crystalline state. The latter can be either glass obtained in the classical way of liquid freeze-in via the metastable state of undercooled liquid (RC) or any other amorphous phase prepared by disintegration of crystalline solids (MD, upwards from solid to dotted line) or deposition from the gaseous state (VD, to thin sloped line). In the latter cases the characteristic region of

178 d. ,~esttik/Thermochimica Acta 280//281 ( 1996 ) 175-- 190

H ; gas

. . t ~ I ~ RC ~ VD .~ I " undercooled ..----,"

~ . / glass I . / ' - " "1" r ,

. . . . . . . I ~ .,H i t

~ f ~_ J l equ,l,br,um ,,

'~'--- Solidification

l

42 undercooled

glass-formation

t equilibirium melting and ~ I ~, recrystallizatien ] j ~ " ~

i glass-transformation ~, ] and crystallization

V " ~.i i L - - . ] enhanced

i i , I g lass -c rys ta l l i za t ion

] ' ' i ' ! ~ f = I i ~ II I

T 'c r Tg Tcr Tmelt

Fig. 1. Schematic diagram of enthalpy, H, versus temperature, T, and its derivative (A T) presented in the actual form which can be used to reconstruct the DTA DSC recording. The solid, dashed and dotted lines indicate the stable, metastable and unstable (glassy) states. Rapid cooling of the melt (RC) can result in equilibrium or non-equilibrium solidification or glass-formation (T g) which on reheating exhibits recrystall- ization. On the other hand an amorphous solid can be formed either by deposition of vapour (VD) against a cooled substrate (thin line) or the mechanical disintegration (MD) of the crystalline state (moving vertically to meet the dotted line)• Reheating of such an amorphous solid often results in early crystallization which overlaps the Tg region•

g lass - t ransformat ion is usual ly ove r l apped by ear ly crys ta l l iza t ion on heat ing which can be dis t inguished only by the basel ine separa t ion occurr ing due to the change of heat capaci ty. Such typical cases exist in oxide glasses but a bet ter example can be found for non-crys ta l l ine chalcogenides p repared by var ious me thods of d i sorder ing [8]. F o r most metal l ic glasses, however, there is negligible change of Cp between the glassy and crystal l ine states and thus the D T A D S C baselines do not show the required displace- ment which even makes it difficult to locate Ty, It is wor th not ing that the entire glassy state [5] can further be classified accord ing its origin, d is t inguishing l iquid glassy crystals, etc. [9], charac ter i sed by their own g lass - t rans format ion regions.

J. S~est~k/Thermochimi~ a Acta 280/281 (1996) 175 190 179

3.2. Basic rules of relaxation and glass transformation

During heat treatment of an as-quenched glass relaxation processes can take place within the short and medium range ordering [10] to cover (i) topological short-range movement of constitutional species and relaxation of structural defects, (ii) compositional short-range rearrangements of the nearest neighbours where usually chemically similar elements can exchange their atomic positions and (iii) diffusional ordering connected again with structural defects and gradients created during quenching of real bodies [2, 10-13]. Such processes are typical during both the isothermal annealing and slow heating which occur below T 0 and can consequently affect the glass transformation region as well as the medium-range ordering, possibly initiating segregation of clusters capable of easier nucleation. Characteristic types of T o region during cooling and heating are illustrated at Fig. 2.

3.3. Crystallization of glasses

There are extensive sources of the basic theory for nucleation-growth controlled processes [14-17]; these form the framework of the resulting description of a centred process of overall crystallization. Schematically, see Fig. 3, the metastable glassy state (dotted) first undergoes glass transformation followed by separation of the closest crystalline phase usually metastable (dot-and-dashed). Consequent and/or overlapping T o (I and II) indicate the possibility of two phase-separated glasses. If considering a more complex case, for example the existence of a second metastable phase, the sequences of processes became multiplied since the metastable phase is probably produced first to precipitate later into more stable phases. This can clearly give an idea of what kind of an equilibrium background [7] is to be considered for the overall crystallization. Thus the partial reaction steps experimentally observed as the individual peaks can be classified to fit the scheme chosen in accordance with the stable metastable boundary lines in the given type of a phase diagram related to the sample composition [4].

4. Non-isothermal crystallization kinetics

4.1. Constitutive equations, derivation methods and applicability checking

The so-called phenomenological description of crystallization kinetics emerged as a separate category to enable utilization of thermoanalytically measured data on the mean and normalised extent of crystallization, ~, for a simplified description of the nucleation-growth-controlled processes [3]. It is conventionally based on a physical-geometrical model, f(~), responsible for crystallization kinetics traditional- ly employing the classical Johnson Mehl Avrami Yerofeeyev Kolgomorov equation (abbreviated JMAYK) read as - ln(1 ~)=(kTt) r. It follows that f ( ~ ) = (1 - ~ ) ( - l n ( 1 -:¢))P where p = 1 1/r [2, 3] has been derived, however, under strict isothermal conditions. Beside the standard requirements of random nucleation and

180 J. S~estdtk/Thermochimica Acta 280/281 (I 996) 175 190

J

H

t

AT

H

AT

H

AT

glass I " ~ - ~ . . J . ~ / . . - ' ~

metals

oxide=

I T I

\

/ " / /~z

_ 5 _ _

annealing

I f / II I ~t

ql

T _'2"-

Fig. 2. Enlarged parts of a hypothetical glass transformation region (cf Fig. 1) shown for several characteristic cases such as cooling two different sorts of material (upper), reheating identical samples at two different heating rates (middle) and reheating after sample annealing (bottom).

T

J. S~estfk/Thermochimica Acta 280/281 (1996) 175 190 181

]

H

~T

I

I Tcr7

I I I I

T~r'a f I T

I . . - - ~ I ~ '~"

" 5 . . . . ;>, ~ ~ - - . - " 5 . . ~ f I ~ ' l " " " - " " " "

/ , T-"j

~ ~ n III

Tgl Tgtt Tcrl Tcr2 Tc,.3 T,

1 1

f J

J J

T

Fig. 3. Hypothetical crystallization of one metastable phase (Tcrl, dot-and-dashed line) or two metastable phases (To,l, Tc~2, bottom)including the possibility of two glasses (Tg~, Tg.), transformation as a possible result of preceding liquid liquid phase separation.

consequent growth, the rate of which is temperature-but not time-dependent, the JMAYK equation is only valid at constant temperature. Some pecularities of non- isothermal kinetics and the general applicability of the JMAYK equation are thus worth reconsidering.

It must be reminded that the extent of crystallization, ~, cannot be considered as a s t a t e f u n c t i o n itself ~=~ f , ( T , t) [3,7]. For the sake of simplicity the truly effective

182 J. S~estdk/Thermochimica Acta 280/28I (1996) 175 190

constitutive equation in kinetics can thus be simplified to be valid for the two basic rates only, i.e., that of crystallization d~/dt=f~(7, T) and of temperature change d T/dt =fr(~, T). Therefore any additional term of the type (d~/dT) t sometimes required to correct the non-isothermal rate in the fundamental Arrhenius equation [18, 19] dct/dt = kTf(~ ) by the additional multiplying term [1 + ( T - To)E/RT 2] is unjustified [7] and incorrect [3]. The resultant isokinetic hypothesis is then accounted for in the construction of TTT and CT diagrams [2], also.

The true temperature-dependent intergration of the basic nucleation growth equation yields very similar dependences [3,20] comparable to the classical JMAYK equation obtained under classical isothermal derivation. It effectively differs only by a constant in the preexponential factor depending on the numerical approximation employed during the integration [3]. This means that the standard form of the JMAYK equation can be used throughout non-isothermal treatments. Therefore a simple preliminary test of its applicability is recommendable. In particular, the value of the multiple (d~/dt)T can be used to check which maximum should be confined to the value 0.63_+0.01 [21]. Another handy test is the shape index, i.e., the ratio of intersections (b I and b:) of the inflection slopes with the linearly interpolated peak baseline which should show a linear relationship [21] ~ =bl /b 2 =0.52+0 .916 [(Ti2/Til ) - 1 ] where T i are respective inflection point temperatures.

This highly mechanistic picture of an idealized process of crystallization bears, however, a little correlation to morphological views and actual reaction mechanism [21,22]. If neither the (d~/dt)T value nor the shape index ~ comply to JMAYK requirements we have to account for real shapes, anisotrophy, overlapping, etc., being usually forced to use a more flexible empirical kinetic model function usually based on two exponents (SB) instead the one (JMAYK) only.

4.2. Apparent values of activation eneryies

Conventional analysis of the basic JMAYK equation shows that the overall value of activation energies, Eapp, usually evaluated from D TA -D S C measurements, can be roughly correlated on the basis of the partial energies of nucleation Enucl, growth Eg r and diffusion Edifi [23]. It follows [24] that E a p p = (13Enu d + OdEgr)/(u + Od) where the denominator equals the power exponent of the JMAYK equation, r, and the coefficients t~ and d indicate the nucleation velocity and the growth dimensionality. The value of ff corresponds to 1 or 1/2 for the movement of the growth front controlled by chemical boundary reaction or diffusion, respectively.

The value of Eap p is easily determined from the classical Kissinger plot [3,4] of In (fl/T 2) versus ( - 1 / R Tp) for a series of the DTA-DSC peak apexes (Ta) at different heating rates (fl) irrespective of the value of exponent r. Adopting the//-dependent concentration of nuclei Matusita and Sakka [25] proposed a modified equation of ln(fl~r+d)/T 2) versus (-dEgr/RTa) applicable to the growth of bulk nuclei where the nuclei number is inversely proportional to the heating rate (~ -~ 1). This is limited, however, to such a crystallization where the nucleation and growth temperature regions are not overlapping. If E,u d = 0 then Eap p simplifies to the ratio (dEg r + 2[t~ + d - 1] R Ta)/(d + t)) where dEg r >> 2 [r + d - 1] R T a. There are various

J. Sestdk/Thermochimica Acta 280/281 (1996) 175 190 183

further explanations of the effective meaning which can be associated with the value of apparent activation energy [4, 20, 21,26, 27].

4.3. Effect of particle size

In the case of surface-controlled crystallization the DTA exotherm is initiated at a lower temperature for the smaller particle size. In contrast the temperature region for the growth of bulk nuclei is independent of particle size. Thus for larger particles the absolute number of surface nuclei is smaller owing to the smaller relative surface area in the particle assembly resulting in predominant growth of bulk nuclei. With decreasing particle size the number of surface nuclei gradually increases, becoming responsible for the DTA exotherm. The particle-size-independent part ofEap p can be explained on the basis of the nucleus formation energy being a function of the curved surface. Concave and convex curvature decreases or increases, respectively, the work of nucleus formation [-24]. Extrapolation OfEap p to the flat surface can be correlated with Egr, U = 1 and d = 3, being typical for silica glasses. It, however, is often complicated by secondary nucleation at the reaction front exhibited by decreasing Eap p with rising temperature for the various particle zones. The invariant part ofEap ~ then falls between the microscopi- cally observed values of Eg r and Enucl the latter being frequently a characteristic value for a yet possible bulk nucleation for the boundary composition of a given glass.

Assuming that the as-quenched glass has a nuclei number equal to the sum of a constant number of quenched-in nuclei and that, depending on the consequent time-temperature heat treatment, the difference between the DTA-DSC apex temperatures for the as-quenched and nucleated glass (T a - T °) becomes proportional to the nuclei number formed during thermal treatment [26, 28]. A characteristic nucleation curve can then be obtained by plotting ( T a - T °) against temperature [26] or even simply using only the DTA DSC peak width at its half maximum [28].

For much less stable metallic glasses the situation becomes more complicated because simultaneous processes associated with the primary, secondary and eutectic crystallization often take place not far from each other's characteristic composition. Typically thin ribbons of iron-based glasses close to the critical content, x, of metalloids (x s + Xsi = 25% with glass-forming limit x R -~ 10%) usually exhibit two consecutive peaks. This is frequently explained by the formation of ~-Fe embryos in the complex matrix containing quench-in sites followed by the formation of the metastable Fe3Si- like compounds which probably catalyze heterogeneous nucleation of more stable compounds. Experimental curves are usually fitted by the classical JMAYK equation describing nucleation and three-dimensional growth. However, small additions of Cu and/or Nb cause the two DTA peaks to overlap each other, essentially decreasing inherent areas (the heat of crystallization is cut down to about one fourth) indicating probably the dopant surfactant effect directed on the critical work of nuclei formation due to the reduced surface energy. The resulting process of associated nanocrystal formation (fin-met~,ls) [29-31] is thus different from the conventional nucleation growth usually characterised by high r(>_ 4). This is still far from being unambiguous and is often a matter of simplified hypotheses on grain growth. Nowadays nano- crystallization is thus another more specific case under increased attention becoming so

184 J. festcik/Thermochimica Acta 280/281 (1996) 175 190

complicated to be explainable in terms of classical physical geometrical theories. It is clear that a sort of fully phenomenological description could here be equally suitable for such a formal description, regardless of the entire mechanism of the reaction in question.

4.4. Phenomenological models in kinetics

There exist different phenomenological treatments, such as the geometric-probabi- listic approach or a purely analytical description by the exponent containing the fraction (a/x + b/x 2) 1. It enables continuous transfer from a quadratic to linear profile [-3] making it possible to identify formally the changes of reaction mechanism by checking characteristics of the peak shape only. Nevertheless, for a general expression of heterogeneous kinetics a most useful approach seems to be the traditional type of a formal description based on an analytical model, f(:¢), regarded as a distorted case of the classical homogeneous-like description (1 - ~)", where n is the so-called reaction order. A recently introduced function, h(~) [22], called the accommodation function (being put equal to either ( - l n ( l (1 - :¢))P or :~"), expresses the discrepancy between the actual, f(:Q, and ideally simplified, (1 - ~)" models. The simplest example of such an accommodation is the application of non-integral exponents (m, n, p) within the one- (JMAYK) or two- (SB) parametric function f(~), in order to match the best fitting of the functional dependences of an experimentally obtained kinetic curve. These non- integral kinetic exponents are then understood as related to thefractal dimensions [22]. From a kinetic point of view the interfacial area plays in crystallization kinetics the same role as that of concentration in homogeneous-like kinetics.

It is apparent that any specification of all (more complicated) processes in solid-state chemistry is difficult [32] to base unambiguously on physical geometrical models although they have been a subject of the popular and widely reported evaluations [32-35]. As already mentioned it can be alternatively replaced by an empirical- analytical (SB) formula [3], f(~) = (1 ~),~m, often called Sestak-Berggren equation [32, 34 36], which exhibits certain interrelation with the JMAYK equation, see Fig. 4. This can be found particularly serviceable in fitting the prolonged reaction tails due to the actual behaviour of real particles and can nicely match the particle non-sphericity, anisotropy, overlapping or other non-ideality [22].

It is worth mentioning that the typical kinetic data derived are strongly correlated with the Arrhenius equation, in the other words, a DTA DSC peak can be interpreted within several kinetic models by simple variation of the activation energy E and preexponential factor A [36]. A correct treatment then requires an independent determination of E from multiple runs and its consequent use to analyse a single DTA-DSC peak in order to determine a more realistic kinetic model and preexponential factor [32, 34].

5. Crystallization of ZnO-Ai203-SiO 2 glasses

An example of complex analysis of the crystallization processes is a complementary investigation of the above-mentioned system which is illustrated in Fig. 5. The upper

d. SestOk/Thermochimica Acta 280/281 (1996) 175-190 185

0.25

n

0.50

0.75

n

J

I 2 3 4 5

5

0 .60

m

0.65

0.70

I 1.00 [ 0.75

I

0 . 0 0 . 2 0 .~ 0 .6 0 .8 1 .0

8

e-1

o

Fig. 4. This diagram demonstrates the relationship between the single exponent r of the JMAYK equation - l n ( l - :~)= (krt) r and the twin exponents n and m of the SB equation (1 -c&c~ m. Upper lines show the gradual development of both exponents m and n with r while the bottom curve illustrates a collective profile of n on m for increasing r.

pa r t represents the s t anda rd da t a ob ta ined by opt ica l measurements by two-s tep me thod and the b o t t o m par t shows the associa ted D T A curves of var ious ly t rea ted samples.

F o r the powdered glass, where the surface crys ta l l iza t ion of/~-SiO 2 is dominan t , the value of Eap p was a l ready repor ted [37] to corre la te to the reciprocal of par t ic le radius, indica t ing enhanced nuclei fo rmat ion on a curved surface. Act iva t ion energy then changes as a function of the surface curva ture and its ex t r apo la t ion to the flat surface reaches Eap p = 50 kcal c o m p a r a b l e to Eg = 66 kcal for l inear g rowth [37].

A more interest ing case is cer ta in ly the bulk crys ta l l iza t ion either spon taneous (gahnite and willemite) or ca ta lysed (Zn-petal i te) a imed at the p r epa ra t i on of low d i l a ta t ion g lass -ceramics . Assuming the crystal g rowth of Zn-pe ta l i te from both the cons tan t number of nuclei ( J M A Y K , r = 3) and at the cons tan t nuclea t ion rate (r = 4) the theoret ica l D T A curve can be ca lcula ted and c o m p a r e d with the exper imenta l one. F o r the ins tan taneous values of g rowth the best agreement was achieved for sa tu ra ted nuclea t ion based on the m a x i m u m nuclea t ion rate. This is shown in Table 1 where the crys ta l l iza t ion rates based on the opt ical and D T A measurements are compared , and agree within the o rde r of magni tude . The appa ren t value of E der ived from D T A traces

186 J. S~estdk/Thermochimica Acta 280/281 (1996) 175-190

log At I

[cm -3 mL,--11

8

6

4

2

0

ATDTA

" ~ ~ //Z~petalite . zro. \ / . zro.

" - ~ ' 7 / - ~ " 7aptic al

Z n oPSt raot2~ / / ~4ahnztrS 2)

- _ / Z , ,, 750 850 950 1050 1150

T---- Crystalization ~" ~ i /

volume / d / /

catalysed / / "T " .~1 // / i

surface / ~ . ~ ' ' J..l I i J ~ " ~" - -I" ~ "7 / " ]7 • ~'" Hypothetical

{J-Si02i'.,~-"/'./"~ ~ ' / V elagram Zn-petahte

Tg I ~ gahnit~

T I . - I

.,o , [ /Wo

powder glad,surface Lt ~

bulk glass

I - - I i

950 1050 1150

galqnit & willemit

' G / 1 0 3

[ c m m i n -1 ]

exo

endo

4 % Zr02

4 % Zr02 ~ ' (annealed 750 °C, 7 h)

6 % ZrO 2 Zn-petalite

l w

DTA Curves

Fig. 5. Complex view to the crystallization behaviour of the 20ZnO 30AIzO 3 70SiO 2 glassy system showing its nucleation (N) and growth (G) data obtained by the classical two-step optical observation of growth (upper), schematic view of the enthaply versus temperature diagram exhibiting the presence of two metastable phases of/~-quartz and Zn-petalite (middle) and respective DTA curves (bottom) obtained either on the powdered glass or on the bulk glass (cast directly into the DTA cell), in-weight about 200 mg, heating rate 10'C min- ~ Netzsch apparatus with classical chart recording.

J. Sestfk/Thermochimica Acta 280/281 (1996) 175--190

Table 1 Crystallization of Zn-petalite

187

Temperature/°C Growth rate, G~ Time of growth, d:~/dTdlr/(min - 1) dct/d TD-rA/(min- 1) (cm min - 1) t/min

820 1.6x10 4 2 1.6×10 2 l x l 0 1 830 1.9x10 4 4 2.2x10 1 4x10 1 840 3.0x10 4 6 2.9x10 ° l x l 0 ° 850 4.6x10 4 8 2.5x101 5x10 ° 860 6.3x10 4 10 1.2×102 8x101

d~/d Ta~ r was evaluated on basis of the JMAYK equation (p = 2/3) and optical data employing the growth rate in (cm min l) and saturation nucleation rate N -~ 5 x 10 l° (cm 3 min ~), do:/dTDx A = (KDTAAT+ Cp d A T / d T ) / A H where the apparatus constant KDT A of the Netzsch instrument used was 5 x 102 (cal K 1 min l) at 850°C and the heat released AH ~ lOcal.

(Eap p'-~ 345 kcal, see T a b l e 2) a lso agrees wi th the s u m of three Eg r (-~ 285 kcal)

i nd i ca t i ng aga in the s a t u r a t i o n of the sites c apab l e of n u c l e a t i o n p r io r the g r o w t h

process . C o n v e n t i o n a l ana lys is of the J M A Y K e x p o n e n t p r o v i d e d an e x p o n e n t va lue

close to three; thus b o t h a b o v e - m e n t i o n e d checks of the J M A Y K va l id i ty were found to

be pos i t ive . S i m u l t a n e o u s l y ca l cu la t ed [32, 34, 35] e x p o n e n t s of the SB e q u a t i o n give

thei r r espec t ive va lues c lose to 3/4. O n the o t h e r h a n d the a n a l o g o u s analys is of gahn i t e c rys t a l l i za t ion shows the

di f ference of a b o u t t w o o rde r s of m a g n i t u d e for b o t h cases u n d e r ca lcu la t ion , see T a b l e

3. T h u s b o t h tests for p r o v i n g the va l id i ty of J M A Y K also shows misfits. At the s a m e

t ime the Eap p ca l cu l a t ed f rom D T A is t o o low ( abou t 130 kcal) to satisfy the t r ip le va lue

of r e spons ib l e Eg r ( "-~ 75 kcal) as s h o w n in T a b l e 2. T o g e t h e r wi th the de l ayed onse t of

the D T A p e a k a n d the shape of its acce l e ra t ing pa r t it c lear ly ind ica tes a m o r e

c o m p l i c a t e d p rocess t h a n a s ingle n u c l e a t i o n - g r o w t h even t for gahni te . It is poss ib ly

affected by the Z r O z - c a u s e d o r d e r i n g which m a y affect p r enuc l eus site f o r m a t i o n

[38 40]. T h e c o r r e s p o n d i n g va lues of r -~ 3.3 and n -~ 0.6 a n d m ~- 0.7 are aga in n o t

Table 2 Apparent kinetic data of the volume crystallization of 20ZnO 30Al:O s 70SiO 2 glass doped with ZrO 2. Nucleation N in cm s min 1, activation energy E in kcal

Nucleator content Gahnite Zn-petalite E from DTA or treatment

Nma x E.u d Eg r Nma x Enucl Egr G a h n i t e Petal i te

4% Z r O 2 6 .4x 104. 115 75 5x 10 ° 45 95 130 Same, 7 h ~ " at 755"C (_~ 108) a ( ~ 107) a 190 6% ZrO 2 5 X 10 l° 350 95 345

a Estimated from the DTA peak, other data from optical measurements, see Fig. 5.

188 J. Sestdk/Thermochimica Acta 280/281 (1996) 175 190

Table 3 Crystallization of gahnite

TemperatureS'C Growth rate, G~ Time of growth d:t/dTdir/(min 1) d~/dTDTA/(mi n 1) (cm min l) t/min

820 ~-10 6 28 ~10 4 ~-10 2 960 9.6×10 5 32 1.7×10 3 6×10 1 980 1.8x10 s 36 1.6×10 2 4×10 °

1000 2.2×10 5 40 3.6×10 z 9×10 ° 1020 2.7×10 5 44 1.0x10-1 2×101 1040 2.9x10 5 48 2.0×10 i 5×10 ~

d~/d Ta~ r was evaluated on basis of the Y MAYK equation (p = 2/3) and optical data employing the growth rate in (cm min 1) and saturation nucleation rate N -~ 6 x 104 (cm 3 min 1), d~/d TDT A = (KDTAA T + Cp dAT/dT)/AH where the apparatus constant KDT A of the Netzsch instrument used was 9 x 102 [cal K 1 min - 1] at 1000°C and the heat released AH ~ 10 cal.

c h a r a c t e r i s t i c e n o u g h for a s ing le p r o c e s s to be s i m p l y u n d e r s t o o d w i t h o u t f u r t h e r

c o m p l e m e n t a r y i n f o r m a t i o n .

Acknowledgement

T h e w o r k was c a r r i e d o u t u n d e r p r o j e c t no. A 2 0 1 0 5 3 2 s u p p o r t e d b y the A c a d e m y

of Sc iences of t h e C z e c h R e p u b l i c .

D e e p t h a n k s a re d u e to P r o f e s s o r Z. S t r n a d of t he L a b o r a t o r y for B i o a c t i v e G l a s s -

C e r a m i c M a t e r i a l s in P r a g u e ( L A S A K ) for k i n d l y s h a r i n g o p t i c a l m e a s u r e m e n t s of t he

Z n O A l z O 3 - S i O 2 sys tem. C o n t i n u o u s c o o p e r a t i o n in s o l v i n g p e c u l a r i t i e s of n o n -

i s o t h e r m a l k i n e t i c s is a l so a p p r e c i a t e d , in p a r t i c u l a r m e n t i o n i n g Dr . J. M ~ l e k o f t he

L a b o r a t o r y for S o l i d - S t a t e C h e m i s t r y a t the P a r d u b i c e U n i v e r s i t y a n d Dr . N. K o g a of

t he C h e m i s t r y L a b o r a t o r y of H i r o s h i m a U n i v e r s i t y .

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