Baixe zanotto 1992 e outras Notas de estudo em PDF para Engenharia de Produção, somente na Docsity! Brazilian Journal of Physics, vol. 22, no. 2, June, 1992 7 Crystallization of Liquids and Glasses Edgar Dutra Zanotto Departamento de Engenharia de Materiais, Universidade Federal de São Carlos 13560, São Carlos, SP, Brasil Received February 10, 1992 The scientific and technological importance of advanced materials are summarized. The governing theories of glass transition, crystal nucleation and crystal growth are combined with the overall theory of transformation kinetics to clarify the phenomenon of glass for- mation from the liquid state. Finally, examples of novel glasses as well as glass-ceramics obtained from the controlled crystallization of certain liquids are given. I. Introduction The contemporary, technology intensive, age with its high technology industries and services demands the use of novel materials with improved properties. For in- stance, in the opinion of the presidents of one hundred Japanese industries the following were the most inno- vative new technologies in the last two decades: VLSI, Biotechnology, Optical Fibers, Robotics, Special Ce- ramics, Interferon, Office Automation, New Materials, Super-Computers and Space Technology*. It is obvious that most of them are directly related to advanced ma- terials. The study and development of useful materials de- mands highly interdisciplinary efforts from physicists, chemists, materials scientists and engineers. Materials Science emphasizes the relationships between the struc- ture and properties of materials, providing a link be- tween the fundamental sciences and applications, while Materials Engineering focus the study of the relation- ships between the processing techniques and the appli- cations. A schematic view of the scope of the various segments of science and materials engineering is pre- sented in Figure 1. Materials can be classified in several ways; i.e., by: i The general behavior: metals, ceramics, polymers and composites; ii Chemical nature: covalent, ionic, metallie, van der Waals, hydrogen, mixed bonding; ii. Some property, e.g.: insulator, semi-conductor, conductor, superconductor, or; iv. Structure: single crystal, polycrystal, vitreous, etc. This article deals with the controlled crystallization of liquids or glasses of any type as a technique to obtain novel materials. MATERIALS SCIENCE MATERIALS ENGINEERING PHYSICS AND CHEMISTRY Emphasis ELECTRONIC MOLECULAR mIGRO Macro STAUGTURE STAUGTURE STRUCTURE STRUCTURE 198 107º 108 10º- 102 Scale (m) Figure 1: Scope of the basic sciences and materials engineering?. II. Types and applications of materials obtained via crystallization The most obvious crystallization process is that fre- quently employed by chemists for the synthesis of purer or new compounds, i.e. the precipitation of powder par- ticles from super-saturated solutions. The geologists rely on the post-mortem study of crystallization to understand the formation of miner- als and solidified magmas. Many solid-state physicists depend on crystal growth from seeded melts to obtain a plethora of single- crystal specimens as well as commercially important materials such as silicon and lithium niobate. Ceramicists and materials scientiste dedicate a lot of time to the synthesis of novel ceramics and glasses employing the sol-gel technology. Tn this case the avoid- ance (or lack) of crystal nucleation and growth in the gel, during the sintering step, can lead to a glass. Finally, the catalyzed crystallization of glass objects volume 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 behaviour. Many commercial glass-ceramic products are available for domestic uses, e.g. vision-TM, rangethops, feed- throughs, electronic substrates, artificial bones and teeth, radomes, etc. III. The glass transition Glasses are amorphous substances which undergo the glass transition. The most striking feature of the glass transition is the abrupt change in the properties of a liquid, such as the thermal expansion coefficient (a) and heat capacity (cp), as it is cooled through the range of temperature where its viscosity approaches 102º Pas. in that range the characteristic time for structural re- laxation is of the order of a few minutes, so Lhe effects of structural reorganization are easily detected by human observers Figure 2 shows the change in volume, V,ofa glass forming liquid during cooling through the transition re- gion. Temperature Figure 2: Schematic representation of glass transition (a) and crystallization of a liquid (b). 1£ the liquid is cooled slowly (path b) it may crys- tallize at the melting point, Tm. If the cooling rate is fast enough to avoid crystal nucleation and growth, a supercooled liquid would be produced (path a). As the temperature drops, the time required to establish the equilibrium configuration of the liquid increases, and eventually the structural change cannot keep pace with the rate of cooling. At that point a transition temper- ature, T;, is reached below which the atoms are frozen into fixed positions (only thermal vibrations remain) and a glass is formed. Thus, glass formation from the liquid state is fea- sible if path (a) is followed. On the other hand, all n'= w: AGp= Activation energy for transport across the nu- E. D. Zanotto glasses heated to a temperature between T, and Tm tend to crystallize to achieve thermodynamical equilib- rium. If crystallization occurs from a large number of sites in the bulk, useful, fine grained, glass-ceramics can be produced. When crystallization occurs in an uncon- trolled way (devitrification) from a few surface impu- rity sites, damage and cracking of the specimen may take place. In the following sections the relevant theo- ries and experimental observations leading to controlled crystallization in the volume of glasses or supercooled liquids will be described. IV. Crystal nucleation When a liquid is cooled below its melting point, crystal nucleation can occur homogeneously (in the vol- ume), by heterophase fiuctuations. The Classical Nu- cleation Theory (CN'T) was derived in the late 50s by Turnbull and Fischer. The homogeneous nucleation rate 1 in condensed systems is given by 1 nn UM /3meT) O x expl-(AGp + W')/kT], (1) where: the number of molecules or formula units of nu- cleating phase per unit volume of parent phase (typically 1028102%m-*); v= vibration frequency (103s-!); n$= number of molecules on the surface of a critical nucleus; number of molecules in the critical nucleus; “Thermodynamic barrier for nucleation; no cleus/matrix interface; k= Boltzmann's constant; The quantity (n$/nº)(W" /8mkT)!2 is within one or two powers of ten for all nucleation problems of in- terest. Therefore, eq. (1) may be written with sufficient accuracy as 1=nvexpl-(AGD + Wº)/kT], (2) where the pre-exponential factor 4 = (nyv) is typically 104! — 1042 m-38"2, Assuming that the molecular re-arrangement for the nucleation process can be described by an effective dif fusion coefficient, D, we have D=vAexp(-AGp/kT), (3) where X is the jump distance, of the order of atomic dimensions. D can be related to the viscosity (7) by means of the Stokes- Einstein equation: D=kT/3mAn. (4) Combining egs. (2), (3) and (4) we have Brazilian Journal of Physics, vol. 22, no. 2, June, 1992 of such processes is usually described by a theory de- rived in the period 1937-1939 by Kolmogorov!2, John- son and Mehl!t and Avramils-1?, best known as the Kolmogorov-Avrami or Johnson-Mehl-Avrami (JMA) theory. Since that time this theory has been intensively used by materials scientists to study the various mech- anisms of phase transformations in metals. More re- cently, the JMA theory has been employed by polymer and glass scientists. Examples of technological impor- tance include the study of stability of glass metals, cur- ing of odontological plasters, devitrification time of rad- wast glasses, glass-ceramics and kinetics calculations of glass formation!8, Avrami!8-17 has 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 num- berofembryos (N) which are gradually exhausted. The number of embryos decreases in two ways; by growing to critical sizes (becoming critical nuclei) with 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 else- where. Under these conditions Avrami!17 has shown that the transformed fraction volume, o”, is given by a'=1-exp — x (estou t+ = EE + CP a where g 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, corre- sponding to very small or very large values of vt. Small values imply that the nucleation rate, 1 = Nvexp(—vt), is constant. Expanding exp(—vt) in eg. (1) and drop- ping fifth and higher order terms gives a'=1-exp(-gu' It? /4), (19) where lg = No. This is the special case treated by Johnson and Mehl? and is valid for N very large when the num ber of embryos is not exhausted until the end of the transformation (homogencous nucleation). Large val- ues of vt, in contrast, means that all nucleation centers are exhausted at an early stage in the reaction. The limiting value of eq. (18) is then a =1-exp(-9Nwt?). (20) Eq. (20) applies for small À, when there is a rapid exhaustion of embryos at the beginning of the reaction (instantaneous heterogeneous nucleation). Avrami has proposed that for a three-dimensional nucleation and 81 Table II Avrami parameters, m, for several mechanisms (Spherical Growth) Interface Diffusion Controlled Controlled Growth Growth Constant 1 4 25 Decreasing 1 3-4 152.5 Constant number of sites 3 15 growth process, the following general relation should be used a'=1-exp(-Kt”), (21) where 3 < m < 4. This expression covers all cases where 1 is some decreasing function of time, up to the limit when Tis constant. Eq. (21) also covers the case of het- erogeneous nucleation from a constant number of sites, which are activated at a constant rate until becoming depleted at some intermediate stage of the transforma- tion. In the more general case, where I and u are time dependent t l-ezp (FL I(r) x 3 dh [ / ' atear) &) É (22) where 7 is the time of birth of particles of the new phase. Table II shows values of m for different transformation mechanisms. Thus, if spherical particles grow in the internal volume of the sample then m should vary from lôto4. If growth proceeds from the external surfaces towards the center (collunar shape) then m will be dif ferent. The above treatment, whilst including the effects of impingement neglects the effect of the free surfaces. This problem was recently treated by Weinberg'?. Eq. (21) is usually written as: q = In In(i=0)1=InK +mnt. (23) This expression is intensively employed by materials scientists to infer the mechanisms of several classes of phase transforinations from the values of m, that is the slopeofln In(1l-a)-! versus In £ plots. The linearity of such plots is taken as an indication of the validity of the JMA equation. It should be emphasized, however, the ln— In plots are insensitive to variations of a” and t and that the value of the intercept K is seldom compared to the theoretical value. This is mainly due to the great 82 difficulty in measuring the high nucleation and growth rates in metallic and ceramic (low viscosity) systems. VIL. Application to glass crystallization The JMA theory can be shown to be exact within the framework of its assumptions. Hence, any viola tion 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 JMA theory has been employed to analyze experimental data for crystallinity versus time in both isothermal and non- isothermal heat treatments of glass systems. Empha- sis was usually given to values of m obtained from the slopes of experimental In In(1-0")-1 versus In t plots. In20-24 for instance, m ranged from 1 for surface nu- cleation to 3 for internal nucleation. In no case has the intercept been compared with the theoretical value. Recently, Zanotto and Galhardi? carried out a se- ries of experiments to test the validity of the Johnson- Mehl-Avrami theory. The isothermal crystallization of a nearly stoichio metric Nas0.2Ca0.3Si0, glass was studied at 6276. and 629ºC by optical microscopy, density measure- ments and X-ray diffraction. Both nucleation and growth rates were measured by single and double stage heat treatments up to very high volume fractions trans- formed and the experimental data for crystallinity were compared with the calculated values at the two temper- atures. The early crystallization stages were well de- scribed by theory for the limiting case of homogeneous nucleation and interface controlled growth. For higher degrees of crystallinity, both growth and overall crystal- lization rate decreased due to compositional changes of the glassy matrix and the experimental kinetics could be described by theory if diffusion controlled growth was assumed. It was also demonstrated that the sole use of numerical fittings to analyse phase transforma- tion kinetics, as very often reported in the literature, can give misleading interpretations. It was concluded that if proper precautions are taken the general theory predicts the glass-crystal transformation well. VHI. Glass formation Turnbull?º noted that there are at least some glass formers in every category of material based on bond type (covalent, ionic, metallic, van der Waals, and hy- drcgen). The cooling rate, density of nuclei and various material properties were suggested as significant factors which affect the tendeney of different liquids to form glasses. This approach leads naturally to posing the ques- tion not whether à material will form an amorphous E. D. Zanotto the estimation of a necessary cooling rate reduces to two questions: (1) how small a volume fraction of crys- tals embedded in a glassy matrix can be detected and identified; and (2) how can the volume fraction of crys- tals be related to the kinetic constants describing the nucleation and growth Processes, and how can these ki- netic constants in turn be related to readily-measurable parameters? Im answering the first of these questions, Uhlmann'8.27 assumed crystals which are distributed randomly through the bulk of the liquid, and a volume fraction of 1078 as a just-detectable concentration of crystals. In answering the second question, Ublmann adopted!827 the formal theory of transformation kinet- ies described in this section. In this paper I shall be concerned with single- component materials or congruently-melting com- pounds, and will assume that the rate of crystal growth and the nucleation frequency are constant with time. For such à case, the volume fraction, a”, crystallized in a time t, may for small a be expressed by a simplified form of Eq. (19): aa lourt, (24) In identifying Jo as the steady-state rate of homo- geneous nucleation, 1 shall neglect heterogeneous nu- cleation events-such as at external surfaces — and will be concerned with minimum cooling rates for glass for- mation, Clearly, a glass cannot be formed if observable amounts of crystals form in the interiors of samples. 1 shall also neglect the effect of transients during which the steady-state concentrations of subcritical embryos are built up by a series of bimolecular reactions. Neglect oftransients in the present analysis is justified whenever the time required to establish the steady-state nucle- ation rate is small relative to the total transformation time. The cooling rate required to avoid a given volume fraction crystallized may be estimated from eq. (24) by the construction of so-called T-T-T (time-temperature- transformation) curves, an example of which is shown in figure 3 for two different volume fractions crystallized. Im constructing these curves, a particular fraction crys- tallized is selected, the time required for that volume fraction to form at a given temperature is calculated and the calculations is Tepeated for other temperatures (and possibly other fractions crystallized). The nose in a T-T-T curvo, corresponding to the least time for the given volume fraction to crystallize, results from a competition between the driving force for crystallization, which increases with decreasing temper- ature, and the atomic mobility, which decreases with decreasing temperature. The transformation times tb, are relatively long in the vicinity of the melting point as well as at low temperatures; and for Purposes of the Present paper, I shall approximate the cooling rate re- Brasilian Journal of Physics, vol. 22, no. 2, June, 1992 quired to avoid a given fraction crystallized by the re- lation a (5) a BEN (25) c dt TN where ATy = Tm — Tw; Tw is the temperature at the nose of the T-T-T curve; rx is equal to the time at the nose of the T-T-T curve, and Ty is the melting point. 20 T T T soh sor Undercooling (ex) soh s 1 1 4 1 1 16! 10 to! 10º Time (sec.) Figure 3: Time-temperature transformation curves for salol: (A) o” = 10; (B) a” = 1058, From the form of eq. (24), as well as from the curves shown in figure 3 which were calculated therefrom, it is apparent that the cooling rate required for glass forma- tion is rather insensitive to the assumed volume fraction crystallized, since the time at any temperature on the T-T-T curve varies only as the one-fourth power of a”. An alternative estimate of the glass-forming char- acteristics of materials may be obtained by considering the thickness of sample which can be obtained as an amorphous solid. Again using the criterion of a vol- ume fraction crystallized less than 107º, and neglect- ing problems associated with heat transfer at the exter- nal surfaces of the sample, the thickness, y., of sample which can be formed without detectable crystallization should be of the order of?” ve = (Dry)? (26) where D is the thermal diffusivity of the sample. To estimate the critical conditions to form a glass of a given material, one can in principle to employ the measured values of the kinetic factors to calculate the T-T-T curves. In practice, however, information on the temperature dependence of the nueleation frequency is seldom available; and in only a portion of the cases of interest there are adequate data available on the varia- tion of the growth rate with temperature. 83 IX. Concluding remarks The kinetic approach of glass formation allows one to conclude that all! materials are capable of forming amorphous solids when cooled in bulk form from the liquid state, The question to be answered is how fast must a given liquid be cooled in order that crystalliza- tion be avoided. Thus novel materials such as metalhc alloys, with unusual properties, have been successfully obtained by very fast quenching?8. On the other hand, if crystal nucleation is controlled to occur uniformely in the bulk of certain glasses, a variety of advanced glass- ceramics can be and, indeed, are being commercially produced? , Deeper insights on the crystallization process, such as precise predictions of TTT curves, and consequently of critical cooling rates for glass formation, based solely on materials properties, will depend critically on new developments concerning the nucleation theory. One interesting attemp on that issue was recently advanced by Meyer with his Adiabatic Nucleation Theory?º. Acknowledgements The author thanks his co-workers and students who collaborated in several phase-transformations problems in the past fifteen years namely: A. Craievich, M. Wein- berg, E. Meyer, P. James, E. Miller, C. Kiminami, A. Galhardi, N. Mora, M. Leite, E. Belini, E. Wittman, E. » Ziemath, Thanks are also due to PADCT (New Materials), contract nº 620058/91-9, for financial support. 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