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Practical limits on up-gradient crystallization
JOURNAL OF
ELSEVIER
Journal of Non-Crystalline Solids 181 (1995) 291-300
Practical limits on up-gradient crystallization S.J. Kim 1, D.P. Birnie III *, B.J.J. Zelinski, D.R. Uhlmann Department of Materials Science and Engineering, University of Arizona, Tucson, AZ 85721, USA Received 17 March 1994; revised manuscript received 2 August 1994
Abstract
The kinetic processes relevant to up-gradient crystallization are analyzed in detail. Crystallization occurs upon moving an amorphous sample up through a temperature gradient which is the reverse of traditional crystallization from the melt. Both nucleation and growth processes are important to the present technique. To obtain a highly oriented microstructure, bulk nucleation must be prevented. Generally, a liquid with small nucleation rate and large crystal growth rate is favorable for the present route. The model was tested for the lithium diborate glass system, which yielded a highly oriented microstructure with only isolated additional nucleation events.
l. Introducfion
In many materials, the microstructure, as defined by the number, type and arrangement of grains and grain boundaries, plays a critical role in determining physical properties. A highly oriented unidirectional microstructure is desirable for many technologically important applications including aligning grains in piezoelectric materials [1], enhancing tensile strength in structural materials [2] and obtaining high critical current density in high temperature ceramic superconductors [3]. Methods used to obtain highly oriented microstructure include melt-solidification [4-6], hot extrusion with applied stress [7,8] and crystallization of a glass by heating the sample in a temperature
* Corresponding author. Tel: + 1-602 322 2960. Telefax: + 1602 322 2993. E-mail: [email protected]. 1 Present address: IBM, Almaden Research Center, San Jose, CA, USA.
gradient [9-18]. This last technique, referred to as 'up-gradient crystallization', is the focus of the present study. Carpay and Cense demonstrated highly oriented microstructures by translating glass samples of various lithium borate compositions (ranging from 72.6 to 91.2 mol% B203) through a temperature gradient [11,12]. In their work, the kinetic processes during the gradient heating were not described, but strong unidirectional characteristics were achieved. Lu and Klein obtained a highly oriented microstructure from a potassium disilicate glass using a similar method [9] 2. Abe et al. also used a temperature gradient to crystallize calcium phosphate glass samples and form a highly oriented microstructure [20]. Other studies have applied up-gradient processing in a similar manner, to achieve solid-state, eutectoidal, directional transformation in binary alloys [21-24].
2 In their study, the temperature gradient did not play a role in suppressing bulk nucleation because potassium disilicate does not exhibit measurable bulk nucleation [19].
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
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SJ. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
We focus on unidirectional crystallization during translation of a pre-formed glass into and through a hot zone such that crystallization occurs during heating of the sample. The applied temperature gradient facilitates the development of a highly oriented microstructure, thus achieving the desired microstructure/ composition/ property combination. This method can produce samples which have specialized microstructures of high density, with geometry limited only by traditional glass-forming practice. One previous study modeled the competition between bulk nucleation and crystal growth rate in an up-gradient geometry [18]. This earlier work based its limiting criterion on a detectable (or minimum allowable) volume of crystallized material formed during a gradient process. By contrast, the present model focuses on the sequential nature of this gradient process. Any bulk nucleus that forms will become an integral part of the growth interface and interrupt the orientation and texture for material that develops later. Depending on the growth rates and applied gradient, the detectibility criterion may be insufficient. The present paper discusses the fundamental aspects of up-gradient crystallization and the kinetic processes it entails. Then, a model based on the avoidance of any nucleation is presented. This model can be used to aid in establishing which types of system and which processing conditions are required to generate highly oriented microstructures.
2. Up-gradient crystallization: general description Initially, a glass sample is prepared by cooling a homogeneous melt having the desired shape. For the present analysis we focus on a glass bar of cross-sectional area A. Following glass preparation, one end of the glass bar is seeded by the attachment of previously formed crystalline material of suitable crystal structure and orientation. During crystallization, the seeded side enters the gradient zone first and the direction of crystal growth is in the direction opposite to sample motion. Growth is occurring while the sample moves up the temperature gradient. The translation of this seeded glass sample is illustrated in Fig. 1. The furnace is controlling at a temperature, Tmax, somewhat below the
Heat
Shield
V
------->
Seeds i<
L
~'
Fig. 1. Schematic diagram of unidirectional crystallization of glass. A seeded sample rod of length, L, and cross-sectional area, A, is translated through a furnace at velocity, V. For this process, the crystal growth interface is moving in the direction opposite to the sample translation.
melting point for the composition which prevents the newly formed crystalline material (and the seeds) from being remelted. Seed crystals at the starting end of the rod serve as locations where crystal growth occurs. At steady state, the interface grows at the temperature, Tint, where the crystal growth rate equals the translation rate of the glass rod into the temperature gradient. This establishes an even and flat crystallization front where the crystalline material is hotter than the amorphous material which is therefore in a reverse orientation from that normally found in melt solidification. The operational variables that control the reverse directional crystallization process are the translation velocity, V, local temperature gradient G, the crosssectional area of the bar being crystallized, A, and the total length of crystalline material to be successfully transformed, L. Note that the local temperature gradient depends both on externally applied conditions and on material factors such as the thermal conductivity and heat of crystallization. In order to generate a unidirectional texture in the final sample microstructure, appreciable bulk nucleation must not occur within the glassy material before the crystallization front passes. This is especially true when the desired orientation is not the fastest growth direction. In this case, a randomly oriented nucleus that forms ahead of the growth front could eventually modify the entire sample orientation through growth velocity competition. Note that
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
this is a more severe limitation than found by determining a minimum volume fraction transformed [18]. The requirement for forming less than a single nucleus in a given sample volume can be formulated as
Unuc,oi =ALf
I(T(t))
dt < 1,
(1)
where AL is the total sample volume to be transformed, I(T) is the temperature dependent nucleation rate and T(t) is the temperature history of the sample. Since the sample is modeled as moving at a uniform rate, every position in the sample experiences the same thermal treatment, although shifted in time. Nnuclei is the total number of nuclei formed in the sample and must be smaller than 1 for successful up-gradient crystallization. For a manufacturing operation a more stringent criterion may need to be imposed. Eq. (1) can be converted to a temperature basis using the identity dT/dt = (dT/dx)(dx/dt)
= G(T)V
(2)
to give
[r~
AL I(T) N"uc'¢i = T Jr, G---~) dT,
(3)
where G(T) is the local temperature gradient and V is the constant translation velocity. Ahead of the growth front, a portion of the sample will be in the supercooled state where it may be vulnerable to homogeneous nucleation. This vulnerability applies to all portions of the sample which are at temperatures where the nucleation rate is significant. Since kinetics of crystallization are usually frozen-out below the glass transition temperature, the lower limit for integration in Eq. (3) will be Tg 3. The upper integration limit, T,, is the maximum temperature for which the nucleation rate is significant. As a practical constraint, this maximum temperature cannot be hotter than the interface temperature, Ti,t. In our model, the achievement of successful upgradient crystallization depends on the control of nucleation alone. Nucleation needs to be prevented
3 For systemslike calciumphosphatewherethe kineticsmay be appreciable even below Tg [20,25], the lower integrationlimit is appropriatelymodified.
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within the glass volume until the seeded crystallization front can engulf the material and transform it to the desired orientation. The temperature gradient and translation velocity combine to reduce the number of homogeneous nuclei that form in a sample by reducing the total amount of material at temperatures in the range between Tg and Tu. Thus, grains formed by edgeseeding can grow continuously to form a highly oriented microstructure.
3. Up-gradient crystallization: physical limits Several experimental and material factors must be considered to determine whether up-gradient crystallization can be used to generate oriented microstructures in a particular system. These include initial homogeneous glass formation ability, translation rate selection, temperature gradient control and a method for selectively inducing nucleation at the leading edge of the sample. In very general terms, it is expected that the most favorable conditions will occur with the fastest translation velocity and steepest temperature gradient because this combination will yield the smallest nucleation time for the supercooled liquid before the crystallization front passes. 3.1. Glass preparation
The starting glass sample must be of high quality to minimize all contributions to bulk nucleation during subsequent processing. It should be homogeneous and free of any internal impurities or inclusions that might cause heterogeneous nucleation. In addition, a glass sample free of liquid-liquid phase separation is preferable because this often leads to copious bulk nucleation [26-28]. Establishing critical cooling rates for glass formation has long been an important problem and is tied directly to the nucleation and growth rates [29-32]. However, many of these treatments use an observability criterion for determining whether glass formation has occurred. Usually this is related to some minimum volume fraction transformed as detectable by XRD. For the present work, it is required that absolutely no nuclei form during formation of the piece. Thus, glass preparation should be performed according to Turn-
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SJ. Kim et aL /Journal of Non-Crystalline Solids 181 (1995) 291-300 180 160 140
E ~2o E E ,~ lOO
"2
(9 40 20
C 6OO
~50
7oo
7so ~o Temperature (C)
8~o
9oo
Fig. 2. Experimental crystal growth velocity data for lithium diborate [33]. The peak growth rate is labeled UTax. The sample must be translated at a velocity smaller than this value to obtain a stable glass/crystal interface.
bull's more rigorous model of glass formation which avoids a single nucleation event [29]. For the purposes of the present paper, we will assume that the time-temperature exposure during glass forming is negligible compared with the exposure to be experienced by the sample during the gradient crystallization step. This is valid because our sample translation rate must be slow enough to match the inherent crystallization velocity of the material. Thus, we believe that Eq. (1) will be more restrictive than the glass formation criterion.
3.2. Translation rate selection The seeded glass sample must be translated into the furnace to induce directional crystallization. In a practical sense, this sample translation should be done as quickly as reasonably possible so that the residence time of the sample passing through the gradient region is minimized. However, the translation velocity has an upper limit that is defined by the maximum crystal growth rate, Ureax, for the sample being tested. Consider for example Fig. 2 which shows the growth rate behavior of lithium diborate [33]. For lithium diborate, if the sample is translated faster than 177 mm/min, the crystallization front cannot keep pace with the translation and ultimately gets pushed completely into the furnace hot zone. At this point, the sample geometry is essentially a seeded
isothermal process which would proceed at the rate governed by the growth velocity and the possible occurrence of heterogeneous or homogeneous nucleation at the temperature of the furnace hot zone. When translating the sample at a velocity smaller than Umax,the system will establish a uniform growth interface location such that the interface temperature will define a growth rate exactly equal to the translation rate. When this condition is met, the translation rate and growth rate will cancel to yield a stationary interface position and therefore fixed interface temperature 4. Note that, because of the seeding, the lower-temperature solution (of the two possible solutions) is established by the crystallizing system. When a very low translation rate is selected, the residence time of the sample becomes larger and the chance of bulk nucleation increases as well. This consideration indicates that whenever possible a sample should be translated at a rate approaching the peak growth rate of the material.
3.3. Temperature gradient selection In addition to increasing the sample translation velocity, it is possible to further reduce the time exposure of the sample to nucleation by controlling the externally applied temperature boundary conditions. This can be done by building an apparatus with heat shields and chill blocks in close proximity [18] so that a steep temperature gradient is applied. If the nucleation and growth behavior have been experimentally determined for the material system of choice, then a critical temperature gradient for uninterrupted growth can be derived. Assuming that the temperature gradient can be approximated as linear (and thus temperature independent), Eq. (3) can be further simplified to Nnuclei = ( AL/GV) ~ T"I(T) dT. T,
(4)
If the number of nuclei is less than one, then we can expect that successful gradient nucleation will occur.
4 The interface temperature may not be defined exactly by Fig. 2. Herron and Bergeron [34] have reported local interface heating in lithium diborate associated with latent heat effects.
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
The critical gradient for this condition, acrit , c a n be defined as acrit = aLIt°tal//v,
(5)
where /total is the nucleation rate integrated over temperature as held in Eq. (4). Note that the final desired sample volume (AL) plays a role in determining how steep the temperature gradient must be. Also, the faster that crystallization can be performed the more likely that this technique will be successful. Finally, the smallest allowable critical temperature gradient, Glimit, can be found when it is possible to translate the sample at the peak crystal growth rate
295
by the sample motion. If the translation velocity is relatively slow, then thermal conductivity will dissipate the latent heat and prevent any dramatic temperature spike at the moving interface. It should be noted that the temperature gradient used to integrate the nucleation rate is the gradient found at temperatures where nucleation is significant. These temperatures may be much lower than Tint . In these cases, temperature perturbations caused by release of the latent heat may not greatly impact acrit. If the nucleation and growth curves have significant overlap, the heat transport equations would have to be solved to determine the relevant gradient.
( V ~,~--Umax )" alimi t = AL/t°tal/Uma x .
(6)
3.4. Heat conduction limitation The physical design of the thermal gradient apparatus is important in that it must take into account sample parameters such as the thermal conductivity, the latent heat of crystallization and the translation velocity. These heat conduction effects may put a practical upper bound on the temperature gradient that can be applied to a given sample geometry. The effect of the thermal conductivity is to smooth out the externally applied temperature gradient; more conductive samples will provide a conduit for heat flow to bypass the heat shields and other factors designed to keep a steep temperature gradient. However, as larger translation velocities are used, then steeper temperature gradients must exist to permit the heat flow to keep up with the translation rate. This will impact interface position within samples and may limit physical crystallization rates due to sample conductivity effects. The effect of latent heat is to perturb temperature distribution near the crystal/liquid interface. This may cause temperature gradients in the crystal and liquid to be different; then, depending on the translation rate and the gradient temperature profile, a local temperature spike may occur [35-37]. The size of this effect will depend on the translation velocity. At larger velocities, there may be a dramatic temperature increase, although the aggregate effect will yield an interface temperature that is compatible with the translation velocity that has been externally selected
3.5. Surface nucleation prevention Since heterogeneous nucleation at free surfaces or sample support interfaces is usually much easier than homogeneous nucleation, it is likely that some surface nuclei will form. Any surface nuclei that occur will grow with random orientations in competition with the seeded crystallization front. However, since these nuclei are growing laterally from the outside surface, it will be difficult for the entire growth front to become occluded by randomly oriented material. We expect a cored morphology with directional texture inside the sample which would have an external shell of small grain size, having randomly oriented texture.
3.6. Crystal seeding The mode of seeding the glass rod is also important. Several modes could be possible including (1) the use of a single crystal in contact with the glass, (2) the use of polycrystalline sample in contact with the leading sample edge, and (3) a physical 'neck' or constriction which would select a single grain growing from a polycrystalline matrix [38,39]. Each of these has its merit, although only the first technique may be used to select the final product orientation. The creation of slow growth velocity orientations using this method has not, to our knowledge, been demonstrated. The other two techniques yield matrices having their orientation aligned with the fastest growth velocity direction. In summary, the processing method of up-gradient directional crystallization can be a useful tech-
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
296
x
._~ 0.6
| g
O.S
\
0.4
~ (1.3 = 0.2
Z
0.1 48o
49o
~o
sio Temperature(O)
s~o
s3o
Fig. 3. Experimental nucleation data for lithium diborate from
Smith and Weinberg [40,41]. Superimposed solid line is a parabola fitted to the measured datapoints. Original error bars have been omitted for simplicity.
nique for the fabrication of oriented single-phase material. The main restriction on this technique is that the cumulative probability of homogeneous nucleation in the glass ahead of the crystallization front must be less than one. If the material and geometric parameters allow this, then the microstructure can be engineered to have texture in the direction of crystallization.
which diffusion was required for separation into two phases during crystallization [11,12]. Appropriate amounts of Li2CO 3 and B203 were mixed, melted at 1050°C for 2 h using a Pt-crucible, and cooled. This glass sample was cut and polished to a 1 Izm finish. Sample seeding was performed by using small amounts of a powder of a previously crystallized lithium diborate sample. Since a random powder was used for seeding, the texture should mirror the fast growth velocity direction. The furnace temperature was held at 725°C, where the crystal growth rate is a maximum. The seeded sample was translated into the furnace at a rate of 45 m m / m i n which is slower than the peak growth rate. Consequently, the interface temperature will be < 725°C. The externally applied thermal gradient was determined to be about 25°C/mm. This gradient was evaluated at 500°C, the temperature at the maximum nucleation rate. After translating the sample so that approximately half had been crystallized, the sample was rapidly pulled back from the gradient zone to freeze the sample and allow inspection of the solid-liquid interface.
5. Results After crystallization, the microstructure was examined using optical microscopy and XRD. Several
4. Experimental procedure The lithium diborate composition, Li20-2B203, was chosen to test the present model. This glass was selected because related compositions in the lithium-borate system have been crystallized successfully using the up-gradient technique, although they yielded two-phase mixtures [11,12]. In addition, lithium diborate glass shows no phase separation and the homogeneous nucleation rate and crystal growth rates have already been characterized as a function of temperature (see Figs. 2 and 3). Finally, the growth rates are fast enough that a reasonable translation rate can be used during processing. Since lithium diborate melts congruently, growth is limited by interface kinetics, not by diffusion ahead of the solid-liquid interface. This is by contrast with the earlier work on lithium borates in
Fig. 4. Fibrous microstructure found after gradient heating the lithium diborate sample. Crystallization proceeded from bottom to top. The fiber axis is aligned parallel to the crystallization front motion direction.
S.I. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
Fig. 5. Location near the solid-liquid growth interface where a single homogeneous nucleation event occurred and then grew along with the crystallization front. This comet-shaped microstructure results from growth velocity competition between the matrix and the randomly oriented, newly formed nucleus. Twinning and growth structure is evident near the center of the grain. The flat growth interface (frozen-in by rapid cooling) appears at the top of the frame.
general microstructural characteristics were noticed. First, only a thin shell of sparse surface nucleation had occurred while the majority of the microstructure was strongly textured. Second, the grain structure in the textured region appeared to be very fibrous with the fiber axis oriented in the translation direction (see Fig. 4). Also, third, occasional homogeneous nucleation had occurred ahead of the crystallization front with subsequent incorporation of the nuclei into the oriented microstructure to varying degrees. One such crystallite is shown in Fig. 5. X-ray diffraction confirmed that the visually textured surface is, in fact, highly oriented. The diffraction pattern, as taken from the cross-section pictured in Fig. 4, displayed no (h00) peaks, demonstrating that none of the crystals had these planes parallel to the surface of the sample. This indicates that strong texture was imposed on the sample by up-gradient crystallization.
6. Discussion
Several aspects of this seeded directional crystallization process are of particular interest. First, the
297
discussion is directed toward the basic foundation of the present model (that nucleation is the key factor that controls the ability to create gradient structures). Then, the general applicability of this technique is discussed. Finally, our experimental crystallization morphology is examined in light of the critical temperature gradient calculation performed above. The justification that nucleation is the key controlling factor for up-gradient crystallization is based on an examination of growth velocity competition effects. One significantly attractive feature of this technique is that it can be used for growth of selected orientations by seeding, even in a material which exhibits anisotropic growth rate. This is important because the orientation of choice may not be the most rapidly growing orientation. In such a case, the randomly oriented faces of a homogeneous nucleus that forms ahead of the interface may be able to grow faster than the interface. Such a nucleus will move at the same rate but be at a colder temperature and thus be ahead of the interface. As sample translation occurs, the homogeneously formed nucleus will crystallize ahead of the oriented interface and will yield a gradually broadening 'wake' of poorly oriented material. This prevents the chosen orientation from establishing itself in the entire gradient-crystallized sample. In the special case where the fastest growth direction is also the orientation of choice, then growth velocity competition will cause improperly oriented nuclei to be engulfed by the crystallization front. This is what must have happened to the grain pictured in Fig. 5. Therefore, only in this special case will the volume fraction crystallized [18] be the criterion for assessing a potential for successful gradient crystallization. The requirement that no homogeneous nucleation occurs during processing may limit this technique to systems that are relatively good glass formers, or to processing techniques where very steep gradients can be applied such as laser firing of amorphous surface layers. Many oxide systems prefer surface nucleation [19] which implies that homogeneous nucleation rates are low, so they may be good candidate systems. It may also be possible to up-gradient process some systems where the homogeneous nucleation rate is too large for isothermal crystallization. It is important that the crystal growth rate for the composition be significant because, if growth is too
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S.J. Kim et al. /Journal of Non-CrystallineSolids 181 (1995) 291-300
slow, isothermal processing could be carried out with careful edge seeding. The primary application will be to systems where the reduction in temperature exposure (by comparison with isothermal processing) will lower the possibility of homogeneous nucleation ahead of crystallization. The nucleation integration presented as Eq. (4) above can be compared with the results found in the present experiment. Experimental nucleation rate data from Smith and Weinberg [40-42] are plotted in Fig. 3 above. These data have been fitted to a parabola for convenience of integration. This line has been superimposed upon the measured datapoints. Then this equation was integrated and evaluated from the first zero crossing temperature to the next to yield the integral required in Eq. (3). The calculation predicts that 180 nuclei per cubic centimeter should be formed by heating in this 25°C/mm temperature gradient and translating the sample at 45 mm/min. Thus, the formation of homogeneous nuclei (an example of which was shown in Fig. 5) is not surprising. Because the total sample volume crystallized was about 0.05 cm 3, about nine nucleation events would be expected. This is very good correspondence between the previously measured nucleation rates and the present experiment. Note that, with so few nuclei, there is little interruption of the growing mass of oriented crystals. A large degree of directionality was achieved, as determined by XRD. At this point we make brief comparison with the earlier up-gradient modeling [18]. Very important differences in built-in assumptions have been made in the two approaches. The most important difference is that each integration element, dV, in their integral is treated independently. Previous work tacitly assumed that the orientation or nucleation extent of any previously crystallized volume element had no influence on the next volume element to enter the gradient zone. That assumption ignores the basic behavior of the system: that the entire growth front is passing through the material and defining the local orientation of subsequent material to be crystallized. This is why we have chosen a pure nucleation criterion; the growth of nuclei and of the interface itself have been assumed to occur at the same rate, so nuclei will interrupt the preferred orientation and defeat the process. The effect of their basic assumption can be noted
in their equation which compares the gradient, G, with the translation velocity, V0, as required for successful up-gradient crystallization. They have analyzed three different cases which depend on the degree of overlap between the nucleation and crystallization curves. Case I had both nucleation and growth becoming gradually more important starting at the same lower bound threshold temperature. Case II corresponds to the condition where nucleation and growth overlap significantly, but nucleation starts at somewhat lower temperatures. Case III covers the case where the peak nucleation rate is found just at the temperature where the growth starts to increase on heating. For these three cases, the following relationships were found: G
--
>>
0.5(C1/4/k5/4),
(7)
Vo 1/4 G / V 3/4 >> 0.7( lmax/k ),
(8)
1/4x / k ) , G / V 3/4 >> O.65(Ima
(9)
where C 1 and k are constants which parameterize the temperature dependencies of nucleation and growth respectively, and /max is the peak nucleation rate. This can be contrasted by rearranging our Eq. (5) above to give GV o > A L I t°tal.
(10)
Our criterion emphasizes that if a slower translation velocity is chosen, then a steeper temperature gradient will be required to keep the residence time in the crystallization zone below a threshold value. By contrast, all of their criteria show that a slower translation velocity will allow a shallower gradient. This appears to be due to the non-sequential nature of their integration. Another difference between the derivations is that, although like us they have also done their integration of the nucleation ahead of the interface, they base their estimation on nucleation and growth curves that are significantly overlapping. This may happen in some selected systems, but generally the temperature dependence of nucleation and growth are not so closely superimposed. Even for systems that have significant overlap, we believe that, by ignoring the sequential crystallization process found with this
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
technique, the gradients predicted will be severely underestimated. Finally, because of the interconnection between thermal conduction and thermal gradient, it is possible that these calculations and processing limits might apply even for laser-melting and regrowth of surface layers. This could then achieve very high effective translation velocity by moving the laser beam rather than the sample. This is relevant to materials with much faster growth rates. This could be used to avoid the faster homogeneous nucleation in these systems.
7. Conclusions Processes relevant to up-gradient directional crystallization of a glass have been described in detail. The model defines a processing window where a highly oriented microstructure may be produced. According to the present model, it is important to impose a steep temperature gradient in combination with a moderate sample translation rate to prevent bulk nucleation. The critical temperature gradient for successful processing can be calculated when the temperature dependence of the homogeneous nucleation rate is known. If homogeneous nucleation is not rapid in a given system, then directional growth may be easy to obtain in a temperature gradient. The model was tested successfully in lithium diborate glass. The authors are grateful to Professors K.A. Jackson, D.R. Poirier and M.C. Weinberg for stimulating discussions and to Dr G.L. Smith for sample preparation assistance.
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ELSEVIER
Journal of Non-Crystalline Solids 181 (1995) 291-300
Practical limits on up-gradient crystallization S.J. Kim 1, D.P. Birnie III *, B.J.J. Zelinski, D.R. Uhlmann Department of Materials Science and Engineering, University of Arizona, Tucson, AZ 85721, USA Received 17 March 1994; revised manuscript received 2 August 1994
Abstract
The kinetic processes relevant to up-gradient crystallization are analyzed in detail. Crystallization occurs upon moving an amorphous sample up through a temperature gradient which is the reverse of traditional crystallization from the melt. Both nucleation and growth processes are important to the present technique. To obtain a highly oriented microstructure, bulk nucleation must be prevented. Generally, a liquid with small nucleation rate and large crystal growth rate is favorable for the present route. The model was tested for the lithium diborate glass system, which yielded a highly oriented microstructure with only isolated additional nucleation events.
l. Introducfion
In many materials, the microstructure, as defined by the number, type and arrangement of grains and grain boundaries, plays a critical role in determining physical properties. A highly oriented unidirectional microstructure is desirable for many technologically important applications including aligning grains in piezoelectric materials [1], enhancing tensile strength in structural materials [2] and obtaining high critical current density in high temperature ceramic superconductors [3]. Methods used to obtain highly oriented microstructure include melt-solidification [4-6], hot extrusion with applied stress [7,8] and crystallization of a glass by heating the sample in a temperature
* Corresponding author. Tel: + 1-602 322 2960. Telefax: + 1602 322 2993. E-mail: [email protected]. 1 Present address: IBM, Almaden Research Center, San Jose, CA, USA.
gradient [9-18]. This last technique, referred to as 'up-gradient crystallization', is the focus of the present study. Carpay and Cense demonstrated highly oriented microstructures by translating glass samples of various lithium borate compositions (ranging from 72.6 to 91.2 mol% B203) through a temperature gradient [11,12]. In their work, the kinetic processes during the gradient heating were not described, but strong unidirectional characteristics were achieved. Lu and Klein obtained a highly oriented microstructure from a potassium disilicate glass using a similar method [9] 2. Abe et al. also used a temperature gradient to crystallize calcium phosphate glass samples and form a highly oriented microstructure [20]. Other studies have applied up-gradient processing in a similar manner, to achieve solid-state, eutectoidal, directional transformation in binary alloys [21-24].
2 In their study, the temperature gradient did not play a role in suppressing bulk nucleation because potassium disilicate does not exhibit measurable bulk nucleation [19].
0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
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We focus on unidirectional crystallization during translation of a pre-formed glass into and through a hot zone such that crystallization occurs during heating of the sample. The applied temperature gradient facilitates the development of a highly oriented microstructure, thus achieving the desired microstructure/ composition/ property combination. This method can produce samples which have specialized microstructures of high density, with geometry limited only by traditional glass-forming practice. One previous study modeled the competition between bulk nucleation and crystal growth rate in an up-gradient geometry [18]. This earlier work based its limiting criterion on a detectable (or minimum allowable) volume of crystallized material formed during a gradient process. By contrast, the present model focuses on the sequential nature of this gradient process. Any bulk nucleus that forms will become an integral part of the growth interface and interrupt the orientation and texture for material that develops later. Depending on the growth rates and applied gradient, the detectibility criterion may be insufficient. The present paper discusses the fundamental aspects of up-gradient crystallization and the kinetic processes it entails. Then, a model based on the avoidance of any nucleation is presented. This model can be used to aid in establishing which types of system and which processing conditions are required to generate highly oriented microstructures.
2. Up-gradient crystallization: general description Initially, a glass sample is prepared by cooling a homogeneous melt having the desired shape. For the present analysis we focus on a glass bar of cross-sectional area A. Following glass preparation, one end of the glass bar is seeded by the attachment of previously formed crystalline material of suitable crystal structure and orientation. During crystallization, the seeded side enters the gradient zone first and the direction of crystal growth is in the direction opposite to sample motion. Growth is occurring while the sample moves up the temperature gradient. The translation of this seeded glass sample is illustrated in Fig. 1. The furnace is controlling at a temperature, Tmax, somewhat below the
Heat
Shield
V
------->
Seeds i<
L
~'
Fig. 1. Schematic diagram of unidirectional crystallization of glass. A seeded sample rod of length, L, and cross-sectional area, A, is translated through a furnace at velocity, V. For this process, the crystal growth interface is moving in the direction opposite to the sample translation.
melting point for the composition which prevents the newly formed crystalline material (and the seeds) from being remelted. Seed crystals at the starting end of the rod serve as locations where crystal growth occurs. At steady state, the interface grows at the temperature, Tint, where the crystal growth rate equals the translation rate of the glass rod into the temperature gradient. This establishes an even and flat crystallization front where the crystalline material is hotter than the amorphous material which is therefore in a reverse orientation from that normally found in melt solidification. The operational variables that control the reverse directional crystallization process are the translation velocity, V, local temperature gradient G, the crosssectional area of the bar being crystallized, A, and the total length of crystalline material to be successfully transformed, L. Note that the local temperature gradient depends both on externally applied conditions and on material factors such as the thermal conductivity and heat of crystallization. In order to generate a unidirectional texture in the final sample microstructure, appreciable bulk nucleation must not occur within the glassy material before the crystallization front passes. This is especially true when the desired orientation is not the fastest growth direction. In this case, a randomly oriented nucleus that forms ahead of the growth front could eventually modify the entire sample orientation through growth velocity competition. Note that
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
this is a more severe limitation than found by determining a minimum volume fraction transformed [18]. The requirement for forming less than a single nucleus in a given sample volume can be formulated as
Unuc,oi =ALf
I(T(t))
dt < 1,
(1)
where AL is the total sample volume to be transformed, I(T) is the temperature dependent nucleation rate and T(t) is the temperature history of the sample. Since the sample is modeled as moving at a uniform rate, every position in the sample experiences the same thermal treatment, although shifted in time. Nnuclei is the total number of nuclei formed in the sample and must be smaller than 1 for successful up-gradient crystallization. For a manufacturing operation a more stringent criterion may need to be imposed. Eq. (1) can be converted to a temperature basis using the identity dT/dt = (dT/dx)(dx/dt)
= G(T)V
(2)
to give
[r~
AL I(T) N"uc'¢i = T Jr, G---~) dT,
(3)
where G(T) is the local temperature gradient and V is the constant translation velocity. Ahead of the growth front, a portion of the sample will be in the supercooled state where it may be vulnerable to homogeneous nucleation. This vulnerability applies to all portions of the sample which are at temperatures where the nucleation rate is significant. Since kinetics of crystallization are usually frozen-out below the glass transition temperature, the lower limit for integration in Eq. (3) will be Tg 3. The upper integration limit, T,, is the maximum temperature for which the nucleation rate is significant. As a practical constraint, this maximum temperature cannot be hotter than the interface temperature, Ti,t. In our model, the achievement of successful upgradient crystallization depends on the control of nucleation alone. Nucleation needs to be prevented
3 For systemslike calciumphosphatewherethe kineticsmay be appreciable even below Tg [20,25], the lower integrationlimit is appropriatelymodified.
293
within the glass volume until the seeded crystallization front can engulf the material and transform it to the desired orientation. The temperature gradient and translation velocity combine to reduce the number of homogeneous nuclei that form in a sample by reducing the total amount of material at temperatures in the range between Tg and Tu. Thus, grains formed by edgeseeding can grow continuously to form a highly oriented microstructure.
3. Up-gradient crystallization: physical limits Several experimental and material factors must be considered to determine whether up-gradient crystallization can be used to generate oriented microstructures in a particular system. These include initial homogeneous glass formation ability, translation rate selection, temperature gradient control and a method for selectively inducing nucleation at the leading edge of the sample. In very general terms, it is expected that the most favorable conditions will occur with the fastest translation velocity and steepest temperature gradient because this combination will yield the smallest nucleation time for the supercooled liquid before the crystallization front passes. 3.1. Glass preparation
The starting glass sample must be of high quality to minimize all contributions to bulk nucleation during subsequent processing. It should be homogeneous and free of any internal impurities or inclusions that might cause heterogeneous nucleation. In addition, a glass sample free of liquid-liquid phase separation is preferable because this often leads to copious bulk nucleation [26-28]. Establishing critical cooling rates for glass formation has long been an important problem and is tied directly to the nucleation and growth rates [29-32]. However, many of these treatments use an observability criterion for determining whether glass formation has occurred. Usually this is related to some minimum volume fraction transformed as detectable by XRD. For the present work, it is required that absolutely no nuclei form during formation of the piece. Thus, glass preparation should be performed according to Turn-
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SJ. Kim et aL /Journal of Non-Crystalline Solids 181 (1995) 291-300 180 160 140
E ~2o E E ,~ lOO
"2
(9 40 20
C 6OO
~50
7oo
7so ~o Temperature (C)
8~o
9oo
Fig. 2. Experimental crystal growth velocity data for lithium diborate [33]. The peak growth rate is labeled UTax. The sample must be translated at a velocity smaller than this value to obtain a stable glass/crystal interface.
bull's more rigorous model of glass formation which avoids a single nucleation event [29]. For the purposes of the present paper, we will assume that the time-temperature exposure during glass forming is negligible compared with the exposure to be experienced by the sample during the gradient crystallization step. This is valid because our sample translation rate must be slow enough to match the inherent crystallization velocity of the material. Thus, we believe that Eq. (1) will be more restrictive than the glass formation criterion.
3.2. Translation rate selection The seeded glass sample must be translated into the furnace to induce directional crystallization. In a practical sense, this sample translation should be done as quickly as reasonably possible so that the residence time of the sample passing through the gradient region is minimized. However, the translation velocity has an upper limit that is defined by the maximum crystal growth rate, Ureax, for the sample being tested. Consider for example Fig. 2 which shows the growth rate behavior of lithium diborate [33]. For lithium diborate, if the sample is translated faster than 177 mm/min, the crystallization front cannot keep pace with the translation and ultimately gets pushed completely into the furnace hot zone. At this point, the sample geometry is essentially a seeded
isothermal process which would proceed at the rate governed by the growth velocity and the possible occurrence of heterogeneous or homogeneous nucleation at the temperature of the furnace hot zone. When translating the sample at a velocity smaller than Umax,the system will establish a uniform growth interface location such that the interface temperature will define a growth rate exactly equal to the translation rate. When this condition is met, the translation rate and growth rate will cancel to yield a stationary interface position and therefore fixed interface temperature 4. Note that, because of the seeding, the lower-temperature solution (of the two possible solutions) is established by the crystallizing system. When a very low translation rate is selected, the residence time of the sample becomes larger and the chance of bulk nucleation increases as well. This consideration indicates that whenever possible a sample should be translated at a rate approaching the peak growth rate of the material.
3.3. Temperature gradient selection In addition to increasing the sample translation velocity, it is possible to further reduce the time exposure of the sample to nucleation by controlling the externally applied temperature boundary conditions. This can be done by building an apparatus with heat shields and chill blocks in close proximity [18] so that a steep temperature gradient is applied. If the nucleation and growth behavior have been experimentally determined for the material system of choice, then a critical temperature gradient for uninterrupted growth can be derived. Assuming that the temperature gradient can be approximated as linear (and thus temperature independent), Eq. (3) can be further simplified to Nnuclei = ( AL/GV) ~ T"I(T) dT. T,
(4)
If the number of nuclei is less than one, then we can expect that successful gradient nucleation will occur.
4 The interface temperature may not be defined exactly by Fig. 2. Herron and Bergeron [34] have reported local interface heating in lithium diborate associated with latent heat effects.
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
The critical gradient for this condition, acrit , c a n be defined as acrit = aLIt°tal//v,
(5)
where /total is the nucleation rate integrated over temperature as held in Eq. (4). Note that the final desired sample volume (AL) plays a role in determining how steep the temperature gradient must be. Also, the faster that crystallization can be performed the more likely that this technique will be successful. Finally, the smallest allowable critical temperature gradient, Glimit, can be found when it is possible to translate the sample at the peak crystal growth rate
295
by the sample motion. If the translation velocity is relatively slow, then thermal conductivity will dissipate the latent heat and prevent any dramatic temperature spike at the moving interface. It should be noted that the temperature gradient used to integrate the nucleation rate is the gradient found at temperatures where nucleation is significant. These temperatures may be much lower than Tint . In these cases, temperature perturbations caused by release of the latent heat may not greatly impact acrit. If the nucleation and growth curves have significant overlap, the heat transport equations would have to be solved to determine the relevant gradient.
( V ~,~--Umax )" alimi t = AL/t°tal/Uma x .
(6)
3.4. Heat conduction limitation The physical design of the thermal gradient apparatus is important in that it must take into account sample parameters such as the thermal conductivity, the latent heat of crystallization and the translation velocity. These heat conduction effects may put a practical upper bound on the temperature gradient that can be applied to a given sample geometry. The effect of the thermal conductivity is to smooth out the externally applied temperature gradient; more conductive samples will provide a conduit for heat flow to bypass the heat shields and other factors designed to keep a steep temperature gradient. However, as larger translation velocities are used, then steeper temperature gradients must exist to permit the heat flow to keep up with the translation rate. This will impact interface position within samples and may limit physical crystallization rates due to sample conductivity effects. The effect of latent heat is to perturb temperature distribution near the crystal/liquid interface. This may cause temperature gradients in the crystal and liquid to be different; then, depending on the translation rate and the gradient temperature profile, a local temperature spike may occur [35-37]. The size of this effect will depend on the translation velocity. At larger velocities, there may be a dramatic temperature increase, although the aggregate effect will yield an interface temperature that is compatible with the translation velocity that has been externally selected
3.5. Surface nucleation prevention Since heterogeneous nucleation at free surfaces or sample support interfaces is usually much easier than homogeneous nucleation, it is likely that some surface nuclei will form. Any surface nuclei that occur will grow with random orientations in competition with the seeded crystallization front. However, since these nuclei are growing laterally from the outside surface, it will be difficult for the entire growth front to become occluded by randomly oriented material. We expect a cored morphology with directional texture inside the sample which would have an external shell of small grain size, having randomly oriented texture.
3.6. Crystal seeding The mode of seeding the glass rod is also important. Several modes could be possible including (1) the use of a single crystal in contact with the glass, (2) the use of polycrystalline sample in contact with the leading sample edge, and (3) a physical 'neck' or constriction which would select a single grain growing from a polycrystalline matrix [38,39]. Each of these has its merit, although only the first technique may be used to select the final product orientation. The creation of slow growth velocity orientations using this method has not, to our knowledge, been demonstrated. The other two techniques yield matrices having their orientation aligned with the fastest growth velocity direction. In summary, the processing method of up-gradient directional crystallization can be a useful tech-
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
296
x
._~ 0.6
| g
O.S
\
0.4
~ (1.3 = 0.2
Z
0.1 48o
49o
~o
sio Temperature(O)
s~o
s3o
Fig. 3. Experimental nucleation data for lithium diborate from
Smith and Weinberg [40,41]. Superimposed solid line is a parabola fitted to the measured datapoints. Original error bars have been omitted for simplicity.
nique for the fabrication of oriented single-phase material. The main restriction on this technique is that the cumulative probability of homogeneous nucleation in the glass ahead of the crystallization front must be less than one. If the material and geometric parameters allow this, then the microstructure can be engineered to have texture in the direction of crystallization.
which diffusion was required for separation into two phases during crystallization [11,12]. Appropriate amounts of Li2CO 3 and B203 were mixed, melted at 1050°C for 2 h using a Pt-crucible, and cooled. This glass sample was cut and polished to a 1 Izm finish. Sample seeding was performed by using small amounts of a powder of a previously crystallized lithium diborate sample. Since a random powder was used for seeding, the texture should mirror the fast growth velocity direction. The furnace temperature was held at 725°C, where the crystal growth rate is a maximum. The seeded sample was translated into the furnace at a rate of 45 m m / m i n which is slower than the peak growth rate. Consequently, the interface temperature will be < 725°C. The externally applied thermal gradient was determined to be about 25°C/mm. This gradient was evaluated at 500°C, the temperature at the maximum nucleation rate. After translating the sample so that approximately half had been crystallized, the sample was rapidly pulled back from the gradient zone to freeze the sample and allow inspection of the solid-liquid interface.
5. Results After crystallization, the microstructure was examined using optical microscopy and XRD. Several
4. Experimental procedure The lithium diborate composition, Li20-2B203, was chosen to test the present model. This glass was selected because related compositions in the lithium-borate system have been crystallized successfully using the up-gradient technique, although they yielded two-phase mixtures [11,12]. In addition, lithium diborate glass shows no phase separation and the homogeneous nucleation rate and crystal growth rates have already been characterized as a function of temperature (see Figs. 2 and 3). Finally, the growth rates are fast enough that a reasonable translation rate can be used during processing. Since lithium diborate melts congruently, growth is limited by interface kinetics, not by diffusion ahead of the solid-liquid interface. This is by contrast with the earlier work on lithium borates in
Fig. 4. Fibrous microstructure found after gradient heating the lithium diborate sample. Crystallization proceeded from bottom to top. The fiber axis is aligned parallel to the crystallization front motion direction.
S.I. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
Fig. 5. Location near the solid-liquid growth interface where a single homogeneous nucleation event occurred and then grew along with the crystallization front. This comet-shaped microstructure results from growth velocity competition between the matrix and the randomly oriented, newly formed nucleus. Twinning and growth structure is evident near the center of the grain. The flat growth interface (frozen-in by rapid cooling) appears at the top of the frame.
general microstructural characteristics were noticed. First, only a thin shell of sparse surface nucleation had occurred while the majority of the microstructure was strongly textured. Second, the grain structure in the textured region appeared to be very fibrous with the fiber axis oriented in the translation direction (see Fig. 4). Also, third, occasional homogeneous nucleation had occurred ahead of the crystallization front with subsequent incorporation of the nuclei into the oriented microstructure to varying degrees. One such crystallite is shown in Fig. 5. X-ray diffraction confirmed that the visually textured surface is, in fact, highly oriented. The diffraction pattern, as taken from the cross-section pictured in Fig. 4, displayed no (h00) peaks, demonstrating that none of the crystals had these planes parallel to the surface of the sample. This indicates that strong texture was imposed on the sample by up-gradient crystallization.
6. Discussion
Several aspects of this seeded directional crystallization process are of particular interest. First, the
297
discussion is directed toward the basic foundation of the present model (that nucleation is the key factor that controls the ability to create gradient structures). Then, the general applicability of this technique is discussed. Finally, our experimental crystallization morphology is examined in light of the critical temperature gradient calculation performed above. The justification that nucleation is the key controlling factor for up-gradient crystallization is based on an examination of growth velocity competition effects. One significantly attractive feature of this technique is that it can be used for growth of selected orientations by seeding, even in a material which exhibits anisotropic growth rate. This is important because the orientation of choice may not be the most rapidly growing orientation. In such a case, the randomly oriented faces of a homogeneous nucleus that forms ahead of the interface may be able to grow faster than the interface. Such a nucleus will move at the same rate but be at a colder temperature and thus be ahead of the interface. As sample translation occurs, the homogeneously formed nucleus will crystallize ahead of the oriented interface and will yield a gradually broadening 'wake' of poorly oriented material. This prevents the chosen orientation from establishing itself in the entire gradient-crystallized sample. In the special case where the fastest growth direction is also the orientation of choice, then growth velocity competition will cause improperly oriented nuclei to be engulfed by the crystallization front. This is what must have happened to the grain pictured in Fig. 5. Therefore, only in this special case will the volume fraction crystallized [18] be the criterion for assessing a potential for successful gradient crystallization. The requirement that no homogeneous nucleation occurs during processing may limit this technique to systems that are relatively good glass formers, or to processing techniques where very steep gradients can be applied such as laser firing of amorphous surface layers. Many oxide systems prefer surface nucleation [19] which implies that homogeneous nucleation rates are low, so they may be good candidate systems. It may also be possible to up-gradient process some systems where the homogeneous nucleation rate is too large for isothermal crystallization. It is important that the crystal growth rate for the composition be significant because, if growth is too
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slow, isothermal processing could be carried out with careful edge seeding. The primary application will be to systems where the reduction in temperature exposure (by comparison with isothermal processing) will lower the possibility of homogeneous nucleation ahead of crystallization. The nucleation integration presented as Eq. (4) above can be compared with the results found in the present experiment. Experimental nucleation rate data from Smith and Weinberg [40-42] are plotted in Fig. 3 above. These data have been fitted to a parabola for convenience of integration. This line has been superimposed upon the measured datapoints. Then this equation was integrated and evaluated from the first zero crossing temperature to the next to yield the integral required in Eq. (3). The calculation predicts that 180 nuclei per cubic centimeter should be formed by heating in this 25°C/mm temperature gradient and translating the sample at 45 mm/min. Thus, the formation of homogeneous nuclei (an example of which was shown in Fig. 5) is not surprising. Because the total sample volume crystallized was about 0.05 cm 3, about nine nucleation events would be expected. This is very good correspondence between the previously measured nucleation rates and the present experiment. Note that, with so few nuclei, there is little interruption of the growing mass of oriented crystals. A large degree of directionality was achieved, as determined by XRD. At this point we make brief comparison with the earlier up-gradient modeling [18]. Very important differences in built-in assumptions have been made in the two approaches. The most important difference is that each integration element, dV, in their integral is treated independently. Previous work tacitly assumed that the orientation or nucleation extent of any previously crystallized volume element had no influence on the next volume element to enter the gradient zone. That assumption ignores the basic behavior of the system: that the entire growth front is passing through the material and defining the local orientation of subsequent material to be crystallized. This is why we have chosen a pure nucleation criterion; the growth of nuclei and of the interface itself have been assumed to occur at the same rate, so nuclei will interrupt the preferred orientation and defeat the process. The effect of their basic assumption can be noted
in their equation which compares the gradient, G, with the translation velocity, V0, as required for successful up-gradient crystallization. They have analyzed three different cases which depend on the degree of overlap between the nucleation and crystallization curves. Case I had both nucleation and growth becoming gradually more important starting at the same lower bound threshold temperature. Case II corresponds to the condition where nucleation and growth overlap significantly, but nucleation starts at somewhat lower temperatures. Case III covers the case where the peak nucleation rate is found just at the temperature where the growth starts to increase on heating. For these three cases, the following relationships were found: G
--
>>
0.5(C1/4/k5/4),
(7)
Vo 1/4 G / V 3/4 >> 0.7( lmax/k ),
(8)
1/4x / k ) , G / V 3/4 >> O.65(Ima
(9)
where C 1 and k are constants which parameterize the temperature dependencies of nucleation and growth respectively, and /max is the peak nucleation rate. This can be contrasted by rearranging our Eq. (5) above to give GV o > A L I t°tal.
(10)
Our criterion emphasizes that if a slower translation velocity is chosen, then a steeper temperature gradient will be required to keep the residence time in the crystallization zone below a threshold value. By contrast, all of their criteria show that a slower translation velocity will allow a shallower gradient. This appears to be due to the non-sequential nature of their integration. Another difference between the derivations is that, although like us they have also done their integration of the nucleation ahead of the interface, they base their estimation on nucleation and growth curves that are significantly overlapping. This may happen in some selected systems, but generally the temperature dependence of nucleation and growth are not so closely superimposed. Even for systems that have significant overlap, we believe that, by ignoring the sequential crystallization process found with this
S.J. Kim et al. /Journal of Non-Crystalline Solids 181 (1995) 291-300
technique, the gradients predicted will be severely underestimated. Finally, because of the interconnection between thermal conduction and thermal gradient, it is possible that these calculations and processing limits might apply even for laser-melting and regrowth of surface layers. This could then achieve very high effective translation velocity by moving the laser beam rather than the sample. This is relevant to materials with much faster growth rates. This could be used to avoid the faster homogeneous nucleation in these systems.
7. Conclusions Processes relevant to up-gradient directional crystallization of a glass have been described in detail. The model defines a processing window where a highly oriented microstructure may be produced. According to the present model, it is important to impose a steep temperature gradient in combination with a moderate sample translation rate to prevent bulk nucleation. The critical temperature gradient for successful processing can be calculated when the temperature dependence of the homogeneous nucleation rate is known. If homogeneous nucleation is not rapid in a given system, then directional growth may be easy to obtain in a temperature gradient. The model was tested successfully in lithium diborate glass. The authors are grateful to Professors K.A. Jackson, D.R. Poirier and M.C. Weinberg for stimulating discussions and to Dr G.L. Smith for sample preparation assistance.
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