Crack generation and avoidance during the growth of sapphire domes ...

Journal of Crystal Growth 204 (1999) 317}324

Crack generation and avoidance during the growth of sapphire domes from an element of shape F. TheH odore , T. Du!ar *, J.L. Santailler , J. Pesenti , M. Keller , P. Dusserre , F. Louchet, V. Kurlov CEA/CEREM/DEM, 17 rue des Martyrs 38054, Grenoble Cedex 9, France INPG/LTPCM, Domaine Universitaire, BP 75, 38402 St Martin d'He% res Cedex, France ISSP/RAS, Chernogolovka, Moscow District 142432, Russia Received 11 January 1999; accepted 22 March 1999 Communicated by D.T.J. Hurle

Abstract A failure mechanism is proposed in order to explain the generation and propagation of cracks during the growth of sapphire domes from an element of shape. According to this model, 10 lm gas bubbles expansion is induced by glide dislocations (Orowan's model), up to a critical size for which the crack is initiated. Numerical simulation of stresses during the growth explains how cracks propagate "rst vertically (V-type cracks) then horizontally (H-type cracks). A criterion based on plastic strain relaxation is de"ned in order to determine the growth parameters (pulling and rotation rates) as a function of the measured thermal gradients in the crystal. This led to the growth of crack-free sapphire hemispheres up to 50 mm in diameter that can be used for infrared dome blanks. 1999 Elsevier Science B.V. All rights reserved. Keywords: Sapphire domes; GES process; Crack; Thermal stresses; Visco-plastic creep; Numerical simulation

1. Introduction Growth from an element of shape (GES) is, among the variety of shaping techniques, a process where the single crystal is grown layer by layer from a liquid meniscus de"ned by a capillary fed die that is only a small part of the growing cross-section [1]. The crystal is shaped by combining accurate displacement and rotation of the seed on which solidi"cation has been initiated.

* Corresponding author. Fax: #33-4-76-88-51-17. E-mail address: [email protected] (T. Du!ar)

Recent studies [2,3] present investigations of distinctive features of this process and successful growth of cone-like sapphire crystals. However, attempts to grow a true hemisphere always failed due to crack generation and material failure. Fig. 1 shows examples of crystals grown at 1 and 30 rpm with a crystallised layer thickness close to 100 lm. Cracks appeared in those crystals suddenly, one by one, in the neighbourhood of the die, as has been seen by direct observation during growth. They start parallel to the pulling direction (optical axis), 3 or 5 mm above the crystal bottom which is wetted by the meniscus. Those vertical cracks (V-type cracks), about 5 mm in length, are oriented radially

0022-0248/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 9 9 ) 0 0 1 5 7 - 8

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F. The& odore et al. / Journal of Crystal Growth 204 (1999) 317}324

Fig. 1. Grown-in cracked GES sapphire. Left: 1 rpm, right: 30 rpm.

to the growth axis and they are almost always emerging on the crystal basis and on both its internal and external sides. The initial vertical cracks do not propagate forward in the crystal. Instead, they steer into a 903 direction and proceed nearly horizontally (H-type cracks). As shown in Fig. 1, where growth went on, the horizontal cracks slowly propagated until they reached other #aws and caused the fall of large parts of the crystal. It should be noted that remelting when cracks passed again on the die never closed the emerging slit at the bottom of the sample. This resulted in long vertical cracks separating parts of the global shape which therefore cannot be used for any application. The present work investigates the failure mechanism and de"nes compatible pulling conditions for crack-free samples based on critical strain rates.

been performed (0.1 mm/min vertical growth rate, 60 lm crystallised layer thickness and 303 half top cone angle). It is however believed that semi-quantitative data useful to understand the hemispherical case can be obtained from this model. A local maximum for the Von Mises stress is found in the vicinity of the die, 5 mm up to the crystal bottom, on its internal face (Fig. 2). However, for a failure scenario, it is necessary to distinguish tensile components from compressive ones. According to the fragile failure theory, the p , p FF XX and p stress components expressed in cylindrical PP coordinate, only are able to cause the initial crack propagation either by tensile opening (Mode I) or by shear along the #aw surface (Modes II and III). But for symmetry reasons and because of stress continuity, p is obviously responsible for the FF crack opening in Mode I. The numerical results con"rm that no shear crack lengthening is acting. Indeed, p and p are FF XX the only stress tensor components that reach high levels: 80 MPa for p and higher than 100 MPa for FF p . XX Fig. 3a shows that the hoop stress p increases FF up to its maximum after 250 min of pulling (25 mm grown). Then, this tensile stress decreases after having possibly been able to cause a vertical crack. Later, it reaches negative values that completely stops further vertical propagation of the #aw. Immediately after, the axial stress p , which is steadily XX increasing from the very beginning of the process, reaches a similar level which may cause crack steering at 903 (Fig. 3b), as observed experimentally. 2.2. Crack initiation

2. Failure modes during the growth 2.1. Numerical calculation of stresses Continuous modelling of the process has been performed as described in Ref. [4] in order to get the stress evolution in the growing crystal. The model includes calculation of the thermal "eld in the growing crystal, resulting in thermal expansion and elastic and visco-plastic mechanical deformation. Due to technical constraints, only full simulation of the growth of a cone-shaped sample has

The stress level able to activate cleavage, without any initiation, is roughly 15 GPa (&k/10, k being the shear constant of the material) and such failure is indeed excluded because stresses as huge as these are never reached during the shaping process. Therefore, the fragile failure requires a starting defect from which a #aw can propagate. Mechanical testing of the crystals have shown that the micro-bubble distribution inside GES sapphire is prone to fragilise the material [5]. They can then be considered as good candidates for crack initiation. In spite of the fact that it is possible to


319

Fig. 2. Thermo-elasto-viscoplastic stress numerical simulation for a GES sapphire cone. Temperature and Von Mises stress "eld, for various crystal lengths.

remove the bubble layers by reducing the crystallised layer thickness appropriately [3], crystals still fail because of sparse bubbles remaining in their volume. It is well known for sapphire and for some other fragile materials that they show subcritical crack initiation by stretching of the pores they contain. This phenomenon is plastically assisted, and each time a dislocation comes in front of the expanding pore the induced elastic interaction increases the size of the pore till it becomes critical for the fragile failure. Pollock [6] uses a model initially proposed by Orowan to explain a 50% lengthening at rates of about 10\ l/s at room temperature. Comparable

observations are reported by Conrad at high temperature with good correlation between the pore lengthening and the development of slip bands [7]. The initial defect size is found to be close at 10 lm, very comparable to the bubble size in shaped sapphire. Moreover, he observed conchoidal crack surfaces, as smooth as glass, exactly as observed by SEM investigations of crystals that failed during the GES process (Fig. 4). A more recent work [8] also argues for the same subcritical failure mechanism. Accordingly, Fig. 5 shows how a subcritical pore is lengthened by prism dislocations moving under the hoop stress in a crystal grown along the optical axis.


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Fig. 3. Stress components responsible for the crack propagation as a function of time (5 mm above the crystal basis).

A prediction of the defect size a critical for the fragile failure can be estimated from the hoop stress p : FF p "K /(pa. FF '!

(1)

Interpolation, from [9], of the stress intensity factor, K , gives a value of 1 MPa m. For p " '! FF 100 MPa, close to the numerical results, relation (1) implies that a 30 lm defect size is able to cause failure during the growth.


Fig. 4. SEM observation of the crack surface in a GES sapphire sample.

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Fig. 6. Enlarged SEM observation of a restricted area of Fig. 4, showing thermal fatigue on the H-type crack surface.

lengthened with the help of plastic relaxation, up to a size of 30 lm which is large enough to generate a longitudinal crack, under a hoop stress of about 100 MPa. After steering at 903, due to the decrease of p and increase of p , the #aw proceeds cycliFF XX cally, probably in relation with the periodic rotation of the sample above the hot die.

3. Critical strain rate criterion

Fig. 5. Subcritical crack initiation (Orowan's model).

2.3. Crack propagation The stress relaxation associated to the fragile cracking stops the #aw propagation. Later, it steers nearly parallel to the basal planes and slowly propagates in mixed brittle and ductile modes. Fig. 6, a magni"cation of Fig. 4, shows how a part of the #aw proceeds periodically, just like what is observed in thermal fatigue. Each #aw lengthening step relaxes the excess of stresses, till they become large enough to lengthen the crack again. In summary, the failure mechanism during the growth of domes from an element of shape is understood in the following way: 10 lm bubbles are

In order to avoid crack initiation, it is necessary to keep the plastic strain rate that relaxes the stresses and lengthens at the same time the initial defect under the critical value. One solution consists in relaxing the hoop opening stress with the same e$ciency than the one with which the defect lengthening is going on. As p is related to the hoop strain via the mateFF rial rigidity, this leads to keeping the elastic strain rate comparable to the plastic one: e e , (2) or, taking into account the total and thermal strains: e !e !e e . (3) In the vicinity of the die, the previous condition gives a necessary restriction of the thermal expansion rate to twice the maximum acceptable plastic


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strain rate: "e !e ")2e (4) the latter depending on the size and density of the micro-bubbles trapped inside the crystal and more or less rapidly becoming critical for the fragile failure. The gas inclusion size commonly observed in the GES samples (10 lm) results in a maximum plastic strain rate of 5;10\ l/s [10]. Solving Eq. (4) is a challenging problem for which no trivial solution is known. The following relation is su$cient and will be considered instead: "e "#"e ")2e . (5) Each term is estimated from in situ temperature measurements [11]: Fig. 7 is a temperature recording of a thermocouple (TC) "xed to the seed and translated from one side to the other with a growing GES ribbon. At the beginning of the process, it records an overheating gradient when arriving on and leaving the die (s is the curvilinear abscissa on the crystal base):

R¹

¹" "15;10 K/m. Q Rs

After a while, the TC gets cooler since leaving upward with the growing crystal. It gets more and more far from the crystallisation front and only records the radial temperature gradient relative to the die position:

R¹

¹" "2;10 K/m. P Rr

(7)

The hoop strain rate would be maximum in the case where the crystal bottom would be at uniform temperature ¹ , actually that of its coldest point: "e ")a "¹Q !¹Q ""a "r " ¹ (8)

P where ¹Q equals zero, r and a being the crys tal base radius and its expansion coe$cient, respectively. Given the seed rotation rate g, the time derivative of the temperature is R¹ R¹ "2pgr Rt Rs

(9)

which leads to (6)

"e ""a "¹Q ""2pga r ¹. Q

Fig. 7. In situ GES temperature measurement during the growth of a ribbon. A thermocouple was attached to the seed.

(10)


Fig. 8. Principle of the growth of a dome from an element of shape (a) and crack-free sapphire hemispheres (b).

As the time derivative of the crystal base radius is the horizontal translation rate of the seed, ; , a no V failure criterion is obtained: 2e !"u " ¹ V P a . g(r)) 2pr ¹ Q

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that is kept constant all along the process in order to keep a constant liquid #ow rate in the die. In practice, crack-free crystals were obtained for linear pulling rates of about twice the latter critical value, a result consistent with the fact that p achieves its FF maximum when the crystal horizontal radius is close to half the ultimate base radius (Fig. 2). The other parameters (; , < ) are derived from V X the Pythagorean relation and from crystallised layers of constant thickness. In practice, (; , < , g) V X are calculated at each time step, connected with latitude increments that describe the hemispherical dome from its top to its base. Those increments (0.53) are "ne enough to get a roughness lower than the surface defect between successive growth layers. Successful crack-free domes were obtained in this way for any diameters up to a maximum of 50 mm, overall dimension achievable in the furnace, with thickness close to 4 mm (Fig. 8b). The criterion established to get these results led to a reduction of the linear growth rate and this lengthened the experiment time. In order to shorten the growth and get a more competitive process, one can think about increasing the growth layers, but the in#uence of the crystallisation height on the gas bubble distribution, so important in the subcritical crack initiation, does not allow a linear gain during growth. Doubled crystallised layers, from 100 to 200 lm, led to only about 2/3 time saving for crack-free crystals. A remaining challenge, actually an old one, is to de"ne growth conditions for gas-bubbles-free crystals that would be more rapidly shaped and would show both high optical quality and high mechanical strength.

(11)

The vertical translation < is combined with the X other displacements to grow sapphire domes in the way presented in Fig. 8a.

4. Results and conclusions According to Eq. (11), g reaches its lowest value, g , at the end of the growth, where the crystal radius S is maximum. This de"nes the linear growth rate

References [1] P.I. Antonov, Y.G. Nosov, S.P. Nikanorov, Bull. Russian Acad. Sci. 49/12 (1985) 2295. [2] V.A. Borodin, V.V. Sidorov, S.N. Rossolenko, T.A. Steriopolo, V.A. Tatarchenko, J. Crystal Growth 104 (1990) 69. [3] V.N. Kurlov, F. TheH odore, Crystal Res. Tech. 34 (1999) 293. [4] F. TheH odore, T. Du!ar, F. Louchet, J. Crystal Growth 198/199 (1999) 232. [5] P.A. Gurjiyants, M.Y. Starostin, V.N. Kurlov, F. TheH odore, J. Delepine, J. Crystal Growth 198/199 (1999) 227.

324 [6] [7] [8] [9]

F. The& odore et al. / Journal of Crystal Growth 204 (1999) 317}324 J.T.A. Pollock, G.F. Hurley, J. Mater. Sci. 8 (1973) 1595. H. Conrad, J. Am. Ceram. Soc. 48/4 (1964) 195. R.W. Rice, J. Mater. Sci. Lett. 16 (1997) 202. T. Bretheau, J. Castaing, J. Rabier, P. Veyssie`re, Adv. Phys. 28 (1979) 835.

[10] G.F. Hurley, J.T.A. Pollock, Metall. Trans. 3 (1972) 397. [11] V. Krymov, V.N. Kurlov, P.I. Antonov, F. TheH odore, J. Delepine, J. Crystal Growth 198/199 (1999) 210.