THERMAL DIFFUSIVITIES Of BBO AND KTP HIGH-TEMPERATURE FLUXES MEASURED BY THE ÅNGSTRÖM'S METHOD
E. Mojaeva,#, G. Orra, E. Dul’kina, M. Tseitlinb and M. Rotha
aDepartment of Applied Physics, The Hebrew University, Jerusalem 91904, Israel
bDepartment of Applied Physics, Ariel University Center of Samaria, Ariel 44837, Israel
ABSTRACT
A modification of the Ångström’s temperature wave method for measuring the thermal properties of materials has been employed for in situ experimental evaluation of thermal diffusivities of high-temperature solutions. High-temperature solutions used for the growth nonlinear optical BBO (BaB2O4) and KTiOPO4 (KTP) crystals have been studied, namely BBO-Na2O, BBO-NaF and KTP-K6P4O13 (KTP-K6). The results show that the thermal diffusivity of the BBO-Na2O solution increases by two orders of magnitude, from = 410-3 to 410-1 cm2/s, along the liquidus line when the Na2O content is changed from 0 (pure BBO melt) to 28%. In the case of the BBO-NaF solution, the thermal diffusivity increases to about 3.510-2 cm2/s when the NaF content reaches 20% and remains practically constant upon further increase of the solvent concentration, namely within the entire range of interest for the BBO crystal growth. The thermal diffusivity of the KTP- K6 solutions is also relatively low, ~ 410-2 cm2/s, and does not change when the K6 concentration is varied from 50 to 70% (which occurs during the crystal growth run from a self-flux). The values of the thermal diffusivities obtained are discussed in terms of the heat transport in the melt and the thermal stability of the crystal/melt interface against temperature fluctuations.
Introduction
Single crystals of barium metaborate -BaB2O4 (BBO) and potassium titanyl phosphate KTiOPO4 (KTP) are among the most widely used materials in nonlinear optical applications, such as second harmonic generation (SHG) and optical parametric oscillation (OPO) of high pulse power lasers (1,2), and electro-optic devices (3). In spite of the recent advances in improving their properties, larger, inclusion-free and optically more uniform crystals are required.
Both BBO and KTP crystals are grown from high-temperature fluxes by the widely used top-seeded solution growth (TSSG) technique (4,5). Successful crystal growth from high-temperature solutions or melts is subject to solving a wide set of physical problems. Full comprehension of the processes occurring in melts during growth requires an integrated knowledge of a wide range of thermodynamic, kinetic, and transport phenomena, and in particular of the momentum, heat and mass transfer in the liquid phase. Thus, in liquids with free surfaces, natural convection can be caused by temperature gradients resulting in buoyancy and nonuniform surface tension (6). The unbalanced buoyancy and surface tension forces result in excessive potential energy. A ratio of this energy to the dissipation factors (viscous drag and heat conductance) is given by the Rayleigh number (in the case of buoyancy) and by the Marangoni number (in the case of nonuniform surface tension) as: (1)
(2)
where is the volume expansion coefficient, is acceleration due to gravity , is the thermal diffusivity, is the kinematic viscosity, is a characteristic length (depth of the flux), is the surface tension, D is the crucible diameter, is the flux density, a and r are the axial and radial temperature differences respectively.
Eqs.(1) and (2) show that the melt thermal diffusivity ( ) is one of the important parameters influencing the fluid flow, since it determines the rate of heat transfer during temporal temperature variation, or simply the rate of temperatures change. The thermal diffusivity is related to the thermal conductivity , specific heat (at constant pressure) Cp and density ρ as follows:
. (3)
If the values of , Cp and are known, it is possible to calculate of any substance from eq. (3), but one should bear in mind that all these parameters are temperature dependent (7).
There are also numerous methods of determining thermal diffusivity from the experiment independently (8-15). One of them is the general dynamic method of measuring thermal parameters, used by Ångström in 1861 (and published in English in 1863 (16)). In his experiments on the thermal conductivity Ångström used a long metallic bar of a small cross-section, one end of which was subjected to periodic changes in temperature (16). The method is based on direct experimental simulation of the finite difference approximation to the one-dimensional heat conduction equation. By analyzing at two points the amplitudes and phases of a sinusoidal heat wave propagated in a semi-infinite metal rod one can determine the thermal diffusivity directly and independently of the external conditions. It can be shown (17) that the thermal diffusivity is then given by
(4)
where is a distance between the points x1 and x2 , is the period of the temperature changes, 1, and B1, C1 are the phases and amplitudes of the heat wave respectively. Various modifications of the Ångström's method have been applied to different solid and liquid materials (18-23).
In the present work, we have applied the Ångström's temperature wave method to determine in situ the thermal diffusivities of several high-temperature oxide fluxes, namely BBO-Na2O, BBO-NaF and KTP-K6P4O13, typically used for the BBO and KTP single crystal growth. The values relevant for the real crystal growth temperature intervals have been determined in order to make the results useful for a phenomenological hydrodynamic flow analysis and further mathematical modeling.
Experimental
The experiment is based on applying a sinusoidally varying thermal flux incident on one face of a one-dimensional specimen and convectively cooling its opposite face. This results in a sinusoidally varying temperature on the cooled face with a measurable phase lag between the incident and transmitted waves that depends on the material properties and the heat transfer coefficient. The block diagram of the experimental setup is presented on Fig.1.
It is based on a resistively heated furnace equipped with a programmable Eurotherm temperature controller with the temperature stable to 0.1°C. The temperature distribution field in the furnace was controlled by Pt-Pt/Rh thermocouples. The high-temperature fluxes under study were synthesized in a Pt crucible of 10 mm diameter and 100 mm height which was ultimately filled to just above 3/4 of its height. The crucible was placed on a pedestal in the central part of the furnace as shown in Fig.1. In order to prevent vertical heat convection, the temperature of the flux surface was kept at about 2-3oC higher temperature relatively to crucible bottom. The radial temperature gradient in the flux did not exceed 0.5oC.
Fig.1. Experimental setup for measuring thermal diffusivities of the molten
solutions.
Periodic cooling of the lower crucible end was achieved by using an air compressor with rotating shutter (stroboscope). An Alumina tube of 6 mm diameter (less than the crucible diameter) was used to transport the pulses of cold air to the crucible bottom through the axial hole drilled in the pedestal. There was no direct contact between the crucible vertical surface and the air blown in. The compressed air was at room temperature, and when the hot crucible bottom was exposed to such air pulse a sinusoidal heat wave propagated through the molten flux along the crucible axis direction. Thermocouples placed along the heat wave path allowed monitoring the time dependence of temperature changes. The shutter was driven by a custom designed controller which permitted to vary the duration of air blow pulses (heat waves) and change their amplitudes at a constant air flow. The amplitude of the heat waves could also be varied by the air blow pressure control. The combined control over the shutter rotation frequency and the air pressure has allowed obtaining heat waves in a sinusoidal form within a 0.4 - 15oC temperature range with time periods from 0.5 to 30 s. An example of typical traces of the temperature waveforms (in voltage units) obtained from the lower thermocouple Tc1 and the upper thermocouple Tc2 immersed in a BBO/Na2O flux is shown in Fig.2, for a measurement with a time period T = 6.3 s. These traces can be very emulated by the sine functions.
Fig.2. Typical temperature waveforms recorded by the thermocouples Tc1 and
Tc2; parameters indicated in the figure are described in the text and by
eq. (4).
The distance between the two Pt-Pt/Rh thermocouples (Tc1 and Tc2) measuring the amplitude and phase difference of the thermal waves, l = x1 – x2, could be varied from 2 to 50 mm, and the accuracy in determining their vertical locations was 0.5 mm. In practice, distances (l) smaller than 10 mm were used for two main reasons. It has been shown for the BBO melt that the amplitude of transient temperature fluctuation decreases exponentially while propagating through the melt (24). We have verified this observation by measuring the amplitude of the thermal waveform as a function of the distance from the crucible bottom (for the constant compressed air pressure and shutter rotation frequency, and the results are shown in Fig.3 for a 80%BBO/20%Na2O flux. Obviously, l = 5-6 mm would be suitable for obtaining well measurable temperature waveforms by both Tc1 and Tc2 thermocouples. Attempts to increase the Tc1 and Tc2 temperature differences by overcooling the crucible bottom showed that when the Tc1 temperature waveform amplitude reached the value of 25°C, the melt became quasi-frozen due to local supercooling, and it was impossible to reveal any differences in the phase shifts in temperature oscillations.
Fig.3. Amplitude attenuation upon thermal wave propagation from the
crucible bottom into the melt (at an oscillation period T = 3.20.1 s).
The problem of extracting weak periodic temperature oscillation signals in high-temperature melts was solved by using an appropriate electronic compensation scheme for monitoring the thermocouple voltage signals. The scheme included a high stability compensation block with a 6 V dry battery used as a source of constant voltage. The compensated signal was amplified and sent to the terminal of the data acquisition unit Model DT322. All experimental data were processed using computer software.
Results and Discussion
Thermal diffusivities of high-temperature solutions with compositions relevant for growing single crystals of BBO and KTP have been determined experimentally. BBO crystals are usually grown from fluxes because they crystallize in the crystallographically noncentrosymmetric -phase, suitable for nonlinear optical applications, far below its 1095°C melting point, namely below 925°C. According to the phase diagrams (25,26), this temperature corresponds to the 0.79BBO/0.21Na2O and 0.69BBO/0.31NaF compositions for the conventional Na2O and more advanced (less viscous) NaF fluxes respectively. In fact, a series of three compositions along the liquidus have been studied for both cases. The compositions were chosen as following: xBBO-(1x)Na2O (x = 0.72 , 0.79, 0.88 ) and xBBO-(1x)NaF (x = 0.58, 0.69, 0.78).
Sinusoidal temperature waveforms corresponding to the periodic cooling, of the type shown in Fig.2, have been used to calculate the thermal diffusivities for each flux composition using eq. (4). The results for the two BBO fluxes using Na2O and NaF as solvents, namely the thermal diffusivities as a function of the solvent concentration, are shown in Fig.4. The thermal diffusivity value of the pure BBO melt is taken from the literature (24), while the measurement have been performed for specific fluxes with composition/temperature combinations corresponding to the respective liquidus curves. The results show that starting from a relatively low value of = 0.004 cm2/s for the BBO melt, the thermal diffusivity increases almost logarithmically along the liquidus line in the case of the Na2O solvent. At 800°C, or a temperature at which the growth BBO crystals from the Na2O flux is typically terminated, the thermal diffusivity is two orders of magnitude higher than that of pure BBO melt. This fact, in addition to the parallel (although not as dramatic) increase in the kinematic viscosity (26), indicates that the Rayleigh becomes smaller at the end of the growth run and the buoyancy driven convection reduces. The flow stability may be thus disposed, also due to the reduction in the axial temperature gradient, and the solid/liquid interface breakdown can be observed (27).
Fig. 4. Thermal diffusivity as a function of the solvent concentration in the
BBO-Na2O (full squares) and BBO-NaF (open squares) solutions.
The liquidus temperatures corresponding to the specific compositions
are indicated as well.
It is noteworthy that the lack of knowledge about the thermal diffusivity variation as a function of composition in the xBBO-(1x)Na2O melts (as well as their viscosity) has lead some workers to erroneous conclusions about the stability of these melts against small transient temperature fluctuations. Thus Wang et al. (24) implied that at a typical growth temperature of 880°C the melt thermal diffusivity was of the order of 410-3 cm2/s and the dynamic viscosity of about 2 P yielding a monstrous Prandtl number (Pr = /) of over 120. According to our new data on the thermal diffusivity (Fig. 4) and also measured viscosities of the BBO-Na2O melt as a function of the solvent concentration (26), the Pr can be calculated for the entire temperature interval employed for growth of BBO crystals. The results of the calculations are given in Fig. 5, and they show that the Pr changes linearly and not dramatically within this interval and is as small as ~ 2.15 at 880°C. The results of Fig. 5 may be important for future computer modeling of the BBO crystal growth.
Fig. 5. Variation of the Pr along the liquidus of the BBO-Na2O binary
phasediagram.
Returning now to the problem of temperature fluctuations, we will refer to starting composition for the BBO crystal growth, namely 80%BBO-20%Na2O, with the = 0.13 and = 0.33 (26). In order for the temperature fluctuations not to affect the crystal growth, the critical width of the temperature wave, l = (8)1/2 (where is the duration of temperature instability), should not exceed the width of the momentum boundary layer, m = 1.613(/)1/2 (where = 2n and n is the crystal rotation rate in the r/s units) (24). With the typical crystal rotation rate of 2 rpm (27), m ≈ 2 cm. Therefore, from the condition
(4)
we calculate that if the duration of transient temperature fluctuation exceeds 1.22 s, the temperature wave can penetrate to the growth interface and affect the quality of the growing BBO crystal.
Fortunately, the thermal diffusivity of the BBO-NaF flux is substantially smaller (the average value is ~ 3.510-2 cm2/s), and it is almost constant along the entire liquidus part that is of interest for the BBO crystal growth. Apparently, the duration of the temperature fluctuation must be four times longer, or about 5 s, in this case in order to cause an instability at the BBO crystal/melt interface. This explains the better quality and the higher yield of BBO crystals that can be grown from the BBO-NaF flux. Table 1 summarizes the exact measured values of thermal diffusivities of the BBO melt and various fluxes together with three KTP-K6Ti4O13 (KTP-K6) self-fluxes and other melts of nonlinear optical crystals, such as LiNbO3 and KNbO3. The latter exhibit clearly smaller values of thermal diffusivities and, therefore, larger Pr, which is the origin of a number of hydrodynamic flow instabilities typical for oxide crystal melts.
Table 1. Summary of measured thermal diffusivity () values of the BBO and KTP
solutions (with indication of the specific solution temperatures) together
with the k values of some pure oxide melts (from literature).
Melt Composition / Melt Temperature, °C / Thermal Diffusivity, cm2/s / SourceBBO / 1,095 / 0.00042 / (24)
88%BBO-12%Na2O / 1,020 / 0.037 0.007 / This work
80%BBO-20%Na2O / 925 / 0.130 0.02 / This work
72%BBO-28%Na2O / 800 / 0.410 0.08 / This work
78%BBO-22%NaF / 982 / 0.036 0.012 / This work
69%BBO-31%NaF / 925 / 0.043 0.005 / This work
58%BBO-42%NaF / 856 / 0.029 0.005 / This work
50%KTP-50%K6 / 920.7 / 0.040 0.012 / This work
60%KTP-40%K6 / 968.4 / 0.036 0.006 / This work
70%KTP-30%K6 / 1,008 / 0.040 0.003 / This work
LiNbO3 / 1,257 / 0.0084 / (24)
KNbO3 / 1,050 / 0.014 / (24)
KTP crystals are mostly grown from self-fluxes, such as the conventional K6 flux, in an almost gradient-free environment with the fluid flow driven entirely by forced convection due to crystal rotation. The fluid dynamics is not governed by thermal diffusion, and the thermal diffusivity is not for the flow modeling. However, the values are still helpful in understanding the influence of possible temperature fluctuations on the crystal/melt interface stability, like discussed above for the case of BBO. The kinematic viscosity of a typical KTP-K6 flux is quite similar to that of BBO, ≈ 0.33 cm2/s (based on the data of (28): dynamic viscosity η = 78 cP and = 2.35 g/cm3). Since the crystal (seed) rotation rate is usually much higher than in the case of BBO, namely: n = 30 rpm (29), the width of the momentum boundary layer is as low as 0.5 cm for the KTP-K6 flux. From Table 1, the flux thermal diffusivity is practically constant, at least for the compositions studied, and we can adopt hereby a = 0.04 cm2/s value. In this case, the calculation based on eq. (4) shows that the growth interface instability may be caused by a very short temperature fluctuation of about 0.15 s. This is important, since periodic inversion of the crystal rotation is routinely used in the KTP crystal growth (29,30), and it may be the main source of compositional growth striations observed in KTP crystals (31). It is still important to note that even if the defects exist, they are evenly distributed along the KTP crystal growing out from a compositionally changing solution (5), since the thermal diffusivity (see Table 1) is practically independent on the solvent concentration.
Conclusions
Ångström’s temperature wave method has been applied to the in situ mesurements of thermal diffusivities of high-temperature solutions routinely used for growing the nonlinear optical BBO and KTP crystals. The difference from the original Ångström’s method was that a vertical molten liquid column was the subject of studies instead of a semi-infinite metal rod. A detailed analysis of the propagation of a sinusoidal heat wave through high-temperature solutions was carried out. The resulting temperature waveforms obtained at two points, namely the differences in their amplitudes and phases, were studied. It was concluded that the distance between the two measurement points should not exceed 5-6 mm due to the fast attenuation of the heat wave in the molten liquid. Surprisingly, the finite difference approximation to solving the one-dimensional heat conduction equation, as suggested by Ångström, could still be applied, and the thermal diffusivities of the various high-temperature solutions could be evaluated successfully. The results show that the thermal diffusivity of the BBO-Na2O solution increases by two orders of magnitude, from = 410-3 cm2/s for the pure BBO melt to 410-1 cm2/s for the 72%BBO-28%Na2O solution. A similar, but slower initial increase of the thermal diffusivity is observed upon adding the solvent in the case of the BBO-NaF solution, but once the solvent reaches the concentration of 20%, the thermal diffusivity stabilizes at a constant level of about 3.510-2 cm2/s. The thermal diffusivity of the KTP-K6 solution is practically constant in the range of the K6 solvent concentrations varying fro 50 to 79%, and its average value, ~ 410-2 cm2/s, is almost as low as that of BBO-NaF. In the case of BBO-NaF, the lower thermal diffusivity (in comparison with the BBO-Na2O solution) is the main reason for the better stability of the crystal/melt interface against temperature fluctuations. In the case of KTP-K6, due to the high rotation rate needed for the compositional homogenization of the solution, the width of the momentum boundary layer is small, and periodic temperature fluctuations lead to the formation of growth striations even though the thermal diffusivity is relatively low.