CRYSTAL GROWTH & DESIGN 2008 VOL. 8, NO. 2 402–406
Articles Analysis of Transport Phenomena during Bridgman Growth of Calcium Fluoride Doped Crystals Carmen Stelian,* Daniela Susan-Resiga, Liliana Lighezan, and Irina Nicoara Department of Physics, West UniVersity of Timisoara, Bd. V. ParVan, No. 4, 300223 Timisoara, Romania ReceiVed February 3, 2007; ReVised Manuscript ReceiVed September 25, 2007
ABSTRACT: The chemical homogeneity of CaF2:Pb crystals, grown by using the vertical Bridgman method, is numerically investigated. The FIDAP commercial code is used to study the heat transfer, convection, and solute transport phenomena during the solidification process. The homogeneity of these doped crystals depends on the solid–liquid interface curvature and the flow intensity. The accurate computation of the thermal field and interface shape for these semitransparent materials requires the modeling of the internal radiative heat transfer. The numerical modeling of the flow field and solute distribution shows a low intensity of the convection and a quasi-diffusive transport regime in the melt. The numerically computed longitudinal solutal profile in the crystal, in agreement with the experimental measurements, shows some significant oscillations of the Pb concentration. For the first time, a correlation between these oscillations and a quasi-diffusive transport regime has been established to explain this phenomenon, which also has been observed in other Bridgman crystal growth experiments. Finally, an optimization of the growth rate to avoid these oscillations and to improve the chemical homogeneity of crystals has been proposed.
1. Introduction The alkaline-earth fluorides CaF2, SrF2, and BaF2 doped with Pb, Tl, Yb, or Er are of large interest because of their attractive laser properties in the infrared range.1,2 The quality of these crystals grown by the vertical Bridgman technique depends on their chemical homogeneity. The aim of this paper is the numerical analysis of the heat transfer, melt flow, and solute distribution during the Bridgman solidification of CaF2:Pb to improve the chemical homogeneity and the quality of these crystals. The numerical simulation is performed by using the finite element FIDAP commercial software. This code has been already used for the transient calculation of the solute distribution in Bridgman growth of the semiconductor alloys3,4 and the modeling of the internal radiative heat transfer during the solidification of semitransparent BaF2 crystals.5 CaF2 belongs to semitransparent materials, which participate in the radiative heat transfer by absorbing and emitting the radiation. The effective thermal conductivity of such materials increases because of the internal radiative heat transport and influences the solid–liquid interface curvature. Calcium fluoride participates in the radiation in both the liquid (L) and solid (S) phases and is characterized by the ratio aL/aS ≈ 10 of the absorption coefficients.6 The thermal conductivities of the * Corresponding author. Tel.: +40256592175; fax: +40256496088; e-mail:
[email protected].
opaque (a f ∞) solid and liquid samples are small: kS ) 2 W m-1 K-1 and kL ) 0.4 W m-1 K-1.7 At this low value of thermal conductivity in the melt, the latent heat effect on the interface curvature becomes important as is shown in ref 5. Therefore, to have an accurate computation of the solid–liquid interface shape, the internal radiative heat transfer and the latent heat released at the solidification front must be taken into account in the simulation. The interface curvature influences the intensity of the melt convection, which interacts with the diffusive transport of the Pb and determines the solute distribution in the sample. Therefore, a complex transient numerical modeling of these coupled transport phenomena is necessary to explain the experimentally measured solute distribution in the sample and to improve the quality of these crystals.
2. Experimental Procedures The modeling is performed for the experimental growth of CaF2 crystals doped with PbF2 by using the vertical Bridgman technique. The sample with the radius R ) 0.5 cm and length L ) 7 cm is contained in a graphite crucible of 0.15 cm thickness. The crucible is pulled at a rateV ) 1.1 µm/s in a furnace characterized by an axial temperature gradient GT ) 20 K/cm. The physical parameters of the CaF2, taken from refs 6 and 7, are given in Table 1. To estimate the values of the segregation coefficient K and the solute diffusivity D, which are unknown for CaF2:Pb, the numerical results are compared to the experimental measurements of the solute concentration in the crystal.8 The Pb distribution in a calcium fluoride crystal
10.1021/cg070125g CCC: $40.75 2008 American Chemical Society Published on Web 01/05/2008
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Table 1. Physical Properties of CaF2 Sample and Graphite Cruciblea density specific heat thermal conductivity heat of fusion thermal expansion coefficient viscosity of melt melting point
Fs,L ) 3200 kg · m-3 FC ) 1700 kg · m-3 CPS,L ) 800 J · kg-1 · K-1 CPC ) 2050 J · kg-1 · K-1 kS ) 2 W · m-1 · K-1 kL ) 0.4 W · m-1 · K-1 kC ) 50 W · m-1 · K-1 ∆H ) 320 000 J · Kg-1 βT ) 2 × 10-5 K-1 µ ) 0.012 kg · m-1 · s-1 TM ) 1653 K
a The index S, L, and C denotes the solid, liquid, and the crucible, respectively.
Figure 2. Comparison of experimental data (ref 8) and numerical results for the longitudinal solute distribution in a CaF2:Pb crystal (C0 ) 1%).
Figure 1. Absorption spectra of CaF2:Pb crystals at 1 and 2% initial solute concentration. (1% initial solute concentration) has been experimentally measured from the UV–vis absorption spectra of 2 mm thin samples obtained after a transversal cutting of the crystal. The absorption coefficient for the band located at λ ) 307 nm depends nearly linearly on the Pb concentration at lower values of the initial solute concentration in the sample (see Figure 1). The longitudinal solute concentration in the crystal has been calculated by measuring the absorption coefficient for each 2 mm piece and by using a calibration curve carried out from the spectra illustrated in the Figure 1. In Figure 2, the numerical computed longitudinal solutal profile in the crystal is compared to the experimental data taken from ref 8. A better agreement is obtained for a segregation coefficient K ) 0.8 and a diffusion coefficient D ) 0.35 × 10-8 m2/s.
3. Results and Discussion The numerical modeling is performed with the commercial code FIDAP. The governing equations solved for the axisymmetric sample-crucible system are Heat transfer equation FcP
( ∂T∂t + u ∇ T) ) k∇ T - ∇ q f
2
f R
(1)
where qR is the radiative heat flux, which is computed for a participating medium by using the P1j approximation. Momentum equation
[
F
f
]
∂u f f f f + (u ∇ )u ) - ∇ p + µ∇2 u + Fg [1 - βT(T ∂t T0)] (2) Solute transport equation: ∂C f + u ∇ C ) D∇2C ∂t
(3)
Figure 3. Numerical results at the solidified fraction fS ) 0.3: (a) thermal field; (b) flow field; (c) solutal field.
In the above equations T, u, C are the temperature, flow velocity, and the solute concentration, respectively. The boundary condition for temperature imposed on the external wall of the crucible is given by
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Stelian et al.
Figure 4. Numerical computed longitudinal solute distribution on the axis and the sample side. Diffusion coefficient D ) 0.4 × 10-8 m2/s, segregation coefficient K ) 0.8, growth rate V ) 1.1 µm/s. f
-kC∇T |C · n ) heff(T - Tref)
(4)
where heff is the effective heat transfer coefficient, which includes the radiative heat exchange between the crucible and the furnace, and the reference thermal profile Tref is imposed according previous experimental measurements and numerical simulations.5 At the solid–liquid interface the heat flux and solute balance conditions are given by f
f
f
(kS∇T |S - kL∇T |L)n ) FS · ∆H · (V · n ) f
f
(5)
f
D∇C |L n ) (1 - K)(V · n )CI
(6)
where b n is the normal vector, and CI is the solute concentration in the liquid at the interface. More details about the FIDAP modeling of the radiative heat transfer and solute transport in vertical Bridgman crystal growth are given in refs 3–5. The numerical computed thermal, flow, and solute fields at the solidified fraction fS ) 0.3 are presented in the Figure 3. The interface has a convex shape with a deflection f ) 1.2 mm (Figure 3a). The melt convection is characterized by two flow cells (Figure 3b). The upper one appears due the radial thermal gradients in this region and has a small intensity. The flow located near the solid–liquid interface has a clockwise direction and influences the solute distribution in the melt adjacent to the interface and the composition of the solidified sample. The maximum convective velocity is umax ) 1.4 × 10-5 m/s. The convective intensity is low because of the high viscosity of the calcium fluoride. Even that, the solute distribution in the melt is strongly affected by the flow because of the huge value of the Schmidt number Sc ) µ/FD ≈ 937 (Figure 3c). The solute is transported toward the crucible wall, and the concentration increases in this region. The longitudinal solute distribution on the axis and the crystal side is presented in Figure 4 (initial concentration of the plumb in the liquid sample was C0 ) 1%wt. The numerical profile, in agreement with the experimental measurements, shows some oscillations of the Pb concentration. The amplitude and the period of these oscillations increase during the solidification process. The oscillations amplitude is greater on the sample axis and increases until ∆Cosc ) 0.15% at the solidified length z = 27 mm. These kinds of oscillations have been also observed in
Figure 5. Axial solute distribution in the crystal computed at various values of the diffusion coefficient: D ) 10-8 m2/s, D ) 0.5 × 10-8 m2/s, and D ) 0.2 × 10-8 m2/s.
other Bridgman solidification experiments,9,10 but a physical explanation has not been provided yet. We assume that these oscillations of the solute concentration are produced by the interaction between the melt convection and the solutal boundary layer near the solid–liquid interface, when a quasi-diffusive regime exists in the melt. More details about the quantitative and qualitative explanation of this phenomenon are given in ref 11. The influence of the convecto-diffusive transport phenomena on the solute distribution in the melt is described by the convecto-diffusive parameter ∆: δV (7) D where δ is the width of the solutal boundary layer which appears near the solid–liquid interface. When ∆ f 0, the solute is well mixed near the interface by the melt flow and a convective regime occurs in the melt. For ∆ f 1, the thickness of the boundary layer is δ ≈ D/V, and a diffusive controlled regime exists in the melt. A quasi-diffusive regime is characterized by a parameter ∆ f 1 and a non-negligible effect of the convection on the solute distribution. The convecto-diffusive parameter can be estimated from the following formula proposed in ref 12: ∆ )
∆ ) 13.2 · Pe · (Gr · Sc)-1⁄3 where the Peclet and Grashof numbers are given by Pe )
V · R D
Gr )
(8)
F2 · βT · ∆T · g · R3
(9) µ2 The radial variation of the temperature at the solid–liquid interface depends on the interface deflection and is given by ∆T ) f · GT
(10)
The value of the convecto-diffusive parameter computed for our simulated problem is ∆ ≈ 0.99, which correlated to the huge value of the Schmidt number (Sc ) 937), indicates a quasidiffusive regime in the melt. In this case, the flow can have a periodical influence on the solute distribution near the interface leading to a significant variation of the solute concentration during the solidification process. The amplitude of these oscillations increases when the ∆ parameter increases (Figure 5). The axial solute profiles in the Figure 5 are computed at various values of the diffusion
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Crystal Growth & Design, Vol. 8, No. 2, 2008 405
Figure 7. Radial segregation versus the solidified length at two growth rates: V ) 1.1 µm/s and V ) 0.6 µm/s (D ) 0.4 × 10-8 m2/s). Figure 6. Axial concentration profile computed at two growth rates: V ) 1.1 µm/s and V ) 0.6 µm/s (D ) 0.4 × 10-8 m2/s).
coefficient: D ) 10-8 m2/s (∆ ) 0.57), D ) 0.5 × 10-8 m2/s (∆ ) 0.91), and D ) 0.2 × 10-8 m2/s (∆ ) 1.68). It should be noted that a value of ∆ ) 1.68 indicates a solutal boundary layer with thickness δ > D/V. The solutal profiles represented in the Figure 5 show that the oscillations amplitude depends on the diffusion coefficient and increases at small values of D. The parameter ∆ ) 0.57 corresponding to a diffusion coefficient D ) 10-8 m2/s indicates a convecto-diffusive regime characterized by very small oscillations of the axial concentration. The same value of ∆ can be obtained at small values of the diffusion coefficient by reducing the growth rate, as can be observed from the eqs 7 and 8. As for example a value ∆ ) 0.58 is carried out from eq 7 for D ) 0.4 × 10-8 m2/s and V ) 0.6 µm/s. Therefore, we can avoid the oscillations of the concentration by reducing the growth rate to have a convectodiffusive regime with ∆ < 0.6. This prediction is confirmed by the numerical simulation. In Figure 6, the axial concentration is plotted at two growth rates: V ) 1.1 µm/s and V ) 0.6 µm/s (diffusion coefficient D ) 0.4 × 10-8 m2/s). It can be observed that the oscillations amplitude is significantly reduced at V ) 0.6 µm/s. So, the chemical homogeneity of these crystals can be improved by reducing the growth rate. The radial variation of the solute concentration is characterized by the quantity: δC )
Cmax - Cmin Cav
(11)
where Cmax, Cmin, and Cav are, respectively, the maximum, the minimum, and the average concentration along the solid–liquid interface. The evolution of the radial segregation δC during the solidification process at V ) 1.1 µm/s and V ) 0.6 µm/s is plotted in Figure 7. The numerical results show low radial variations of the concentration. The oscillations of δC are smaller at the growth rate V ) 0.6 µm/s than at V ) 1.1 µm/s. The maximum value of the radial segregation (δCmax ) 0.14) obtained at V ) 1.1 µm/s is about two times greater than the value predicted by the Coriell’s formula13 for a diffusive controlled regime: ∆C ) C0(1 - K)f
V D
(12)
where ∆C ) Cmax - Cmin. This formula describing a linear
dependence ∆C ∼ f cannot explain our numerical results which show the same maximum value of the radial segregation at different values of the interface deflection: f ) 1.2 and 2 mm. This result can be explained by taking into account the convex shape of the interface. In this case, the solute is transported by the convection toward the crucible wall, and the solute concentration should be greater in this zone as compared to the sample axis. But the quantity of the solute rejected at the interface is smaller near the crucible wall because of the f
angle between the interface velocity V and the normal vector f
at the solidification front n (see eq 6). This leads to a nearly homogeneous distribution of the solute at the interface when the interface curvature increases. So, the samples having a convex shape of the solid–liquid interface are more homogeneous on the radial direction than the samples with a concave interface, characterized by a significant increase of the radial segregation for a quasi-diffusive regime, when the interface deflection increases.14
4. Conclusions The solute distribution in CaF2:Pb crystals grown by the vertical Bridgman method, has been for the first time numerically investigated. Previous experimental measurements of the longitudinal Pb distribution in a low doped (1%) CaF2 crystal, show some oscillations of the solute concentration. The same result has been obtained from the transient numerical modeling of the transport phenomena during the solidification process. The numerical results show that the amplitude of these oscillations depends on the convecto-diffusive parameter ∆ and becomes huge when ∆ f 1. In this case a quasi-diffusive regime, characterized by a low intensity of the convection and a significant influence of the flow on the solute distribution, occurs in the melt. Other Bridgman experiments,9,10 performed in the same conditions show the same oscillations of the concentration on the longitudinal direction. In this paper, a physical explanation of this phenomenon has been proposed for the first time, by considering the interaction between the flow and the solutal boundary layer near the solid–liquid interface. The amplitude of these oscillations becomes negligible for a convecto-diffusive regime characterized by ∆ < 0.6. To obtain a convecto-diffusive transport regime at low values of the diffusion coefficient, the growth rate must be reduced. The numerical computations performed for a growth rate two times smaller than the experimental one show a significant reduction
406 Crystal Growth & Design, Vol. 8, No. 2, 2008
of the oscillations amplitude. Because of the high value of the segregation coefficient K g 0.8, the solute longitudinal profile is nearly uniform for the crystals grown at low pulling rates. Moreover, the convex shape of the solid–liquid interface has influence on the radial segregation, which is very small for these crystals. In conclusion, numerical modeling shows that the CaF2:Pb crystals grown at the optimal value of the pulling rate are characterized by a high chemical homogeneity. Acknowledgment. The present research is supported from Romanian Government Ministry of Education and Research, National Authority for Scientific Research project CEEX-M1C2-1185, C64/2006 iSMART-flow.
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CG070125G
