On the Creation of a Stable Drop-Like Static Meniscus, Appropriate for the Growth of a Single Crystal Tube with Prior Specified Inner and Outer Radii

Abstract

A theoretical procedure for the creation of a stable drop-like static meniscus, appropriate for the growth of a single crystal tube, with a priori specified inner and outer radius, is presented. The method locates the controllable part 𝑝 of the pressure difference across the free surface. It consists in a set of calculus, which leads to the determination of the melt column height (between the horizontal crucible melt level and the shaper top level) in function of the pressure of the gas flow (introduced in the furnace for release the heat) in order to obtain the desired meniscus. The procedure is presented in general and is numerically illustrated for InSb tubes. The novelty is the algorithm for the exact determination of 𝑝, which has to be used, the determination of the melt column height, and the evaluation of the effect of shaper radii. The setting of the thermal conditions, which assure that for the obtained static meniscus the solidification conditions are satisfied at the “right” places, is not considered here.

1. Introduction

The conventional melt growth techniques, as Bridgman growth [1–3] or Czochralski pulling [4–6] of single crystals, typically produce ingots of circular or square cross-sections which need to be cut in hundreds of slices to produce wafers. Using these processes, it is difficult to produce thin wafers from an ingot without wasting 40%–50% of material as kerfs during the cutting process. For this reason the E.F.G. technology can be more appropriate to produce single crystals with prescribed shapes and sizes which can be used without additional machining.

The growth of silicon tubes by E.F.G. process was first reported by Erris et al. [7]. In [7] a theory of tube growth by E.F.G. process is developed to show the dependence of tube wall thickness on the growth variables. The theory uses approximation reported in [8, 9], and it has been shown to be a useful tool understanding the feasible limits of the wall thickness control. A more accurate predictive model would require an increase of the acceptable tolerance range introduced by approximation.

Later, the heat flow in a tube growth system was analyzed in [10–19].

The state of the arts at the time 1993-1994, concerning the calculation of the meniscus shape in general in the case of the growth by E.F.G. method is summarized in [20]. According to [20], for the general differential equation describing the free surface of a liquid meniscus, possessing axial symmetry, there are no complete analysis and solution. For the general equation only numerical integrations were carried out for a number of process parameter values that were of practical interest at the moment. The authors of [21, 22] consider automated crystal growth processes based on weight sensors and computers. They give an expression for the weight of the meniscus, contacted with crystal and shaper of arbitrary shape, in which there are two terms related to the hydrodynamic factor.

In [23] it is shown that the hydrodynamic factor is too small to be considered in the automated crystal growth. In [24] a theoretical and numerical study of meniscus dynamics, under symmetric and asymmetric configurations, is presented. A meniscus dynamics model is developed to consider meniscus shape and its dynamics, heat and mass transfer around the die top and meniscus. Analysis reveals the correlations among tube thickness, effective melt height, pull rate, die top temperature, and crystal environmental temperature.

In [25] the effect of the controllable part of the pressure difference on the free surface shape of the static meniscus is analyzed for the tube growth by E.F.G. method for materials for which 0<𝛼_𝑐<𝜋/2; 0<𝛼_𝑔<𝜋/2; 𝛼_𝑐>𝜋/2−𝛼_𝑔.

The present paper concerns also the shape and the stability of the free surface of a static meniscus (pulling rate equal to zero). More precisely, it is shown in which kind the explicit formulas reported in [25] can be combined in order to create a stable static meniscus having a free surface with prescribed size and shape, which is appropriate for the growth of a single crystal tube having a priori specified inner and outer radii. The free surface of a static meniscus is appropriate for the growth of a single crystal tube of constant inner radius 𝑟_𝑖 and constant outer radius 𝑟_𝑒 if the angle between the tangent lines to the free surface at the points (𝑟_𝑖,𝑧_𝑖(𝑟_𝑖)),(𝑟_𝑒,𝑧_𝑒(𝑟_𝑒)) (Figure 1 where the solidification conditions have to be assured) and the vertical is equal to the growth angle 𝛼_𝑔. Moreover, the function describing the free surface has to minimize the energy functional of the melt column (i.e., the meniscus has to be stable). In this paper we give a procedure for the choice of the melt column height, between the horizontal crucible melt level and shaper top level and of the pressure of the gas flow introduced in the furnace (for release the heat), in order to create a static meniscus of which free surface is appropriate for the growth of a single crystal tube of constant inner radius 𝑟_𝑖 and outer radius 𝑟_𝑒.The thermal problem concerning the setting of the thermal conditions, which assure that for the obtained static meniscus at the level 𝑧_𝑖(𝑟_𝑖),𝑧_𝑒(𝑟_𝑒) the solidification conditions are satisfied is not considered in this paper. The novelty consists in the fact that the free surface is not approximated by an arc with constant curvature, the computation takes into account the pressure of the gas flow, and the stability of the free surface is assured.

Figure 1 

Axisymmetric meniscus geometry in the tube growth by E.F.G. method.

2. The Free Surfaces Equations and the Pressure Difference Limits

For a single crystal tube growth by E.F.G. technique, in hydrostatic approximation, the outer free surface equation of the static meniscus is

𝑧^𝑒=𝜌⋅𝑔⋅𝑧^𝑒−𝑝_𝑒𝛾1+𝑧^𝑒²^3/2−1𝑟⋅1+𝑧^𝑒²⋅𝑧^𝑒;𝑟∈𝑅^𝑔𝑖+𝑅_𝑔𝑒2,𝑅^𝑔𝑒,(2.1) and the inner free surface equation is 𝑧^𝑖=𝜌⋅𝑔⋅𝑧^𝑖−𝑝_𝑖𝛾1+𝑧^𝑖²^3/2−1𝑟⋅1+𝑧^𝑖²⋅𝑧^𝑖;𝑟∈𝑅_𝑔𝑖,𝑅^𝑔𝑖+𝑅_𝑔𝑒2.(2.2) Here, γ is the surface tension of the melt; ρ is the melt density; 𝑔 is the gravitational acceleration; 𝑧_𝑒,𝑧_𝑖 are the coordinates with respect to the Oz axis, directed vertically upwards; 𝑟 is the radial coordinate with respect to the 𝑂𝑟 axis, oriented horizontal; 𝑅_𝑔𝑒,𝑅_𝑔𝑖 are the outer and inner radius of the shaper, respectively; 𝑝_𝑒,𝑝_𝑖 are the pressure difference across the outer and inner free surface, respectively:

𝑝_𝑒=𝑝_𝑚−𝑝^𝑒𝑔−𝜌⋅𝑔⋅𝐻,𝑝_𝑖=𝑝_𝑚−𝑝^𝑖𝑔−𝜌⋅𝑔⋅𝐻.(2.3) In (2.3), 𝑝_𝑚 denotes the hydrodynamic pressure in the meniscus melt due to the thermal and Marangoni convection; 𝑝^𝑒𝑔,𝑝^𝑖𝑔 are the pressure of the gas flow introduced in the exterior and in the interior of the tube, respectively, for releasing the heat from the inner and outer side of the tube wall; 𝐻 denotes the melt column “height” between the horizontal crucible melt level, and the shaper top level (Figure 1). 𝐻 is positive when the crucible melt level is under the shaper top level and it is negative when the shaper top level is under the crucible melt level.

The solution 𝑧_𝑒=𝑧_𝑒(𝑟) of (2.1) has to satisfy the following conditions:

𝑧^𝑒𝑟_𝑒=−tan𝜋𝑧2−𝛼^𝑔,(2.4a)_𝑒𝑧𝑅_𝑔𝑒=−tan𝛼_𝑐,(2.4b)_𝑒𝑅_𝑔𝑒=0and𝑧_𝑒(𝑟)isstrictlydecreasingon𝑟_𝑒,𝑅_𝑔𝑒,(2.4c) where 𝑟_𝑒∈((𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2,𝑅_𝑔𝑒) is the tube outer radius; 𝛼_𝑔 is the growth angle; 𝛼_𝑐is the contact angle between the outer free surface and the outer edge of the shaper top and 0<𝛼_𝑐<𝜋/2;0<𝛼_𝑔<𝜋/2;𝜋/2−𝛼_𝑔<𝛼_𝑐 (Figure 1).

Condition (2.4a) expresses that at the point (𝑟_𝑒,𝑧_𝑒(𝑟_𝑒)) (the left end of the outer free surface), where the solidification has to be realized, the angle between the tangent line to the free surface and the vertical is equal to the growth angle 𝛼_𝑔 (i.e., the tangent to the tube outer wall is vertical).

Condition (2.4b) expresses that at the point (𝑅_𝑔𝑒,0) (the right end of the outer free surface, where the free surface is attached to the shaper edge), the angle between the tangent line to the free surface and horizontal (i.e., the contact angle) is equal to 𝛼_𝑐.

Condition (2.4c) expresses that at the point (𝑅_𝑔𝑒,0) the free surface is attached to the shaper edge.

Moreover, the solution 𝑧_𝑒=𝑧_𝑒(𝑟) has to minimize the energy functional of the melt column: 𝐼_𝑒(𝑧)=^𝑅𝑔𝑒_𝑟𝑒𝛾⋅1+𝑧^²^1/2+12⋅𝜌⋅𝑔⋅𝑧²𝑧𝑟−𝑝_𝑒⋅𝑧⋅𝑟⋅𝑑𝑟,_𝑒=ℎ_𝑒>0,𝑧𝑅_𝑔𝑒=0.(2.5) The solution 𝑧_𝑖=𝑧_𝑖(𝑟) of (2.2) has to satisfy the following conditions: 𝑧𝑧^𝑖𝑅_𝑔𝑖=tan𝛼_𝑐,(2.6a)_𝑖𝑧𝑟_𝑖=tan𝜋2−𝛼^𝑔,(2.6b)_𝑖𝑅_𝑔𝑖=0and𝑧_𝑖(𝑟)isstrictlyincreasingon𝑅_𝑔𝑖,𝑟_𝑖,(2.6c) where: 𝑟_𝑖∈(𝑅_𝑔𝑖,(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2) is the tube inner radius; 𝛼_𝑔 is the growth angle; 𝛼_𝑐is the contact angle between the inner free surface and the inner edge of the shaper top and 0<𝛼_𝑐<𝜋/2;0<𝛼_𝑔<𝜋/2;𝜋/2−𝛼_𝑔<𝛼_𝑐 (Figure 1).

Condition (2.6a) expresses that at the point (𝑅_𝑔𝑖,0) (the left end of the inner free surface, where the free surface is attached to the shaper edge), the angle between the tangent line to the free surface and horizontal (i.e., the contact angle) is equal to 𝛼_𝑐.

Condition (2.6b) expresses that at the point (𝑟_𝑖,𝑧_𝑖(𝑟_𝑖)) (the right end of the inner free surface), where the solidification has to be realized, the angle between the tangent line to the free surface and the vertical is equal to the growth angle 𝛼_𝑔 (i.e., the tangent to the tube inner wall is vertical).

Condition (2.6c) expresses that at the point (𝑅_𝑔𝑖,0) the free surface is attached to the shaper edge.

Moreover, the solution 𝑧_𝑖=𝑧_𝑖(𝑟) has to minimize the energy functional of the melt column: 𝐼_𝑖(𝑧)=^𝑟𝑖_𝑅𝑔𝑖𝛾⋅1+𝑧^²^1/2+12⋅𝜌⋅𝑔⋅𝑧²𝑧𝑅−𝑝_𝑖⋅𝑧⋅𝑟⋅𝑑𝑟,_𝑔𝑖=0,𝑧𝑟_𝑖=ℎ_𝑖>0.(2.7)

Based on the mathematical theorems, rigorously proven in [25], the following statements, regarding the creation of an appropriate meniscus, can be formulated.

Statement 1
If the solution of the initial value problem (IVP) 𝑧^𝑒=𝜌⋅𝑔⋅𝑧^𝑒−𝑝_𝑒𝛾1+𝑧^𝑒²^3/2−1𝑟⋅1+𝑧^𝑒²⋅𝑧^𝑒𝑧_𝑒,𝑅_𝑔𝑒=0,𝑧^𝑒𝑅_𝑔𝑒=−tan𝛼_𝑐(2.8) is convex, then it does not represent the outer free surface of an appropriate drop-like static meniscus.

Comment 1. The above statement shows that the solutions of IVP (2.8), which represent the outer free surface of an appropriate drop-like static meniscus, can be obtained only for those values of 𝑝_𝑒 for which the solution is not globally convex.

Statement 2
If 𝑝_𝑒<(𝛾/𝑅_𝑔𝑒)⋅sin𝛼_𝑐, then the solution of the IVP (2.8) is globally convex, and it does not represent the outer free surface of an appropriate drop-like static meniscus.

Comment 2. The above statement locates a set of 𝑝_𝑒 values which are not appropriate for the creation of a drop-like meniscus.

Statement 3
If the solution 𝑧_𝑒=𝑧_𝑒(𝑟) of the nonlinear boundary value problem (NLBVP) (2.1), (2.4a), (2.4b), and (2.4c) represents the outer free surface of an appropriate concave static meniscus on the closed interval [𝑅_𝑔𝑒/𝑛,𝑅_𝑔𝑒], with 1<𝑛<(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2, then the following inequalities hold: 𝑛𝑛−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒⋅cos𝛼_𝑐+𝛾𝑅_𝑔𝑒⋅cos𝛼_𝑔≤𝑝_𝑒≤𝑛𝑛−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒⋅sin𝛼_𝑔+𝑛−1𝑛⋅𝜌⋅𝑔⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒⋅sin𝛼_𝑐.(2.9)

Comment 3. The above statement shows that in order to obtain the outer free surface of a concave static meniscus, appropriate for the growth of a tube of outer radius 𝑟_𝑒=𝑅_𝑔𝑒/𝑛, 𝑝_𝑒 has to be searched in the range defined by the inequalities (2.9).

Consequence 1
If 𝑛=2⋅𝑅_𝑔𝑒/(𝑅_𝑔𝑖+𝑅_𝑔𝑒) (𝑟_𝑒 is the middle point of the interval [𝑅_𝑔𝑖,𝑅_𝑔𝑒]), then 𝑟_𝑒=𝑅_𝑔𝑒/𝑛=(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2 and inequalities (2.9) become 2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅cos𝛼_𝑐+𝛾𝑅_𝑔𝑒⋅cos𝛼_𝑔≤𝑝_𝑒≤2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅sin𝛼_𝑔+𝜌⋅𝑔⋅𝑅^𝑔𝑒−𝑅_𝑔𝑖2⋅tan𝛼^𝑐+2⋅𝛾𝑅_𝑔𝑒+𝑅_𝑔𝑖⋅sin𝛼_𝑐.(2.10)

Comment 4. The above consequence shows that in order to obtain the outer free surface of a concave static meniscus, appropriate for the growth of a tube of outer radius 𝑟_𝑒=(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2, 𝑝_𝑒 has to be searched in the range defined by the inequalities (2.10).

Consequence 2
If 𝑝_𝑒 verifies the inequality 𝑝_𝑒<2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅cos𝛼_𝑐+𝛾𝑅_𝑔𝑒⋅cos𝛼_𝑔,(2.11) then there is no 𝑟_𝑒 in the closed interval [(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2,𝑅_𝑔𝑒] for which the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) possesses a concave solution.

Comment 5. The above consequence shows that if 𝑝_𝑒 is in the range defined by the inequality (2.11), then it is impossible to obtain a static meniscus having concave outer free surface which is appropriate for the growth of a tube of outer radius 𝑟_𝑒 situated in the range [(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2,𝑅_𝑔𝑒].

Consequence 3
If 𝑛→1, then 𝑟_𝑒=𝑅_𝑔𝑒/𝑛→𝑅_𝑔𝑒 and, according to (2.9), 𝑝_𝑒→+∞.

Comment 6. The above consequence shows that in order to obtain the outer free surface of a concave static meniscus, appropriate for the growth of a tube of outer radius 𝑟_𝑒≈𝑅_𝑔𝑒, 𝑝_𝑒 has to be very high.

Statement 4
If 𝑛 and 𝑝_𝑒 verify the inequalities 1<𝑛<2⋅𝑅^𝑔𝑒𝑅_𝑔𝑒+𝑅_𝑔𝑖𝑝_𝑒,>𝑛𝑛−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒⋅sin𝛼_𝑔+𝑛−1𝑛⋅𝜌⋅𝑔⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒⋅sin𝛼_𝑐,(2.12) then there exist 𝑟_𝑒∈[𝑅_𝑔𝑒/𝑛,𝑅_𝑔𝑒] and a concave solution of the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) on the interval [𝑟_𝑒,𝑅_𝑔𝑒].

Comment 7. The above consequence shows that if 𝑛 and 𝑝_𝑒 verify (2.12), then a static meniscus having concave outer free surface, appropriate for the growth of a tube of outer radius 𝑟_𝑒 situated in the range [𝑅_𝑔𝑒/𝑛,𝑅_𝑔𝑒], is obtained.

Consequence 4
If for 1<𝑛^<𝑛<2⋅𝑅_𝑔𝑒/(𝑅_𝑔𝑖+𝑅_𝑔𝑒) and 𝑝_𝑒 the following inequalities hold: 𝑛𝑛−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒⋅sin𝛼_𝑔+𝑛−1𝑛⋅𝜌⋅𝑔⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒⋅sin𝛼_𝑐<𝑝_𝑒<𝑛^𝑛^−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒⋅cos𝛼_𝑐+𝛾𝑅_𝑔𝑒⋅cos𝛼_𝑔,(2.13) then there exist 𝑟_𝑒 in the interval [𝑅_𝑔𝑒/𝑛,𝑅_𝑔𝑒/𝑛^] and a concave solution of the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) on the interval [𝑟_𝑒,𝑅_𝑔𝑒].

Comment 8. The above consequence shows that if 𝑝_𝑒 is in the range defined by the inequalities (2.13), then a static meniscus having concave outer free surface, appropriate for the growth of a tube of outer radius 𝑟_𝑒 situated in the range [𝑅_𝑔𝑒/𝑛,𝑅_𝑔𝑒/𝑛^], is obtained.

Statement 5
A concave solution 𝑧_𝑒=𝑧_𝑒(𝑟) of the NLBVP (2.1), (2.4a), (2.4b), and (2.4c) is a weak minimum of the energy functional of the melt column.

Comment 9. The above consequence shows that a static meniscus having a concave outer free surface, appropriate for the growth of a tube of outer radius 𝑟_𝑒 situated in the range [(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2,𝑅_𝑔𝑒], is stable.

Statement 6
If 𝑛 and 𝑝_𝑒 satisfy the inequalities 1<𝑛<2⋅𝑅^𝑔𝑒𝑅_𝑔𝑒+𝑅_𝑔𝑖,𝑝_𝑒>𝑛−1𝑛⋅𝑔⋅𝜌⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒,(2.14) then the solution 𝑧_𝑒=𝑧_𝑒(𝑟) of the IVP (2.8) is concave on the interval 𝐼∩[𝑅_𝑔𝑒/𝑛,𝑅_𝑔𝑒] where 𝐼 is the maximal interval of the existence of 𝑧_𝑒(𝑟).

Comment 10. The above consequence shows that if 𝑝_𝑒 is in the range defined by the inequalities (2.14), then eventually it can be used for the creation of a drop-like static meniscus, appropriate for the growth of a tube of outer radius 𝑟_𝑒 in the range [𝑅_𝑔𝑒/𝑛,𝑅_𝑔𝑒].

Statement 7
If 𝑝_𝑒>(𝛾/𝑅_𝑔𝑒)⋅sin𝛼_𝑐 and for a value 𝑟_𝑒=𝑅_𝑔𝑒/𝑛, which satisfies 𝑟_𝑒∈((𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2,𝑅_𝑔𝑒), a static meniscus appropriate for the growth of a tube of outer radius 𝑟_𝑒 exists, then for 𝑝_𝑒 the following inequalities hold: 𝛾𝑅_𝑔𝑒⋅sin𝛼_𝑐<𝑝_𝑒<𝜌⋅𝑔⋅𝑛−1𝑛⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒⋅cos𝛼_𝑐.(2.15)

Comment 11. The above statement locates those values of 𝑝_𝑒 for which eventually nonglobally concave (drop-like) static meniscus, appropriate for the growth of a tube of outer radius 𝑟_𝑒=𝑅_𝑔𝑒/𝑛, exists.

Statement 8
If the solution of the IVP 𝑧^𝑖=𝜌⋅𝑔⋅𝑧^𝑖−𝑝_𝑖𝛾1+𝑧^𝑖²^3/2−1𝑟⋅1+𝑧^𝑖²⋅𝑧^𝑖𝑧_𝑖,𝑅_𝑔𝑖=0,𝑧^𝑖𝑅_𝑔𝑖=tan𝛼_𝑐(2.16) is convex, then it does not represent the inner free surface of an appropriate drop-like static meniscus.

Comment 12. The above statement shows that the inner free surface of an appropriate drop-like static meniscus can be obtained only for those values of 𝑝_𝑖 for which the IVP (2.16) is not globally convex.

Statement 9
If 𝑝_𝑖 satisfies the inequality 𝑝_𝑖<−𝛾𝑅_𝑔𝑖,(2.17) then the solution 𝑧_𝑖=𝑧_𝑖(𝑟) of the IVP (2.15) is convex on the interval 𝐼∩[𝑅_𝑔𝑖,(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2], where 𝐼 is the maximal interval of the existence of 𝑧_𝑖(𝑟).

Comment 13. The above statement shows that if 𝑝_𝑖 is in the range defined by the inequality (2.17), then it is impossible to obtain a static meniscus, appropriate for the growth of a tube of inner radius 𝑟_𝑖 situated in the range [𝑅_𝑔𝑖,(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2].

Statement 10
If the solution 𝑧_𝑖=𝑧_𝑖(𝑟) of the NLBVP (2.2) and (2.5) represents the inner free surface of an appropriate concave static meniscus on the closed interval [𝑅_𝑔𝑖,𝑚⋅𝑅_𝑔𝑖] with 1<𝑚<(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2⋅𝑅_𝑔𝑖, then the following inequalities hold: 1𝑚−1𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑖⋅cos𝛼_𝑐−𝛾𝑅_𝑔𝑖⋅sin𝛼_𝑐≤𝑝_𝑖≤1(𝑚−1)𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑖⋅sin𝛼_𝑔+(𝑚−1)⋅𝜌⋅𝑔⋅𝑅_𝑔𝑖⋅tan𝛼_𝑐−𝛾𝑚⋅𝑅_𝑔𝑖⋅cos𝛼_𝑔.(2.18)

Comment 14. The above statement shows that in order to obtain the inner free surface of a concave static meniscus, appropriate for the growth of a tube of inner radius 𝑟_𝑖=𝑚⋅𝑅_𝑔𝑖, 𝑝_𝑖 has to be searched in the range defined by the inequalities (2.18).

Consequence 5
If 𝑚=(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2⋅𝑅_𝑔𝑖, then 𝑟_𝑖=(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2 (i.e., 𝑟_𝑖 is the middle point of the interval ⌊𝑅_𝑔𝑖,𝑅_𝑔𝑒⌋) and the inequalities (2.18) become: 2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅cos𝛼_𝑐−𝛾𝑅_𝑔𝑖⋅sin𝛼_𝑐≤𝑝_𝑖≤2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅sin𝛼_𝑔+𝜌⋅𝑔⋅𝑅^𝑔𝑒−𝑅_𝑔𝑖2⋅tan𝛼^𝑐−2⋅𝛾𝑅_𝑔𝑒+𝑅_𝑔𝑖⋅cos𝛼_𝑔.(2.19)

Comment 15. The above consequence shows that in order to obtain the inner free surface of a concave static meniscus, appropriate for the growth of a tube of inner radius 𝑟_𝑖=(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2, 𝑝_𝑖 has to be searched in the range defined by the inequalities (2.19).

Consequence 6
If 𝑝_𝑖 verifies: 𝑝_𝑖<2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅cos𝛼_𝑐−𝛾𝑅_𝑔𝑖⋅sin𝛼_𝑐,(2.20) then there is no 𝑟_𝑖 in the closed interval [𝑅_𝑔𝑖,(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2] for which the NLBVP (2.2) and (2.5) possesses a concave solution.

Comment 16. The above consequence shows that if 𝑝_𝑖 is in the range defined by the inequalities (2.19), then it is impossible to obtain a static meniscus, having concave inner free surface and appropriate for the growth of a tube of inner radius 𝑟_𝑖 situated in the range [𝑅_𝑔𝑖,((𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2)].

Consequence 7
If 𝑚→1, then 𝑟_𝑖=𝑚⋅𝑅_𝑔𝑖→𝑅_𝑔𝑖 and 𝑝_𝑖→+∞.

Comment 17. The above consequence shows that in order to obtain the inner free surface of a concave static meniscus appropriate for the growth of a tube of inner radius 𝑟_𝑖≈𝑅_𝑔𝑖, 𝑝_𝑖 has to be very high.

Statement 11
If 𝑚 and 𝑝_𝑖 verify the inequalities 1<𝑚<𝑅^𝑔𝑖+𝑅_𝑔𝑒2⋅𝑅_𝑔𝑖𝑝_𝑖,>1𝑚−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑖⋅sin𝛼_𝑔+(𝑚−1)⋅𝜌⋅𝑔⋅𝑅_𝑔𝑖⋅tan𝛼_𝑐−𝛾𝑚⋅𝑅_𝑔𝑖⋅cos𝛼_𝑔,(2.21) then there exist 𝑟_𝑖 in the closed interval [𝑅_𝑔𝑖,𝑚⋅𝑅_𝑔𝑖] and a concave solution of the NLBVP (2.2) and (2.5) on the interval [𝑅_𝑔𝑖,𝑟_𝑖].

Comment 18. The above statement shows that if 𝑚 and 𝑝_𝑖 verify (2.21), then a static meniscus having concave inner free surface and appropriate for the growth of a tube of inner radius 𝑟_𝑖 in the range (𝑅_𝑔𝑖,𝑚⋅𝑅_𝑔𝑖), is obtained.

Consequence 8
If for 1<𝑚^<𝑚<(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2⋅𝑅_𝑔𝑖 and 𝑝_𝑖 the following inequalities hold: 1𝑚−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑖⋅sin𝛼_𝑔+(𝑚−1)⋅𝜌⋅𝑔⋅𝑅_𝑔𝑖⋅tan𝛼_𝑐−𝛾𝑚⋅𝑅_𝑔𝑖⋅cos𝛼_𝑔<𝑝_𝑖<1𝑚^−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑖⋅cos𝛼_𝑐+𝛾𝑅_𝑔𝑖⋅sin𝛼_𝑐,(2.22) then there exist 𝑟_𝑖 in the interval [𝑚^⋅𝑅_𝑔𝑖,𝑚⋅𝑅_𝑔𝑖] and a concave solution of the NLBVP (2.2) and (2.5) on the interval [𝑅_𝑔𝑖,𝑟_𝑖].

Comment 19. The above consequence shows that if 𝑝_𝑖 is in the range defined by the inequalities (2.22), then a static meniscus having a concave inner free surface and appropriate for the growth of a tube of inner radius 𝑟_𝑖, situated in the range [𝑚^⋅𝑅_𝑔𝑖,𝑚⋅𝑅_𝑔𝑖], is obtained.

Statement 12
A concave solution 𝑧_𝑖=𝑧_𝑖(𝑟) of the NLBVP (2.2) and (2.5) is a weak minimum of the energy functional of the melt column.

Comment 20. The above statement shows that a static meniscus having a concave inner free surface appropriate for the growth of a tube of inner radius 𝑟_𝑖, situated in the range [𝑅_𝑔𝑖,(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2], is stable.

Statement 13
If 𝑝_𝑖 satisfies 𝑝_𝑖<(−𝛾/𝑅_𝑔𝑖)⋅sin𝛼_𝑐 and there exists 𝑟_𝑖∈(𝑅_𝑔𝑖,(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2) such that on the interval [𝑅_𝑔𝑖,𝑟_𝑖] an appropriate static meniscus exists, then 𝑝_𝑖 verifies: −𝛾𝑅_𝑔𝑖<𝑝_𝑖<−𝛾𝑅_𝑔𝑖⋅sin𝛼_𝑐.(2.23)

Comment 21. Inequality (2.23) locates the values of 𝑝_𝑖 for which eventually nonglobally concave (convex-concave) static meniscus can be obtained.

Statement 14
If (−𝛾/𝑅_𝑔𝑖)⋅sin𝛼_𝑐<𝑝_𝑖 and there exists 𝑟_𝑖=𝑚⋅𝑅_𝑔𝑖;𝑚∈(1,(𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2⋅𝑅_𝑔𝑖) such that on [𝑅_𝑔𝑖,𝑟_𝑖] an appropriate non-globally concave static meniscus exists, then 𝑝_𝑖 satisfies the inequalities −𝛾𝑅_𝑔𝑖⋅sin𝛼_𝑐<𝑝_𝑖<𝜌⋅𝑔⋅(𝑚−1)⋅𝑅_𝑔𝑖⋅tan𝛼_𝑐+𝛾𝑅_𝑔𝑖.(2.24)

Comment 22. Inequality (2.24) locates the values of 𝑝_𝑖 for which eventually concave-convex static menisci can be obtained.

Statement 15
If 𝑚 and 𝑝_𝑖 satisfy the inequalities 1<𝑚<𝑅^𝑔𝑖+𝑅_𝑔𝑒2⋅𝑅_𝑔𝑖,𝑝_𝑖>𝛾𝑅_𝑔𝑖+𝜌⋅𝑔⋅(𝑚−1)⋅𝑅_𝑔𝑖⋅tan𝛼_𝑐,(2.25) then the solution 𝑧_𝑖(𝑟) of the IVP (2.16) is concave on the interval 𝐼∩[𝑅_𝑔𝑖,𝑚⋅𝑅_𝑔𝑖] where 𝐼 is the maximal interval of the existence of 𝑧_𝑖(𝑟).

Comment 23. Inequality (2.25) locates the values of 𝑝_𝑖 for which we can obtain eventually appropriate static menisci.

3. Creation of an Appropriate Drop-Like Static Meniscus for the Growth of a Tube of Inner Radius 𝑟_𝑖and Outer Radius 𝑟_𝑒

In this section it will be shown in which kind the explicit formulas presented in the above section can be used for the creation of an appropriate drop-like meniscus when 𝛼_𝑐,𝛼_𝑔,𝜌,𝛾,𝑅_𝑔𝑖,𝑅_𝑔𝑒,𝑟_𝑖and𝑟_𝑒 are given a priori. In the same time the melt column “height” 𝐻, which has to be used, is found in function of the pressure of the gas flow introduced in the furnace for release the heat.

3.1. Creation of the Outer Free Surface

For the creation of the appropriate outer free surface, the following limits were considered: 𝐿^𝑒1(𝑛)=𝑛𝑛−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒⋅cos𝛼_𝑐+𝛾𝑅_𝑔𝑒𝐿_𝑒2⋅cos𝛼_𝑔,(𝑛)=𝑛𝑛−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒⋅sin𝛼_𝑔+𝑛−1𝑛⋅𝜌⋅𝑔⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒𝐿_𝑒3⋅sin𝛼_𝑐,(𝑛)=𝑛−1𝑛⋅𝜌⋅𝑔⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒𝐿_𝑒4⋅cos𝛼_𝑐,(𝑛)=𝑛−1𝑛⋅𝜌⋅𝑔⋅𝑅^𝑔𝑒⋅tan𝛼_𝑐+𝑛⋅𝛾𝑅_𝑔𝑒𝑙_𝑒1,=2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅cos𝛼_𝑐+𝛾𝑅_𝑔𝑒𝑙_𝑒2⋅cos𝛼_𝑔,=2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅sin𝛼_𝑔+𝜌⋅𝑔⋅𝑅^𝑔𝑒−𝑅_𝑔𝑖2⋅tan𝛼^𝑐+2⋅𝛾𝑅_𝑔𝑒−𝑅_𝑔𝑖𝑙_𝑒3⋅sin𝛼_𝑐,=𝛾𝑅_𝑔𝑒⋅sin𝛼_𝑐(3.1) for 𝑛>1.

These limits are represented for 𝑛∈(1;1.1) in Figure 2. The computations were performed in MathCAD V.13 using the following numerical data 𝑅_𝑔𝑒=4.8⋅10𝛼⁻³[m];𝑅_𝑔𝑖=4.2⋅10⁻³[m];𝛼_𝑐=63.8⁰=1.1135[rad_𝑔];=𝛾2=8.94⁰.2=⋅100.50₋₁44[rad];𝜌=6582kg/m³;[N/m];𝑔=9.81m/s².(3.2) When 𝑛₁=1.03226, that is, 𝑟^1𝑒=4.65⋅10⁻³[m], we have the following.

Figure 2 

The possible 𝑝𝑒 ranges which have to be investigated for 𝑛∈(1;1.1).

(i)If there exists a concave outer free surface, appropriate for the growth of a tube of outer radius 𝑟^1𝑒=4.65⋅10⁻³[m], then according to Statement 3 this can be obtained for a value of 𝑝_𝑒 which is in the range (𝐿^𝑒1(𝑛),𝐿^𝑒2(𝑛))=(134.85;164.49)[Pa].(ii)Taking into account the above fact, in order to create a concave outer free surface, appropriate for the growth of a tube of which outer radius is equal to 𝑟^1𝑒=4.65⋅10⁻³[m], we have solved the IVP (2.8) for different values of 𝑝_𝑒in the range (134.85;164.49)[Pa].

More precisely, we have integrated the following system: 𝑑𝑧_𝑒𝑑𝑟=−tan𝛼^𝑒𝑑𝛼_𝑒𝑑𝑟=−cos1𝛼_𝑒⋅𝑔⋅𝜌⋅𝑧^𝑒−𝑝_𝑒𝛾+1𝑟⋅sin𝛼^𝑒(3.3) for 𝑧_𝑒(𝑅_𝑔𝑒)=0, 𝑧^𝑒(𝑅_𝑔𝑒)=−tan𝛼_𝑐 and different 𝑝_𝑒. The obtained outer radii 𝑟_𝑒 versus 𝑝_𝑒 are represented in Figure 3, which shows that the desired outer radius 𝑟^1𝑒=4.65⋅10⁻³[m] is obtained for 𝑝^𝑒=149.7[Pa].

Figure 3 

Outer radii 𝑟𝑒 versus 𝑝𝑒 in the range (134.85;164.49)[Pa].

Actually, as it can be seen on the same figure, for 𝑝^𝑒=149.7[Pa], we can obtain also a second outer radius 𝑟^2𝑒=3.8⋅10⁻³[m], which is not anymore in the desired range ((𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2,𝑅_𝑔𝑒). Moreover, the outer free surface of this meniscus is not globally concave; it is a convex-concave meniscus (Figure 4).

(a)

(b)

Figure 4 

Nonglobally concave outer free surface obtained for 𝑝𝑒=149.7[Pa].

Taking into account 𝑝_𝑚≈0 [7, 23, 24], the melt column height in this case is 𝐻^𝑒=−[1/𝜌⋅𝑔]⋅[𝑝^𝑒+𝑝^𝑒𝑔], where 𝑝^𝑒𝑔≥0 is the pressure of the gas flow (introduced in the furnace for release the heat from the outer side of the tube wall). When 𝑝^𝑒𝑔=0 [7, 24], then 𝐻^𝑒 is negative, 𝐻^𝑒=−2.31⋅10⁻³[m]; that is, the crucible melt level has to be with −𝐻^𝑒=2.31⋅10⁻³[m] above the shaper top level. When 𝑝^𝑒𝑔=800[Pa], then 𝐻^𝑒=−15⋅10⁻³[m]; that is, the crucible melt level has to be with −𝐻^𝑒=15⋅10⁻³[m] above the shaper top level.

Additional remarks are as follows.

(i)For 𝑝_𝑒 in the range (−∞,𝑙^𝑒3)=(−∞,78.51)[Pa], according to Statement 2, the outer free surface of the meniscus is globally convex, and, according to Statement 1, such a meniscus is not appropriate for the growth of a tube of outer radius 𝑟^1𝑒=4.65⋅10⁻³[m]. (ii)For 𝑝_𝑒 in the range (𝑙^𝑒3,𝑙^𝑒1)=(78.51;105.73)[Pa], according to Consequence 2, it is impossible to obtain concave outer free surface, which is appropriate for the growth of a tube of outer radius 𝑟^1𝑒=4.65⋅10⁻³[m].(iii)For 𝑝_𝑒 in the range (𝑙^𝑒1,𝐿^𝑒1(𝑛))=(105.73;134.85)[Pa], according to the Statement 3, it is impossible to obtain concave outer free surface, which is appropriate for the growth of a tube of outer radius 𝑟^1𝑒=4.65⋅10⁻³[m].(iv)For 𝑝_𝑒 in the range (𝐿^𝑒2(𝑛);+∞)=(164.49;+∞)[Pa], according to the Statement 4, the outer free surface is concave and is appropriate for the growth of a tube which outer radius 𝑟_𝑒 is higher than 𝑟^1𝑒=4.65⋅10⁻³[m].

3.2. Creation of an Appropriate Inner Free Surface

For the creation of the appropriate inner free surface, the following limits were considered 𝐿^𝑖1(𝑚)=1𝑚−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑖⋅cos𝛼_𝑐−𝛾𝑅_𝑔𝑖𝐿_𝑖2⋅sin𝛼_𝑐,(𝑚)=1𝑚−1⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑖⋅sin𝛼_𝑔+(𝑚−1)⋅𝜌⋅𝑔⋅𝑅_𝑔𝑖⋅tan𝛼_𝑐−1𝑚⋅𝑅𝛾_𝑔𝑖𝐿_𝑖3⋅cos𝛼_𝑔,(𝑚)=𝜌⋅𝑔⋅(𝑚−1)⋅𝑅_𝑔𝑖⋅tan𝛼_𝑐+𝛾𝑅_𝑔𝑖𝑙_𝑖1,=2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅cos𝛼_𝑐−𝛾𝑅_𝑔𝑖𝑙_𝑖2⋅sin𝛼_𝑐,=2⋅𝛾⋅𝛼^𝑐+𝛼_𝑔−𝜋/2𝑅_𝑔𝑒−𝑅_𝑔𝑖⋅sin𝛼_𝑔+𝜌⋅𝑔⋅𝑅^𝑔𝑒−𝑅_𝑔𝑖2⋅tan𝛼^𝑐−2⋅𝛾𝑅_𝑔𝑒+𝑅_𝑔𝑖𝑙_𝑖3⋅cos𝛼_𝑔,=−𝛾𝑅_𝑔𝑖𝑙_𝑖4⋅sin𝛼_𝑐,=−𝛾𝑅_𝑔𝑖,(3.4) for 𝑚>1.

These limits are represented for 𝑚∈(1;1.1) in Figure 5. The computations were performed in MathCAD V.13 using the same numerical data as for the creation of the appropriate outer free surface.

Figure 5 

The possible 𝑝𝑖 ranges which have to be investigated for 𝑚∈(1;1.1).

When 𝑚₁=1.03571, that is, 𝑟^𝑖=0.00435[m], we obtain the following.

(i)If there exists a concave inner free surface for the growth of a tube of inner radius 𝑟^𝑖=0.00435[m], then, according to the Statement 10, this can be obtained for a value of 𝑝_𝑖 which is in the range (𝐿^𝑖1(𝑚₁),𝐿^𝑖2(𝑚₁))=(−31.46,−1.07)[Pa].

Taking into account the above fact, in order to create a concave inner free surface, appropriate for the growth of a tube which inner radius is equal to 𝑟^𝑖=0.00435[m], we have solved the IVP (2.16) for different values of 𝑝_𝑖 in the range (𝐿^𝑖1(𝑚₁),𝐿^𝑖2(𝑚₁))=(−31.46,−1.07)[Pa]. More precisely, we have integrated the following system: 𝑑𝑧_𝑖𝑑𝑟=tan𝛼^𝑖𝑑𝛼_𝑖,𝑑𝑟=co1s𝛼_𝑖⋅𝑔⋅𝜌⋅𝑧^𝑖−𝑝_𝑖𝛾−1𝑟⋅sin𝛼^𝑖(3.5) for 𝑧_𝑖(𝑅_𝑔𝑖)=0, 𝑧^𝑖(𝑅_𝑔𝑖)=tan𝛼_𝑐 and different 𝑝_𝑖. The obtained inner radii 𝑟_𝑖 versus 𝑝_𝑖 are represented in Figure 6, which shows that the desired inner radius 𝑟^1𝑖=4.35⋅10⁻³[m] is obtained for 𝑝^𝑖=−16.2[Pa].

Figure 6 

Inner radii 𝑟𝑖 versus 𝑝𝑖 in the range (𝐿𝑖1(𝑚1),𝐿𝑖2(𝑚1))=(−31.46,−1.07)[Pa].

Taking 𝑝_𝑚≈0 [7, 23, 24], the melt column height in this case is 𝐻^𝑖=−(1/𝜌⋅𝑔)⋅[𝑝^𝑖+𝑝^𝑖𝑔], where 𝑝^𝑖𝑔≥0 is the pressure of the gas flow (introduced in the furnace for release the heat from the inner side of the tube wall). When 𝑝^𝑖𝑔=0 [7, 24], then 𝐻^𝑖 is positive, 𝐻^𝑖=0.25⋅10⁻³[m]; that is, the crucible melt level has to be with 𝐻^𝑖=0.25⋅10⁻³[m] under the shaper top level. When 𝑝^𝑖𝑔=800[Pa], then 𝐻^𝑖=−12.1⋅10⁻³[m]; that is, the crucible melt level has to be with −𝐻^𝑒=12.1⋅10⁻³[m] above the shaper top level.

Additional remarks are as follows.

(i)For 𝑝_𝑖 in the range (−∞,𝑙^𝑖4)=(−∞,−100)[Pa], according to Statement 9, the meniscus inner free surface is convex on the interval 𝐼∩[𝑅_𝑔𝑖,((𝑅_𝑔𝑖+𝑅_𝑔𝑒)/2)], and according to Statement 8, such a meniscus is not appropriate for the growth of a tube of inner radius 𝑟^𝑖=0.00435[m].(ii)For 𝑝_𝑖 in the range (𝑙^𝑖4,𝑙^𝑖3)=(−100,89.72)[Pa] if an appropriate static meniscus exists, then the inner free surface of the meniscus is not globally concave.(iii)If 𝑝_𝑖 is in the range (𝑙^𝑖3,+∞)=(89.72,+∞)[Pa] and an appropriate nonglobally concave inner free surface exists, then 𝑝_𝑖 is less than 𝐿^𝑖3(𝑚₁)=119.68[Pa].(iv)For 𝑝_𝑖 in the range (𝐿^𝑖2(𝑚),+∞)=(−1.07,+∞)[Pa] the obtained inner free surface is appropriate for the growth of a tube of inner radius 𝑟_𝑖 which is less than the desired radius 𝑟^𝑖=0.00435[m].

3.3. Creation of a Concave Meniscus, Appropriate for the Growth of a Tube of Inner Radius 𝑟_𝑖 and Outer Radius 𝑟_𝑒

For creating a concave meniscus, appropriate for the growth of a tube with outer radius 𝑟^𝑒=0.00465 [m] (𝑛₁=𝑅_𝑔𝑒/𝑟_𝑒=1.03226) and inner radius 𝑟^𝑖=0.00435 [m] (𝑚₁=𝑅_𝑔𝑖/𝑟_𝑖=1.03571) the melt column heights (with respect to the crucible melt level) have to be 𝐻^1𝑖=−1𝜌⋅𝑔⋅𝑝^1𝑖+𝑝^𝑖𝑔=−1𝐻_1𝑒𝜌⋅𝑔⋅−16.2+𝑝^𝑖𝑔,=−1𝜌⋅𝑔⋅𝑝^1𝑒+𝑝^𝑒𝑔=−1𝜌⋅𝑔⋅149.7+𝑝^𝑒𝑔.(3.6)

In the following we will present three different procedures to create such a meniscus.

Procedure A [𝐻^1𝑒=𝐻^1𝑖; 𝑝^𝑖𝑔≠𝑝^𝑒𝑔]
When the shaper outer top is at the same level as the shaper inner top, with respect to the crucible melt level, then the relation 𝐻^1𝑒=𝐻^1𝑖 holds and hence: (1/𝜌⋅𝑔)⋅[−16.2+𝑝^𝑖𝑔]=(1/𝜌⋅𝑔)⋅[149.7+𝑝^𝑒𝑔]. It follows that the pressure of the gas flow, introduced in the furnace for releasing the heat from the inner wall of the tube, 𝑝^𝑖𝑔, has to be higher than the pressure of the gas flow introduced in the furnace for release the heat from the outer wall of the tube, 𝑝^𝑒𝑔; 𝑝^𝑖𝑔−𝑝^𝑒𝑔=149.7+16.2=165.9[Pa].The results of the integration of the system (3.3) and (3.5) for 𝑝_𝑒=𝑝^1𝑒=149.7[Pa] and 𝑝_𝑖=𝑝^1𝑖=−16.2[Pa] when the shaper outer top is at the same level as the shaper inner top, with respect to the crucible melt level, are represented in Figure 7.Figure 7 shows also that the inner free surface is higher than the outer free surface: 𝑧_𝑖(𝑟^1𝑖)=ℎ^𝑎𝑖=2.865⋅10⁻⁴[m];𝑧_𝑒(𝑟^1𝑒)=ℎ^𝑎𝑒=2.860⋅10⁻⁴[m];ℎ^𝑎𝑖−ℎ^𝑎𝑒=0.005⋅10⁻⁴[m].

Figure 7 

𝑧𝑖(𝑟) and 𝑧𝑒(𝑟) for 𝑝𝑒=149.7[Pa];𝑝𝑖=−16.2[Pa] and 𝐻1𝑒=𝐻1𝑖.

Procedure B [𝑝^𝑒𝑔=𝑝^𝑖𝑔; 𝐻^1𝑒≠𝐻^1𝑖]
When the pressure of the gas flow introduced in the furnace for releasing the heat from the outer wall of the tube, 𝑝^𝑒𝑔, is equal to the pressure of the gas flow introduced in the furnace for release the heat from the inner wall of the tube 𝑝^𝑖𝑔, then the equalities 𝑝^𝑒𝑔=−𝑝^1𝑒−𝜌⋅𝑔⋅𝐻^1𝑒=−149.7−𝜌⋅𝑔⋅𝐻^1𝑒 and 𝑝^𝑖𝑔=−𝑝^1𝑖−𝜌⋅𝑔⋅𝐻^1𝑖=16.2−𝜌⋅𝑔⋅𝐻^1𝑖 imply the equality 𝐻^1𝑖−𝐻^1𝑒=(1/𝜌⋅𝑔)⋅165.9=2.569⋅10⁻³[m].(3.7)

In other words, when 𝑝^𝑒𝑔=𝑝^𝑖𝑔, then the level difference between the shaper inner top and outer top has to be 𝐻^1𝑖−𝐻^1𝑒=2.569⋅10⁻³[m]. In this condition the inner surface top level ℎ^𝑏𝑖 with respect to the shaper outer top level is ℎ^𝑏𝑖=𝐻^1𝑖−𝐻^1𝑒+ℎ^𝑎𝑖=(1/𝜌⋅𝑔)⋅(−𝑝^1𝑒+𝑝^1𝑖)+ℎ^𝑎𝑖=2.855⋅10⁻³[m] (see Figure 8).

Figure 8 

𝑧𝑖(𝑟) and 𝑧𝑒(𝑟) for 𝑝𝑖=−16.2[Pa] and 𝑝𝑒=149.7[Pa] and 𝑝𝑒𝑔=𝑝𝑖𝑔.

Hence the level difference between the inner free surface top ℎ^𝑏𝑖 and outer free surface top ℎ^𝑏𝑒=ℎ^𝑎𝑒 in this case is ℎ^𝑏𝑖−ℎ^𝑏𝑒=(1/𝜌⋅𝑔)⋅(−𝑝^1𝑒+𝑝^1𝑖)+(ℎ^𝑎𝑖−ℎ^𝑎𝑒)=2.5695⋅10⁻³[m].

Due to the difficulties which can appear in this case in the creation of an appropriate thermal field, which assures the solidification in the right places, it is more appropriate to use 𝐻^1𝑒=𝐻^1𝑖 and gas flows having the property (Procedure A) 𝑝^𝑖𝑔=𝑝^𝑒𝑔+(𝑝^1𝑖−𝑝^1𝑒)=𝑝^𝑒𝑔+165.9[Pa]. For this choice the difference between the inner free surface height and outer free surface height with respect to the shaper top level is only ℎ_𝑖−ℎ_𝑒=0.0005⋅10⁻³[m], and the creation of the thermal field, which assures the solidification in the right places, can be easier.

Procedure C [𝑝^𝑒𝑔≠𝑝^𝑖𝑔; 𝐻^1𝑒≠𝐻^1𝑖]
Finally, to create an appropriate concave meniscus for which the inner meniscus is equal to the outer meniscus height, we will proceed by changing both the pressures 𝑝^𝑖𝑔, 𝑝^𝑒𝑔 of the gas flows and the inner and outer top levels 𝐻_𝑖,𝐻_𝑒 of the shaper.
For example, if we take 𝑝^𝑒𝑔=100[Pa] for creating an appropriate outer free surface, then the crucible melt level has to be with −𝐻^1𝑒=(1/𝜌⋅𝑔)⋅[149.7+100][m]=38.67⋅10⁻⁴[m] above the shaper outer top level.
Now we take 𝐻^1𝑖=𝐻^1𝑒−(ℎ^𝑎𝑖−ℎ^𝑎𝑒)=−38.67⋅10⁻⁴−0.005⋅10⁻⁴=−38.675⋅10⁻⁴[m]. After that, in order to satisfy 𝑝_𝑖=−𝑝^𝑖𝑔−𝜌⋅𝑔⋅𝐻^1𝑖=−16.2[Pa], we take the pressure 𝑝^𝑖𝑔 equal to 𝑝^𝑖𝑔=16.2−𝜌⋅𝑔⋅(−38.675⋅10⁻⁴)=265.92[Pa]. For this choice, we will have an appropriate drop-like meniscus for which ℎ^𝑐𝑖=ℎ^𝑐𝑒, as it is shown in Figure 9, obtained by integration of systems (3.3) and (3.5) for the given numerical data.

Figure 9 

𝑧𝑖(𝑟) and 𝑧𝑒(𝑟) for 𝑝𝑖=−16.2[Pa] and 𝑝𝑒=149.7[Pa]; 𝑝𝑒𝑔=100[Pa];𝑝𝑖𝑔=265.92[Pa];𝐻1𝑒=−38.67⋅10−4[m];𝐻1𝑖=−38.675⋅10−4[m].

4. The Effect of the Choice of the Shaper Radii

In this sequence we intend to evaluate numerically the effect of the choice of the shaper radii for the growth of a tube which outer radius is equal to 𝑟_𝑒, and inner radius is equal to 𝑟_𝑖.

For this purpose, the procedure described in sequence 3 will be applied for the growth of a tube having the same outer radius 𝑟^𝑒=0.00465 [m] and inner radius 𝑟^𝑖=0.00435 [m], but the shaper radii changed: the inner radius of the shaper now is 𝑅_𝑔𝑖=4⋅10⁻³[m] (instead of 4.2⋅10⁻³[m]), and the outer radius of the shaper now is 𝑅_𝑔𝑒=5⋅10⁻³[m] (instead of 4.8⋅10⁻³[m]).

Applying the procedure in this case we find that to create a static meniscus which outer free surface is appropriate for the growth, 𝑝_𝑒 has to be 𝑝_𝑒=124.38[Pa], and to create a static meniscus which inner free surface is appropriate for the growth, 𝑝_𝑖 has to be 𝑝_𝑖=−41.35[Pa]. The so-obtained outer and inner free surfaces are stable. The inner free surface height is ℎ_𝑖=6.57⋅10⁻⁴[m], and the outer free surface height is ℎ_𝑒=6.603⋅10⁻⁴[m].

When we take the shaper inner top at the same level as the shaper outer top, then the pressure difference between the pressure of the inner gas flow 𝑝^𝑖𝑔 and the outer gas flow 𝑝^𝑒𝑔 has to be 𝑝^𝑖𝑔−𝑝^𝑒𝑔=𝑝_𝑒−𝑝_𝑖=165.73[Pa], and the level difference between the inner and outer free surface top is equal to ℎ_𝑖−ℎ_𝑒=−0.033⋅10⁻⁴[m].

When we take the pressure of the inner gas flow 𝑝^𝑖𝑔 equal to the pressure of the outer gas flow 𝑝^𝑒𝑔, then the level difference between the shaper inner top 𝐻_𝑖 and the shaper outer top 𝐻_𝑒 has to be 𝐻_𝑖−𝐻_𝑒=(1/𝜌⋅𝑔)⋅[𝑝_𝑖−𝑝_𝑒]=2.567⋅10⁻³[m], and the level difference between the inner and outer free surface top is equal to the difference 𝐻_𝑖+ℎ_𝑖−(𝐻_𝑒+ℎ_𝑒)=(1/𝜌⋅𝑔)⋅[𝑝_𝑖−𝑝_𝑔]+ℎ_𝑖−ℎ_𝑒=2.5637⋅10⁻³[m].

5. Conclusions

(i)Stable and drop-like static meniscus, appropriate for the growth of a single-crystal cylindrical tube with a priori specified inner and outer radii, can be created by the choice of the difference between the inner top level and outer top level of the shaper, or by the choice of the pressure of the gas flows, introduced inside and outside of the tube for release the heat from the tube walls.(ii)The values of the differences between the inner and outer top level of the shaper, or the pressure in the inner and outer gas flows, depend on the shaper radii. This dependence can be relevant.(iii)The realization of the thermal conditions, which assure that for the obtained static meniscus the solidification conditions are satisfied at the right places, is easier when we take the shaper inner top at the same level as the shaper outer top, and we use inner and outer gas flows having different pressures (𝑝^𝑖𝑔≠𝑝^𝑒𝑔). The setting of the thermal conditions is not discussed in this paper.

Acknowledgments

The authors thank Professor Koichi Kakimoto for the useful discussions and suggestions concerning the tube growth process. They are grateful to the Romanian National Authority for Research for supporting the research under the Grant ID 354 no. 7/2007.