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Inequalities for Single Crystal Tube Growth by EdgeDefined FilmFed Growth Technique
Journal of Inequalities and Applications volume 2009, Article number: 732106 (2009)
Abstract
The axisymmetric YoungLaplace differential equation is analyzed. Solutions of this equation can describe the outer or inner free surface of a static meniscus (the static liquid bridge free surface between the shaper and the crystal surface) occurring in single crystal tube growth. The analysis concerns the dependence of solutions of the equation on a parameter which represents the controllable part of the pressure difference across the free surface. Inequalities are established for which are necessary or sufficient conditions for the existence of solutions which represent a stable and convex outer or inner free surfaces of a static meniscus. The analysis is numerically illustrated for the static menisci occurring in silicon tube growth by edgedefined filmfed growth (EFGs) technique. This kind of inequalities permits the adequate choice of the process parameter . With this aim this study was undertaken.
1. Introduction
The first successful Si tube growth was reported in [1]. Also a theory of tube growth by E.F.G. process is developed there to show the dependence of the tube wall thickness on the growth variables. The theory concerns the calculation of the shape of the liquidvapor interface (i.e., the free surface of the meniscus) and of the heat flow in the system. The inner and the outer free surface shapes of the meniscus (Figure 1) were calculated from YoungLaplace capillary equation, in which the pressure difference across a point on the free surface is considered to be constant, where represents the effective height of the growth interface (Figure 1). The above approximation of is valid when , where is the height of the growth interface above the shaper top. Another approximation used in [1] is that the outer and inner free surface shapes are approximated by circular segments. With these relatively tight tolerances concerning the menisci in conjunction with the heat flow calculation in the system, the predictive model developed in [1] has been shown to be 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.
The growth process was scaled up by Kaljes et al. in [2] to grow 15 cm diameter silicon tubes. It has been realized that theoretical investigations are necessary for the improvement of the technology. Since the growth system consists of a small die type (1 mm width) and a thin tube (order of m wall thickness), the width of the melt/solid interface and the meniscus are accordingly very small. Therefore, it is essential to obtain accurate solution for the free surface of the meniscus, the temperature, and the liquidcrystal interface position in this tinny region.
For single crystal tube growth by edgedefined filmfed growth (E.F.G.) technique, in hydrostatic approximation the free surface of a static meniscus is described by the YoungLaplace capillary equation [3]:
Here is the melt surface tension, denotes the melt density, is the gravity acceleration, denote the main normal curvatures of the free surface at a point of the free surface, is the coordinate of with respect to the axis, directed vertically upwards, and is the pressure difference across the free surface. For the outer free surface, and for the inner free surface, .
Here denotes the hydrodynamic pressure in the meniscus melt, denote the pressure of the gas flow introduced in the furnace in order to release the heat from the outer and inner walls of the tube, respectively, and denotes the melt column height between the horizontal crucible melt level and the shaper top level. When the shaper top level is above the crucible melt level, then , and when the crucible melt level is above the shaper top level, then (see Figure 1).
To calculate the outer and inner free surface shapes of the static meniscus, it is convenient to employ the YoungLaplace (1.1) in its differential form. This form of the (1.1) can be obtained as a necessary condition for the minimum of the free energy of the melt column [3].
For the growth of a single crystal tube of inner radius and outer radius the axisymmetric differential equation of the outer free surface is given by
which is the Euler equation for the energy functional
The axisymmetric differential equation of the inner free surface is given by
which is the Euler equation for the energy functional
The state of the arts at the time 19931994, concerning the dependence of the shape of the meniscus free surface on the pressure difference for small and large bond numbers, in the case of the growth of single crystal rods by E.F.G. technique, are summarized in [4]. According to [4], for the general differential equation (1.2), (1.4) describing the free surface of a liquid meniscus, 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.
Later, in 2001, Rossolenko shows in [5] that the hydrodynamic factor is too small to be considered in the automated single crystal tube growth. Finally, in [6] the authors present theoretical and numerical study of meniscus dynamics under axisymmetric and asymmetric configurations. In [6] the meniscus free surface is approximated by an arc of constant curvature, and a meniscus dynamics model is developed to consider meniscus shape and its dynamics, heat and mass transfer around the dietop and meniscus. Analysis reveals the correlations among tube wall thickness, effective melt height, pullrate, dietop temperature, and crystal environmental temperature.
In the present paper the shape of the inner and outer free surfaces of the static meniscus is analyzed as function of , the controllable part of the pressure difference across the free surface, and the static stability of the free surfaces is investigated. The novelty with respect to the considerations presented in literature consists in the fact that the free surface is not approximated as in [1, 6], by an arc with constant curvature, and the pressure of the gas flow introduced in the furnace for releasing the heat from the tube wall is taken in consideration. The setting of the thermal conditions is not considered in this paper.
2. Meniscus Outer Free Surface Shape Analysis in the Case of Tube Growth
Consider the differential equation
and such that .
Definition 2.1.
A solution of the (2.1) describes the outer free surface of a static meniscus on the interval if possesses the following properties:
 (a)

(b)
and

(c)
and is strictly decreasing on .
The described outer free surface is convex on if in addition the following inequality holds:
 (d)
Theorem 2.2.
If there exists a solution of (2.1), which describes a convex outer free surface of a static meniscus on the closed interval , then the following inequalities hold:
Proof.
Let be a solution of (2.1), which describes a convex outer free surface of a static meniscus on the closed interval and . The function verifies the equation
and the boundary conditions
Hence, there exists such that the following equality holds:
Since , is strictly increasing and is strictly decreasing on , therefore the values of are in the range and the following inequalities hold:
Equality (2.5) and inequalities (2.6) imply inequalities (2.2).
Corollary 2.3.
If with , then inequalities (2.2) become
Corollary 2.4.
If , then and (2.7) becomes
If , then and .
Theorem 2.5.
Let be such that . If satisfies the inequality
then there exists in the closed interval such that the solution of the initial value problem
on the interval describes the convex outer free surface of a static meniscus.
Proof.
Consider the solution of the initial value problem (2.10). Denote by the maximal interval on which the function exists and by the function defined on . Remark that for the equality (2.3) holds.
Since
there exists , such that for any the following inequalities hold:
Let be defined by
It is clear that and for any inequalities (2.12) hold.
From (2.12) and (2.13) it follows that is strictly increasing and bounded on Therefore exists and satisfies
Moreover, since is strictly decreasing and possesses bounded derivative on , exists too, it is finite, and satisfies
We will show now that and . In order to show that we assume the contrary, that is, that . Under this hypothesis we have
for some . Hence This last inequality is impossible, since according to the inequality (2.14), we have . Therefore, , defined by (2.14), satisfies .
In order to show that we remark that from the definition (2.14) of it follows that at least one of the following three equalities holds:
Since for any it follows that the equality is impossible. Hence, we obtain that at at least one of the following two equalities holds: , . We show now that the equality is impossible. For that we assume the contrary, that is, . Under this hypothesis, from (2.12) we have:
what is impossible.
In this way we obtain that the equality holds.
For the solution of the initial value problem (2.8) on the interval describes a convex outer free surface of a static meniscus.
Corollary 2.6.
If for the following inequality holds:
then there exists in the interval such that the solution of the initial value problem (2.10) on the interval describes a convex outer free surface of a static meniscus.
Corollary 2.7.
If for the following inequalities hold:
then there exists in the interval such that the solution of the initial value problem (2.10) on the interval describes a convex outer free surface of a static meniscus. The existence of and the inequality follows from Theorem 2.5. The inequality follows from Corollary 2.3.
Remark 2.8.
The solution of the initial value problem (2.10) is convex at (i.e., ) if and only if
That is because if and only if , that is,
Moreover, if , the solution of the initial value problem (2.8) is convex everywhere (i.e., for ). That is because the change of convexity implies the existence of such that and , what is impossible.
Theorem 2.9.
If a solution of (2.1) describes a convex outer free surface of a static meniscus on the interval , then it is a weak minimum for the energy functional of the melt column:
Proof.
Since (2.1) is the Euler equation for (2.23), it is sufficient to prove that the Legendre and Jacobi conditions are satisfied in this case.
Denote by the function defined as
It is easy to verify that we have
Hence, the Legendre condition is satisfied.
The Jacobi equation
in this case is given by
For (2.27) the following inequalities hold:
Hence, the equation
is a Sturm type upper bound for (2.27) [7].
Since every nonzero solution of (2.29) vanishes at most once on the interval , the solution of the initial value problem
has only one zero on the interval [7]. Hence the Jacobi condition is satisfied.
Definition 2.10.
A solution of (2.1) which describes the outer free surface of a static meniscus is said to be stable if it is a weak minimum of the energy functional of the melt column.
Remark 2.11.
Theorem 2.9 shows that if describes a convex outer free surface of a static meniscus on the interval , then it is stable.
Theorem 2.12.
If the solution of the initial value problem (2.10) is concave (i.e., ) on the interval , then it does not describe the outer free surface of a static meniscus on .
Proof.
on implies that is strictly decreasing on . Hence .
Theorem 2.13.
If and there exists such that the solution of the initial value problem (2.10) is the outer free surface of a static meniscus on , then for the following inequalities hold:
Proof.
Denote by the solution of the initial value problem (2.10) which is assumed to represent the outer free surface of a static meniscus on the closed interval . Let be defined as in Theorem 2.2. for . Since , we have . Hence and therefore for , close to . Taking into account the fact that it follows that there exists such that .
Therefore . Since and , the following inequality holds: . On the other hand . Using the above evaluations we obtain inequalities (2.31).
Remark 2.14.
If appearing in Theorem 2.13 is represented as , , then inequality (2.31) becomes
For inequality (2.32) becomes
Theorem 2.15.
Let be . If for the following inequality holds:
then the solution of the initial value problem (2.10) is concave on the interval where is the maximal interval of the existence of .
Proof.
Consider and remark that for the following relations hold:
Hence: for .
3. Meniscus Inner Free Surface Shape Analysis in the Case of Tube Growth
Consider now the differential equation
and such that .
Definition 3.1.
A solution of (3.1) describes the inner free surface of a static meniscus on the interval if possesses the following properties:
 (a)

(b)
and

(c)
and is strictly increasing on .
The described inner free surface is convex on if in addition the following inequality holds:

(d)
.
Theorem 3.2.
If there exists a solution of (3.1), which describes a convex inner free surface of a static meniscus on the closed interval, then the following inequalities hold:
Proof.
Let be a solution of (3.1), which describes a convex inner free surface of a static meniscus on the closed interval and . The function verifies the equation
and the boundary conditions
Hence, there exists such that the following equality holds:
Since , is strictly increasing and is strictly increasing on , therefore the following inequalities hold:
Equality (3.5) and inequalities (3.6) imply inequalities (3.2).
Corollary 3.3.
If with , then inequalities (3.2) become
Corollary 3.4.
If , then and (3.7) becomes
If , then and .
Theorem 3.5.
Let m be such that . If satisfies the inequality
then there exists in the closed interval such that the solution of the initial value problem
on the interval describes the convex inner free surface of a static meniscus.
Proof.
Consider the solution of the initial value problem (3.10). Denote by the maximal interval on which the function exists and by the function defined on . Remark that for the following equality holds:
Since
there exists , such that for any the following inequalities hold:
Let be defined by
It is clear that and for any inequalities (3.13) hold. Moreover, exists and satisfies, and . Hence is finite, it is strictly positive, and for every the following inequalities hold:
We will show now that and .
In order to show that , we assume the contrary, that is, . Under this hypothesis we have
for some . Hence and it follows that there exists such that and . This last inequality is impossible according to the definition of .
Therefore, defined by (3.14) satisfies .
In order to show that we remark that from the definition (3.14) of it follows that at least one of the following three equalities holds:
Since for , it follows that the equality is impossible.
Hence we obtain that in at least one of the following two equalities holds:
We show now that the equality is impossible. For this we assume the contrary, that is, . Under this hypothesis, from (3.11), we have
what is impossible.
In this way we obtain that the equality holds.
For the solution of the initial value problem (3.10) on the interval describes a convex inner free surface of a static meniscus.
Corollary 3.6.
If for the following inequality holds
then there exists in the interval such that the solution of the initial value problem (3.10) on the interval describes a convex inner free surface of a static meniscus.
Corollary 3.7.
If for the following inequalities hold:
then there exists in the interval such that the solution of the initial value problem (3.10) on the interval describes a convex inner free surface of a static meniscus.
The existence of and the inequality follows from Theorem 3.5. The inequality follows from the Corollary 3.3.
Theorem 3.8.
If a solution of (3.1) describes a convex inner free surface of a static meniscus on the interval , then it is a weak minimum for the energy functional of the melt column:
Proof.
It is similar to the proof of Theorem 2.9.
Definition 3.9.
A solution of (3.1) which describes the inner free surface of a static meniscus is said to be stable if it is a weak minimum of the energy functional of the melt column.
Remark 3.10.
Theorem 3.8 shows that if describes a convex inner free surface of a static meniscus on the interval , then it is stable.
Remark 3.11.
The solution of the initial value problem (3.10) is convex at (i.e., ) if and only if
Theorem 3.12.
If represents the inner free surface of a static meniscus on the closed interval which possesses the following properties:

(a)
is convex at , and

(b)
the shape of changes once on the interval , that is, there exists a point such that for for ,
then there exists such that and for the following inequality holds:
Proof.
Since increases on and decreases on , and , there exists such that . The maximum value is less than . From (3.11) we have
and therefore
Remark 3.13.
If the solution of the initial value problem (3.10) is concave (i.e., ) on the interval, then it does not describe the inner free surface of a static meniscus on .
Theorem 3.14.
Let be . If for the following inequality holds:
then the solution of the initial value problem (3.10) is concave on the interval where is the maximal interval of the existence of .
Proof.
Consider and remark that for the following relations hold:
Hence for .
4. Numerical Illustration
Numerical computations were performed for a Si cylindrical tube for the following data:
The objective was to verify if the necessary conditions are also sufficient, or if the sufficient conditions are also necessary. Moreover, the above data were used in experiments and the computed results can be tested against the experiments in order to evaluate the accuracy of the theoretical predictions. This test is not the subject of this paper.
For the considered numerical data, inequality (2.7) becomes
Integration of (2.10) shows that for there exists such that the solution is a convex outer free surface of a static meniscus on , but for there is no such that the solution is a convex outer free surface of a static meniscus on (Figures 2 and 3). Hence, inequality (2.7) is not a sufficient condition.
For the same numerical data inequality (2.9) becomes . We have already obtained that for there exists in the interval such that the solution of (2.10) describes a convex outer free surface of a static meniscus on the interval . Hence, inequality (2.9) is not a necessary condition.
For the same numerical data inequality (2.19) becomes . Integration shows that for there exists , such that the solution of (2.10) describes a convex outer free surface of a static meniscus on the closed interval (Figures 4 and 5). Hence, inequality (2.19) is not a necessary condition.
For the same numerical data inequality (2.20) becomes . Integration of (2.10) illustrates the above phenomenon for (Figures 6 and 7) and also the fact that the condition is not necessary (see ).
For the same numerical data inequality (2.21) becomes . Integration of (2.10) illustrates the above phenomenon for (Figures 8 and 9).
For the considered numerical data inequality (2.32) becomes . Integration shows that for there is no such that the solution of (2.10) is a nonglobally convex outer free surface of a static meniscus on . Moreover, for it is not anymore the outer free surface of a static meniscus (Figures 10 and 11). Hence, inequality (2.32) is not a sufficient condition.
For the same numerical data inequality (2.33) becomes . We have already obtained that for (Figure 10) the solution of (2.10) is not anymore the outer free surface of a static meniscus. Hence, inequality (2.33) is not a sufficient condition.
For the same numerical data inequality (2.34) becomes . Integration of (2.10) for proves that the solution is globally concave on (Figures 12 and 13). Hence, inequality (2.34) is not a necessary condition.
For the considered numerical data inequality (3.7) becomes . Integration of (3.10) shows that for there exists such that the solution is a convex inner free surface of a static meniscus on , but for there is no such that the solution is a convex inner free surface of a static meniscus on (Figures 14 and 15). Hence, inequality (3.7) is not a sufficient condition.
For the considered numerical data (3.9) becomes . We have already obtained that for there exists such that the solution of (3.10) describes a convex inner free surface of a static meniscus on . Hence, inequality (3.9) is not a necessary condition.
For the considered numerical data (3.20) becomes Integration of (3.10) shows that for there exists such that the solution describes a convex inner free surface of a static meniscus on (Figures 16 and 17). Hence, inequality (3.20) is not a necessary condition.
For the considered numerical data (3.21) becomes . Integration of (3.10) illustrates the above phenomenon for and also the fact that the condition is not necessary (Figures 18 and 19) (see .
For the same numerical data inequality (3.23) becomes . Integration of (3.10) illustrates the above phenomenon for (Figures 20 and 21).
For the considered numerical data inequality (3.24) becomes . Integration shows that for there exists such that the solution of (3.10) is a nonglobally convex inner free surface of a static meniscus on , but for it is not anymore the inner free surface of a nonglobally static meniscus (Figures 20 and 21). Hence, (3.24) is not a sufficient condition.
For the same numerical data (3.27) becomes . Integration of (3.8) for proves that the solution is globally concave on (Figures 22 and 23). Hence, (3.27) is not a necessary condition.
5. Conclusions
(1) Inequalities (2.2) and (2.7) localize regions on the pressure axis where the outer pressure has to be taken in order to obtain convex outer free surface.
Inequalities (2.9), (2.19), and (2.20) localize regions on the pressure axis, having the property that if the outer pressure is taken in this region, then a convex outer free surface is obtained.
Inequalities (2.31) and (2.32) localize region on the pressure axis, where the outer pressure has to be taken in order to obtain a convexconcave outer free surface.
(2) Inequalities (3.2) and (3.7) localize regions on the pressure axis where the inner pressure has to be taken in order to obtain convex inner free surface.
Inequalities (3.5), (3.20), and (3.21) localize regions on the pressure axis, having the property that if the inner pressure is taken in this region, then a convex inner free surface is obtained.
Inequality (3.24) localizes region on the pressure axis, where the inner pressure has to be taken in order to obtain a convexconcave inner free surface.
(3) By computation these values are found in a real case, and the "accuracy" (sufficiency or necessity) of the reported inequalities is discussed.
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Acknowledgments
The authors thank the anonymous referees for their valuable comments and suggestions, which led to an improvement of the manuscript. They are grateful to the Romanian National Authority for Research supporting this research under the Grant ID 354 no. 7/2007.
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Balint, S., Balint, A.M. Inequalities for Single Crystal Tube Growth by EdgeDefined FilmFed Growth Technique. J Inequal Appl 2009, 732106 (2009). https://doi.org/10.1155/2009/732106
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DOI: https://doi.org/10.1155/2009/732106
Keywords
 Free Surface
 Tube Growth
 Closed Interval
 Pressure Axis
 Tube Wall Thickness