Inequalities for Single Crystal Tube Growth by Edge-Defined Film-Fed Growth Technique

Journal of Inequalities and Applications Volume 2009, Article ID 732106,28pages doi:10.1155/2009/732106

Research Article

Inequalities for Single Crystal Tube Growth by Edge-Defined Film-Fed Growth Technique

Stefan Balint

and Agneta M. Balint

1Department of Computer Science, Faculty of Mathematics and Computer Science, West University of Timisoara, Blv. V.Parvan 4, 300223 Timisoara, Romania

2Faculty of Physics, West University of Timisoara, Blv. V.Parvan 4, 300223 Timisoara, Romania

Correspondence should be addressed to Agneta M. Balint,balint@physics.uvt.ro Received 3 January 2009; Accepted 29 March 2009

Recommended by Yong Zhou

The axi-symmetric Young-Laplace differential equation is analyzed. Solutions of this equation can describe the outer or inner free surface of a static meniscusthe static liquid bridge free surface between the shaper and the crystal surfaceoccurring in single crystal tube growth. The analysis concerns the dependence of solutions of the equation on a parameterp which represents the controllable part of the pressure difference across the free surface. Inequalities are established forp 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 edge-defined film-fed growthEFGs technique. This kind of inequalities permits the adequate choice of the process parameterp. With this aim this study was undertaken.

Copyrightq2009 S. Balint and A. M. Balint. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The first successful Si tube growth was reported in1. 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 liquid-vapor interfacei.e., the free surface of the meniscusand of the heat flow in the system. The inner and the outer free surface shapes of the meniscusFigure 1were calculated from Young-Laplace capillary equation, in which the pressure differenceΔpacross a point on the free surface is considered to be Δp ρ·g ·H_eff constant, whereH_effrepresents the effective height of the growth interface Figure 1. The above approximation of Δp is valid when H_eff h, whereh 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 in1has 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. in2to 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 type1 mm widthand a thin tubeorder of 200μm 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 liquid-crystal interface position in this tinny region.

For single crystal tube growth by edge-defined film-fed growthE.F.G.technique, in hydrostatic approximation the free surface of a static meniscus is described by the Young- Laplace capillary equation3:

γ· 1 R1 1

R2

ρ·g·z−p. 1.1

Hereγ is the melt surface tension,ρdenotes the melt density,g is the gravity acceleration, 1/R₁,1/R₂ denote the main normal curvatures of the free surface at a pointMof the free surface,z is the coordinate ofMwith respect to theOz axis, directed vertically upwards, andpis the pressure difference across the free surface. For the outer free surface, p p_e p_m−p^e_g−ρ·g·Hand for the inner free surface,p p_i p_m−pⁱ_g−ρ·g·H.

Herepmdenotes the hydrodynamic pressure in the meniscus melt,p_g^e, p_gⁱ 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, andHdenotes 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, thenH >0, and when the crucible melt level is above the shaper top level, thenH <0seeFigure 1.

To calculate the outer and inner free surface shapes of the static meniscus, it is convenient to employ the Young-Laplace1.1in its differential form. This form of the1.1 can be obtained as a necessary condition for the minimum of the free energy of the melt column3.

For the growth of a single crystal tube of inner radiusri ∈ Rgi,RgiRge/2and outer radiusr_e∈RgiR_ge/2, Rgethe axi-symmetric differential equation of the outer free surface is given by

z ρ·g·z−pe

γ

1 z₂3/2

−1 r ·

1 z₂

·z forr∈

re, Rge , 1.2

which is the Euler equation for the energy functional

Iez _Rge

re

γ·

1 z_21/2

1

2·ρ·g·z²−pe·z

·r·dr, zre he>0, z

Rge

0.

1.3

z Inner gas flow

Outer gas flow

ri

re

Tube

Outer free surface Inner

free surface Meniscus melt

Shaper Capillary channel

Crucible melt R_gi

Rge

H H_eff hihe r

0

zir z_er

αc αc

αg αg

Figure 1: Axisymmetric meniscus geometry in the tube growth by E.F.G. method.

The axi-symmetric differential equation of the inner free surface is given by

z ρ·g·z−pi

γ

1 z₂3/2

−1 r ·

1 z₂

·z forr∈

Rgi, ri , 1.4

which is the Euler equation for the energy functional

Iiz _ri

Rgi

γ·

1 z_21/2

1

2·ρ·g·z²−pi·z

·r·dr, z

Rgi

0, zri hi>0.

1.5

The state of the arts at the time 1993-1994, concerning the dependence of the shape of the meniscus free surface on the pressure difference p 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 to4, for the general differential equation1.2,1.4describing 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 axi-symmetric 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 die-top and meniscus. Analysis reveals the correlations among tube wall thickness, effective melt height, pull-rate, die-top 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 ofp, 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 in1,6, 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

z ρ·g·z−p_e γ

1

z_23/2

− 1 r ·

1 z₂

·z for R_giR_ge

2 ≤r ≤R_ge, 2.1

andαc, αgsuch that 0< αc< π/2−αg, αg∈0, π/2.

Definition 2.1. A solution z zx of the 2.1 describes the outer free surface of a static meniscus on the interval re, R_ge Rgi R_ge/2 < r_e < R_ge if possesses the following properties:

azre −tanπ/2−α_g, bzRge −tanα_c,and

czRge 0 andzris strictly decreasing onre, R_ge.

The described outer free surface is convex onre, R_geif in addition the following inequality holds:

dzr>0∀r∈re, Rge.

Theorem 2.2. If there exists a solution of2.1, which describes a convex outer free surface of a static meniscus on the closed intervalre, Rge, then the following inequalities hold:

−γ·π/2−

α_cα_g

R_ge−r_e ·cosαc γ

R_ge·sinαc

≤pe≤ −γ·π/2−

α_cα_g

R_ge−r_e ·sinαgρ·g·

Rge−re ·tanπ 2 −αg

γ

r_e ·cosαg. 2.2

Proof. Letzrbe a solution of2.1, which describes a convex outer free surface of a static meniscus on the closed intervalre, R_geandαr −arctanzr. The functionαrverifies the equation

αr p_e−ρ·g·zr

γ · 1

cosαr−1

r ·tanαr 2.3

and the boundary conditions

αre π

2 −α_g, α R_ge

α_c. 2.4

Hence, there existsr∈re, Rgesuch that the following equality holds:

pe −γ·π/2−

α_cα_g

R_ge−r_e ·cosα r

ρ·g·z r

γ

r ·sinα r

. 2.5

Sincezr > 0 on re, R_ge,zr is strictly increasing and αr −arctanzr is strictly decreasing onre, Rge, therefore the values ofαrare in the rangeαc, π/2−αgand the following inequalities hold:

αc≤α r

≤ π 2 −αg, sinαg≤cosα

r

≤cosαc, sinα_c≤sinα

r

≤cosα_g,

−tanπ 2 −αg

≤zr≤ −tanαc,

ρ·g·

R_ge−r

·tanα_c≤ρ·g·zr≤ρ·g·

R_ge−r

·tanπ 2 −α_g

.

2.6

Equality2.5and inequalities2.6imply inequalities2.2.

Corollary 2.3. Ifre Rge/nwith 1< n <2·Rge/RgiRge, then inequalities2.2become

−γ·π/2−

α_cα_g

R_ge · n

n−1 ·cosα_c γ

R_ge·sinα_c

≤p_e≤ −γ·π/2−

α_cα_g

R_ge · n

n−1 ·sinα_gρ·g·R_ge·n−1

n ·tanπ

2 −α_g γ

R_ge ·n·cosα_g.

2.7

Corollary 2.4. Ifn → 2·Rge/RgiRge, thenre → RgiRge/2 and2.7becomes

−2·γ·π/2−

α_cα_g

R_ge−R_gi ·cosα_c γ

R_ge·sinα_c

≤p_e ≤ −2·γ·π/2−

α_cα_g

R_ge−R_gi ·sinα_gρ·g·

R_ge−R_gi

2 ·tanπ

2 −α_g 2·γ

R_giR_ge ·cosα_g.

2.8

Ifn → 1, thenr_e → R_geandp_e → −∞.

Theorem 2.5. Letnbe such that 1< n <2·Rge/RgiRge. If p_esatisfies the inequality

p_e<−γ·π/2−

α_cα_g

R_ge · n

n−1 ·cosα_c γ

R_ge·sinα_c, 2.9

then there exists r_e in the closed intervalRge/n, R_ge such that the solution of the initial value problem

z ρ·g·z−p_e

γ ·

1 z2_3/2

−1 r ·

1 z²

·z for R_giR_ge

2 < r≤R_ge z

R_ge

0, z R_ge

−tanα_c, 0< α_c< π

2 −α_g, α_g∈ 0,π

2

2.10

on the intervalre, R_gedescribes the convex outer free surface of a static meniscus.

Proof. Consider the solutionzrof the initial value problem2.10. Denote byIthe maximal interval on which the function zr exists and by αr the function αr −arctanzr defined onI. Remark that forαrthe equality2.3holds.

Since z

R_ge

>0, z R_ge

−tanα_c<0, z R_ge

>−tanπ 2 −α_g

, 2.11

there existsr∈I, 0< r< R_gesuch that for anyr ∈r, R_gethe following inequalities hold:

zr>0, zr≤ −tanαc, zr≥ −tanπ 2 −αg

. 2.12

Letr_∗ be defined by r_∗ inf

r∈I |0< r< Rgesuch that for anyr∈

r, Rge inequalities2.12hold

. 2.13 It is clear thatr_∗≥0 and for anyr∈r_∗, R_geinequalities2.12hold.

From 2.12 and 2.13 it follows that zr is strictly increasing and bounded on r_∗, R_ge.Thereforezr_∗0 lim_{r→r∗, r>r∗}zrexists and satisfies

−tanπ 2 −α_g

≤zr∗0≤ −tanα_c. 2.14

Moreover, since zr is strictly decreasing and possesses bounded derivative on r∗, Rge, zr_∗0 limr→r∗, r>r∗zrexists too, it is finite, and satisfies

0<

Rge−r_∗

·tanαc≤zr∗0≤

Rge−r_∗

·tanπ 2 −αg

<∞. 2.15

We will show now thatr_∗ > Rge/nandzr∗0 −tanπ/2−αg. In order to show that r_∗ > Rge/n we assume the contrary, that is, that r_∗ ≤ Rge/n . Under this hypothesis we have

αr∗0−α Rge

−α r

· Rge−r_∗

−p_e

γ ρ·g·zr

γ sinαr

r

· R_ge−r_∗ cosαr

>

π/2−

α_cα_g

R_ge · n

n−1·cosα_c− 1

R_ge·sinα_cρ·g·zr

γ sinαr

r

· Rge−r_∗ cosαr

> π 2 −

αcαg

2.16

for somer ∈ r∗, R_ge. Henceαr∗0 > π/2−α_g.This last inequality is impossible, since according to the inequality2.14, we haveαr∗0 ≤ π/2−αg. Therefore,r_∗, defined by 2.14, satisfiesr_∗> R_ge/n.

In order to show thatzr∗0 −tanπ/2−α_gwe remark that from the definition 2.14ofr_∗it follows that at least one of the following three equalities holds:

zr_∗0 −tanα_c, zr_∗0 −tanπ 2 −α_g

, zr_∗0 0. 2.17

Sincezr_∗0< zr≤ −tanα_cfor anyr ∈r_∗, R_geit follows that the equalityzr_∗0

−tanα_cis impossible. Hence, we obtain that atr_∗at least one of the following two equalities holds:zr∗0 0,zr∗0 −tanπ/2−αg. We show now that the equalityzr∗0 0 is impossible. For that we assume the contrary, that is,zr_∗0 0. Under this hypothesis,

from2.12we have:

p_e g·ρ·zr_∗0 γ

r_∗ ·sinαr_∗0

> g·ρ·

R_ge−r_∗

·tanα_c γ

R_ge·sinα_c

> γ

R_ge ·sinα_c−γ·π/2−

α_cα_g

R_ge · n

n−1 ·cosα_c

> p_e

2.18

what is impossible.

In this way we obtain that the equalityzr_∗0 −tanπ/2−α_gholds.

For r_e r_∗ the solution of the initial value problem 2.8 on the interval re, R_ge describes a convex outer free surface of a static meniscus.

Corollary 2.6. If forp_ethe following inequality holds:

p_e<−2·γ·π/2−

α_cα_g

R_ge−R_gi ·cosα_c γ

R_ge·sinα_c, 2.19

then there existsr_e in the intervalRgiR_ge/2, Rgesuch that the solution of the initial value problem2.10on the intervalre, R_gedescribes a convex outer free surface of a static meniscus.

Corollary 2.7. If for 1< n< n <2·Rge/Rgi Rgethe following inequalities hold:

−γ·π/2− αcαg

Rge · n

n−1 ·sinα_gρ·g·R_ge·n−1

n ·tanπ 2 −α_g

γ

Rge·ncosα_g

< p_e<−γ·π/2− αcαg

Rge · n

n−1·cosα_c γ

Rge ·sinα_c,

2.20 then there existsr_ein the intervalRge/n, R_ge/nsuch that the solution of the initial value problem 2.10on the intervalre, Rgedescribes a convex outer free surface of a static meniscus. The existence ofre and the inequalityre ≥ Rge/nfollows fromTheorem 2.5. The inequalityre ≤ Rge/nfollows fromCorollary 2.3.

Remark 2.8. The solution of the initial value problem2.10is convex atRgei.e.,zRge>0 if and only if

p_e< γ

R_ge ·sinα_c. 2.21

That is becausezRge>0 if and only ifαRge −zRge·cos²αc<0, that is, pe

γ −sinα_c Rge

<0⇐⇒p_e< γ

Rge ·sinα_c. 2.22

Moreover, ifp_e < γ/R_ge·sinα_c, the solutionzrof the initial value problem2.8is convex everywherei.e.,zr > 0 forr ∈ I, 0 < r ≤ R_ge. That is because the change of convexity implies the existence ofr∈I, 0< r< Rgesuch thatαr> αc, zr>0 andpe ρ·g·zr γ/r·sinαr> γ/R_ge·sinα_c, what is impossible.

Theorem 2.9. If a solution z₁ z₁r of 2.1 describes a convex outer free surface of a static meniscus on the intervalre, Rge RgiRge/2 < re < Rge, then it is a weak minimum for the energy functional of the melt column:

Iez _Rge

re

γ·

1 z₂1/2

1

2·ρ·g·z²−pe·z

·r·dr zre z1re, z

Rge

z1

Rge

0.

2.23

Proof. Since 2.1is the Euler equation for2.23, it is sufficient to prove that the Legendre and Jacobi conditions are satisfied in this case.

Denote byFr, z, zthe function defined as

F r, z, z

r· 1

2 ·ρ·g·z²−p_e·zγ· 1

z2_1/2

. 2.24

It is easy to verify that we have

∂²F

∂z²

r·γ

1 z²3/2 >0. 2.25

Hence, the Legendre condition is satisfied.

The Jacobi equation ∂²F

∂z² − d dr

∂²F

∂z ∂z

·η− d dr

∂²F

∂z² ·η

0 2.26

in this case is given by

d dr

⎛

⎜⎝ r·γ

1 z²3/2 ·η

⎞

⎟⎠−ρ·g·r·η 0. 2.27

For2.27the following inequalities hold:

r·γ

1 z²_3/2 ≥r·γ·cos³π 2 −αg

r·γ·sin³αg, −ρ·g·r≤0. 2.28

Hence, the equation

η·r·γ·sin³αg

0 2.29

is a Sturm type upper bound for2.27 7.

Since every nonzero solution of2.29vanishes at most once on the intervalre, R_ge, the solutionηrof the initial value problem

d dr

⎛

⎜⎝ r·γ

1 z²_3/2 ·η

⎞

⎟⎠−ρ·g·r·η 0,

ηre 0, ηre 1

2.30

has only one zero on the intervalre, R_ge 7. Hence the Jacobi condition is satisfied.

Definition 2.10. A solutionz zrof2.1which 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 2.9shows that ifz zrdescribes a convex outer free surface of a static meniscus on the intervalre, Rge, then it is stable.

Theorem 2.12. If the solutionz zrof the initial value problem2.10is concave (i.e.,z”r<0) on the intervalre, Rge RgiRge/2< re< Rge, then it does not describe the outer free surface of a static meniscus onre, Rge.

Proof. zr<0 onre, R_geimplies thatzris strictly decreasing onre, R_ge. Hencezre>

zRge −tanαc>−tanπ/2−αg.

Theorem 2.13. Ifp_e > γ/R_ge·sinα_cand there existsr_e∈RgiRge/2, Rgesuch that the solution of the initial value problem2.10is the outer free surface of a static meniscus onre, R_ge, then forp_e the following inequalities hold:

γ

R_ge ·sinαc< pe ≤ρ·g· Rge−re

·tanπ 2 −αg

γ

r_e ·cosαg. 2.31 Proof. Denote by zr the solution of the initial value problem2.10which is assumed to represent the outer free surface of a static meniscus on the closed intervalre, R_ge. Letαr be defined as inTheorem 2.2. forr ∈ re, Rge. Sincepe > γ/Rge·sinαc, we havezRge

−1/cos²α_c·αRge <0. HenceαRge> 0 and thereforeαr< αRge α_cforr < R_ge,r

close toRge. Taking into account the fact thatαre π/2−αe> αcit follows that there exists r∈re, R_gesuch thatαr 0.

Thereforep_e ρ·g·zr γ/r·sinαr. Since 0≤αr≤π/2−α_gandr_e < r, the following inequality holds:γ/r·sin αr< γ/re·cosαg. On the other handzr< zre≤ Rge−re·tanπ/2−αg. Using the above evaluations we obtain inequalities2.31.

Remark 2.14. If re appearing in Theorem 2.13 is represented as re Rge/n, 1 < n <

2·R_ge/RgiR_ge, then inequality2.31becomes γ

R_ge ·sinα_c< p_e≤ρ·g·n−1

n ·R_ge·tanπ 2 −α_g

γ

R_ge·n·cosα_g. 2.32 Forn → 2·R_ge/RgiR_geinequality2.32becomes

γ

Rge ·sinαc< pe≤ρ·g·Rge−Rgi

2·Rge ·tanπ 2 −αg

2·γ

R_giR_ge·cosαg 2.33

Theorem 2.15. Letnbe 1< n <2·R_ge/RgiR_ge. If forp_ethe following inequality holds:

pe> ρ·g·Rge·n−1

n ·tanαc n

R_ge ·γ, 2.34

then the solutionzrof the initial value problem2.10is concave on the intervalI∩Rge/n, Rge whereIis the maximal interval of the existence ofzr.

Proof. Considerαr −arctanzrand remark that forr ∈ I∩Rge/n, R_gethe following relations hold:

αr 1

cosαr ·pe

γ −ρ·g·zr

γ −sinαr

r

≥ 1

cosαr ·

ρ·g·Rgen−1

γ·n ·tanαc n

Rge −ρ·g·Rgen−1

γ·n ·tanαc− n Rge

≥0.

2.35

Hence:zr −1/cos²αr·αr≤0 forr∈I∩Rge/n, R_ge.

3. Meniscus Inner Free Surface Shape Analysis in the Case of Tube Growth

Consider now the differential equation

z ρ·g·z−pi

γ

1 z₂3/2

−1 r ·

1 z₂

·z forr∈

Rgi,RgiRge

2

, 3.1

andα_c, α_gsuch that 0< α_c< π/2−α_g,α_g∈0, π/2.

Definition 3.1. A solutionz zxof3.1describes the inner free surface of a static meniscus on the intervalRgi, r_i Rgi< r_i<RgiR_ge/2if possesses the following properties:

azRgi tanαc,

bzri tanπ/2−αg,and

czRgi 0 andzris strictly increasing onRgi, r_i.

The described inner free surface is convex onRgi, r_iif in addition the following inequality holds:

dzr>0∀r∈Rgi, r_i.

Theorem 3.2. If there exists a solution of3.1, which describes a convex inner free surface of a static meniscus on the closed intervalRgi, r_i, then the following inequalities hold:

−γ·π/2−

α_cα_g

r_i−R_gi ·cosα_c− γ

R_gi·cosα_g

≤pi≤ −γ·π/2− αcαg

r_i−R_gi ·sinαgρ·g·

ri−Rgi ·tanπ 2 −αg

− γ

r_i ·sinαc. 3.2

Proof. Letzrbe a solution of3.1, which describes a convex inner free surface of a static meniscus on the closed intervalRgi, r_iandαr arctanzr. The functionαrverifies the equation

αr ρ·g·zr−pi

γ · 1

cosαr−1

r ·tanαr 3.3

and the boundary conditions α

Rgi

αc, αri π

2 −αg. 3.4

Hence, there existsr∈Rgi, risuch that the following equality holds:

p_i −γ·π/2−

α_cα_g

r_i−R_gi ·cosα r

ρ·g·z r

− γ

r ·sinα r

. 3.5

Since zr > 0 onRgi, r_i, zr is strictly increasing and αr arctanzr is strictly increasing onRgi, ri, therefore the following inequalities hold:

α_c≤α r

≤ π 2 −α_g, sinα_g≤cosα

r

≤cosα_c, sinαc≤sinα

r

≤cosαg, ρ·g·

r−Rgi

·tanαc≤ρ·g·z r

≤ρ·g· r−Rgi

·tanπ 2 −αg

.

3.6

Equality3.5and inequalities3.6imply inequalities3.2.

Corollary 3.3. Ifri m·Rgiwith 1< m <RgiRge/2·Rgi, then inequalities3.2become

−γ·π/2−

α_cα_g

m−1·R_gi ·cosαc− γ

R_gi·cosαg

≤pi≤ −γ·π/2−

α_cα_g

m−1·R_gi ·sinαgρ·g·Rgi·m−1·tanπ 2 −αg

− γ

m·R_gi ·sinαc. 3.7 Corollary 3.4. Ifm → R_giRge/2·Rge, thenri → RgiRge/2 and3.7becomes

−2·γ·π/2− αcαg

Rge−Rgi ·cosα_c− γ

Rgi ·cosα_g

≤p_i≤ −2·γ·π/2− αcαg

Rge−Rgi ·sinα_gρ·g·

Rge−Rgi

2 ·tanπ

2 −α_g

− 2·γ

RgiRge·sinα_c.

3.8

Ifm → 1, thenr_i → R_giandp_i → −∞.

Theorem 3.5. Let m be such that 1< m <RgiRge/2·Rgi. Ifpisatisfies the inequality

pi<−γ·π/2− αcαg

m−1·R_gi ·cosαc γ

R_gi ·cosαg, 3.9

then there existsr_iin the closed intervalRgi, m·Rgisuch that the solution of the initial value problem

z ρ·g·z−pi

γ ·

1 z_23/2

−1 r ·

1 z₂

·z forRgi< r ≤ RgiRge

2 ,

z Rgi

0, z Rgi

tanαc, 0< αc< π

2 −αg, αg ∈ 0,π

2

3.10

on the intervalRgi, r_idescribes the convex inner free surface of a static meniscus.

Proof. Consider the solutionzrof the initial value problem3.10. Denote byIthe maximal interval on which the functionzrexists and byαrthe functionαr arctanzrdefined onI. Remark that forαrthe following equality holds:

αr 1 cosαr·

ρ·g·zr−p_i

γ −sinαr

r

. 3.11

Since

z R_gi

>0, z R_gi

tanα_c>0, z R_gi

<tanπ 2 −α_g

, 3.12

there exists r ∈ I,R_gi < r < RgiR_ge/2 such that for any r ∈ r, R_githe following inequalities hold:

zr>0, zr>0, zr<tanπ 2 −αg

. 3.13

Letr^∗ be defined by r^∗ sup

r∈I|R_gi< r< R_giRge

2 such that for anyr∈

R_gi, r inequalities 3.13hold

.

3.14 It is clear thatr^∗≤RgiR_ge/2 and for anyr ∈Rgi, r^∗inequalities3.13hold. Moreover, zr^∗−0 limr→r^∗, r<r^∗zrexists and satisfies,zr^∗−0>0 andzr^∗−0≤tanπ/2−αg. Hencezr^∗−0 limr→r^∗, r<r^∗zris finite, it is strictly positive, and for everyr ∈Rgi, r^∗the following inequalities hold:

tanαc≤zr^∗−0≤tanπ 2 −αg

, r^∗−R_gi

·tanα_c≤zr^∗−0≤

r^∗−R_gi

·tanπ 2 −α_g

.

3.15

We will show now thatr^∗≤m·R_giandzr^∗−0 tanπ/2−α_g.

In order to show thatr^∗≤m·R_gi, we assume the contrary, that is,r^∗ > m·R_gi. Under this hypothesis we have

αr^∗−0−α R_gi α

r

· r^∗−Rgi

1

cosαr·

−p_i

γ ρ·g·zr

γ −sinαr r

· r^∗−Rgi

> r^∗−R_gi cosαr·

π/2−

α_cα_g

R_gi · 1

m−1·cosαc 1

R_gi·cosαgρ·g·zr

γ −sinαr r

> π 2 −

α_cα_g

3.16 for somer ∈ Rgi, r^∗. Henceαr^∗−0 > π/2−α_g and it follows that there existsr₁ such thatRgi < r1 < r^∗ andαr1 π/2−αg. This last inequality is impossible according to the definition ofr^∗.

Therefore,r^∗defined by3.14satisfiesr^∗< m·Rgi.

In order to show thatzr^∗−0 tanπ/2−α_gwe remark that from the definition 3.14ofr^∗it follows that at least one of the following three equalities holds:

zr^∗−0 tanα_c, zr^∗−0 tanπ 2 −α_g

, zr^∗−0 0. 3.17

Sincezr^∗−0> zr>tanα_cforr∈Rgi, r^∗, it follows that the equalityzr^∗−0 tanα_c is impossible.

Hence we obtain that inr^∗at least one of the following two equalities holds:

zr^∗−0 0, zr^∗−0 tanπ 2 −αg

. 3.18

We show now that the equalityzr^∗−0 0 is impossible. For this we assume the contrary, that is,zr^∗−0 0. Under this hypothesis, from3.11, we have

p_i ρ·g·zr^∗−0− γ

r^∗ ·sin αr^∗−0

>− γ

R_gi·cosα_g

>− γ

R_gi·cosα_g−γ· 1 m−1 ·

π/2− α_cα_g

R_gi ·cosα_c

> p_i,

3.19

what is impossible.

In this way we obtain that the equalityzr^∗−0 tanπ/2−αgholds.

For r_i r^∗ the solution of the initial value problem 3.10 on the interval Rgi, r_i describes a convex inner free surface of a static meniscus.

Corollary 3.6. If forpithe following inequality holds

pe<−2·γ·π/2−

α_cα_g

R_ge−R_gi ·cosαc− γ

R_gi ·cosαg, 3.20

then there existsri in the interval Rgi,RgiRge/2such that the solution of the initial value problem3.10on the intervalRgi, r_idescribes a convex inner free surface of a static meniscus.

Corollary 3.7. If for 1< m< m <RgiRge/2·Rgithe following inequalities hold:

−γ·π/2−

α_cα_g

m−1·R_gi ·sinα_gρ·g·R_gi· m−1

·tanπ 2 −α_g

− γ

m·R_gi·sinα_c

< p_i<−γ·π/2−

α_cα_g

m−1·R_gi ·cosα_c− γ

R_gi ·cosα_g,

3.21

then there existsriin the intervalm·Rgi, m·Rgisuch that the solution of the initial value problem 3.10on the intervalRgi, r_idescribes a convex inner free surface of a static meniscus.

The existence of r_i and the inequality r_i ≤ m·R_gi follows from Theorem 3.5. The inequalityri≥m·Rgifollows from theCorollary 3.3.

Theorem 3.8. If a solution z₁ z₁r of 3.1describes a convex inner free surface of a static meniscus on the intervalRgi, r_i ri ∈ Rgi,RgiR_ge/2, then it is a weak minimum for the energy functional of the melt column:

I_iz _ri

Rgi

γ·

1 z₂1/2

1

2·ρ·g·z²−p·z

·r·dr, z

Rgi

z1

Rgi

0, zri z1ri.

3.22

Proof. It is similar to the proof ofTheorem 2.9.

Definition 3.9. A solutionz zrof3.1which 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 3.8shows that ifz zrdescribes a convex inner free surface of a static meniscus on the intervalRgi, ri, then it is stable.

Remark 3.11. The solution of the initial value problem3.10is convex atR_gii.e.,zRgi>0 if and only if

p_i<− γ

R_gi·sinα_c. 3.23

Theorem 3.12. Ifzrrepresents the inner free surface of a static meniscus on the closed interval Rgi, ri ri<RgiRge/2which possesses the following properties:

azris convex atRgi, and

bthe shape ofzrchanges once on the intervalRgi, ri, that is, there exists a point r ∈ Rgi, risuch thatzr>0 forr ∈Rgi, ri, zr 0 andzr<0 forr ∈r, ri, then there existsr_i¹ ∈Rgi, risuch thatzr_i¹ tanπ/2−αgand forpi the following inequality holds:

− γ Rgi

< p_i<− γ

Rgi ·sinα_c. 3.24

Proof. Since αr arctanzr increases on Rgi, r and decreases on r, ri, and αri π/2−α_g> α_c, there existsr_i¹∈Rgi, r_isuch thatαr_i¹ π/2−α_g. The maximum valueαr is less thanπ/2. From3.11we have

pi ρ·g·z r

− γ

r ·sinα r

, 3.25