Abstract—Nonlinear boundary value problem of the YoungLaplace equation which describes the meniscus free surface in semiconductor crystals grown by Dewetted Bridgman technique is considered. The statically stability of the menisci, via the conjugate point criterion of the calculus of variations, is investigated in the cases of the classical semiconductors grown in (i) uncoated crucibles (i.e., the wetting angle θ c and growth angle α e satisfy the inequality θ c +α e <180°), and (ii) coated crucibles or pollution (θ c +α e ≥ 180°). Necessary or sufficient conditions for the existence of the statically stable convex (or concave, convexconcave, concaveconvex) solutions of the considered BVP are established. Nonlinear boundary value problem, YoungLaplace equation, growth from the melt, dewetted Bridgman crystal growth technique. I. INTRODUCTION major problem to which crystal growth researchers have been confronted was the development of techniques capable to monitor and control the external shape of meltgrown crystals, and simultaneously to improve the crystal structures. In the crystal growth processes based on the principle of capillary shaping (Czochralski, Floating zone, Edgedefined filmfed growth, Dewetted Bridgman techniques, etc.), the shape and the dimensions of the crystal are determined by the liquid meniscus and by the heat transfer at the meltcrystal interface. Historically, the physical origin and the shape of a liquid meniscus have been among the first phenomena studied in capillarity, in particular by Hauksbee (1709) [1], as cited by Maxwell [2] in his introduction to the Capillary Action written for the Encyclopaedia Britannica: „the first accurate observations of the capillary action of tubes and glass plates were made by Hauksbee. He ascribes the action to an attraction between the glass and the liquid". Manuscript received October 20, 2009: Revised version received October 24, 2009. This work was supported by the North Atlantic Treaty Organisation under Grant CBP.EAP.CLG 982530/20072009 and Romanian National University Research Council under Grant PN II 131/20092011. L. Braescu is with the Computer Science Department, West University of Timisoara, Blv. V Parvan 4, Timisoara 300223, ROMANIA (corresponding author phone: +40256592221; fax: +40256592316; email: lilianabraescu@ balint1.math.uvt.ro). The first formal analytical expression was given by Laplace [3], after introduction of the mean curvature κ defined as average (arithmetic mean) of the principal curvatures [4]: 1 2 1 1 1 2 . R R κ = + (1) Laplace showed that the mean curvature of the free surface is proportional to the pressure change across the surface. In crystal growth processes, the proportionality coefficient contains the surface tension γ and the pressure change across the surface (the pressure of the external gas on the melt p v ; the internal pressure applied on the liquid, which can generally be defined at the origin, p O ; the hydrostatic pressure gz l ρ ; the pressure determined by the centrifugal force due to a possible liquid rotation ( ) 2 2 2 2 l l x y / ρ + where l is the angular velocity of the liquid; and when the magnetic fields is used, the Maxwell pressure which is proportional to the square of the magnetic induction ( ) 2 2 B x,y / ) [5]. Thus, the following equality known as YoungLaplace’s equation must hold: ( ) ( ) 2 2 2 2 1 2 1 1 1 2 2 O v l l l B x,y p p gz x y . R R ρ ρ γ − − + + − + = (2) Denoting the meniscus surface by z(x,y), it is known from differential geometry, that the mean curvature is expressed as: ( ) 2 2 2 I I I II I II I II I F G E E G F F G E − + − = κ (3) where 2 1 ∂ ∂ + = x z E I , y z x z F I ∂ ∂ ⋅ ∂ ∂ = , 2 1 ∂ ∂ + = y z G I represent the coefficients of the first fundamental form of the surface, and 2 2 2 2 1 II z x E , z z x y ∂ ∂ = ∂ ∂ + + ∂ ∂ Nonlinear boundary value problem of the meniscus for the dewetted Bridgman crystal growth process L. Braescu A INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Issue 1, Volume 4, 2010 42