W.E. Langlois

Diffusion of gas from a liquid into an expanding bubble

The growth of a bubble within a volume of isothermal viscous liquid containing uniformly distributed dissolved gas is considered. The problem of characterizing this growth-by-mass-transfer is reduced fo an integro-differential equation for the bubble radius as a function of time, and a computer solution is obtained. The initial and final stages of growth are treated analytically.

Czochralski crystal growth in an axial magnetic field: effects of Joule heating

Abstract The feasibility of using magnetohydrodynamic effects to improve the quality of silicon crystals obtained by Czochralski growth from the melt has been demonstrated by several experimenters. An earlier publication pointed out that certain approximations widely used in liquid metal MHD make it relatively easy to incorporate magnetomotive effects into numerical models for Czochralski flow. Specifically, the low value of the magnetic Reynolds number gives rise to two useful simplifications: the induced magnetic field is negligible, and the electric field is irrotational. These lead to the result that the magnetic part of the calculation is diagnostic rather than prognostic. The earlier paper simulated the effect of a strong axial field applied to the melt, under the assumption that Joule heating is negligible. The validity of that assumption is confirmed here: except under extremely high fields, Joule heating represents only a miniscule part of the energy budget in magnetic Czochralski growth. Its effect upon the melt circulation can be ignored.

Effects of the finite electrical conductivity of the crystal on hydromagnetic Czochralski flow

Abstract There has recently been much interest in using hydromagnetic damping to reduce the turmoil of Czochralski silicon melts. Since molten silicon conducts electricity nearly as well as mercury, applied magnetic fields in the range of a few kilogauss have a pronounced effect on the melt motion. Digital simulations of magnetic Czochralski flow have taken the solid crystal to be a perfect electrical insulator. However, asymptotic considerations have shown this to be questionable: even though the electrical conductivity of solid silicon is only 3% that of the melt, current leakage into the crystal can significantly modify the melt flow. Numerical studies described here confirm quantitatively the predictions of the asymptotic theory. Accounting for crystal conductivity in the computational model leads to the expected major changes in the azimuthal motion. The meridional circulation and, concomitantly, the dopant transport are only slightly modified for the rotation rates considered.

Melt motion in a Czochralski puller with a weak transverse magnetic field

Abstract The melt motion in a Czochralski silicon crystal puller with a uniform transverse magnetic field is intrinsically three-dimensional. When the magnetic field is relatively weak, the motion consists of an axisymmetric base solution, plus two perturbations which behave as sin 2θ and cos 2θ, neglecting terms which are proportional to the fourth power of the magnetic field strength. A 700 G magnetic field stabilizes a melt motion which is unsteady without a magnetic field. As the strength of the magnetic field increases, the magnitudes of the axial and radial velocities increase to maxima at some field strength above 1000 G. This contrasts with an axial magnetic field for which the axial and radial velocities always decrease as the field strength increases. We estimate that 1500 G is the maximum field strength for which our weak-field asymptotic solution is valid.

Computer simulation of Czochralski melt convection in a magnetic field

Abstract The practical value of growing silicon crystals in a large magnetic field has recently been demonstrated, and computational investigations of the hydromagnetic melt flow have begun to appear. These investigations, and some current work, are reviewed here.

Hydromagnetic flows and effects on Czochralski silicon crystals

Abstract The most important method of producing single crystals in large quantities is Czochralski growth. Fluid motion within the melt affects the transport of heat and solutes to the growth interface, and has therefore been the subject of much study. Some Czochralski melts, such as silicon, are excellent conductors of electricity, so that the flow can be modified hydromagnetically. Flow under the influence of an axial magnetic field has received the most attention because the configuration remains rotationally symmetric. In the case of a uniform field, the theory has been extensively developed, using both asymptotic and numerical methods. The advantage of quiescent flow is offset by an unfavorable radial distribution of solutes in the finished crystal. Consequently, alternate configurations are being explored. A transverse field destroys the axial symmetry, but can sometimes be investigated by a combination of asymptotic and numerical methods. A non-uniform axial field is another possibility that may offer advantages. This can be investigated with minor modifications of the theory for uniform fields.

The lightly loaded foil bearing at zero angle of wrap

A method is developed for determining, to the first order, the deflection from a straight path of a perfectly flexible tape moving near a rigid cylinder. The case of a parabolic cylinder is considered.

Effect of co-rotation and counter-rotation on suppression of melt convection in magnetic Czochralski growth

Abstract Melt flow phenomena of magnetic Czochralski growth are simulated numerically for a 5 inch diameter crystal grown from a 14 inch diameter crucible. It is found that, in conjunction with an axial magnetic field, co-rotation of the crystal with the crucible suppresses meridional circulation more thoroughly than counter-rotation.