Scott Beckman
January 2003

University of California at Berkeley
Department of Materials Science and Engineering
577 Evans Hall #1760
Berkeley, CA 94720
sbeckman@uclink.berkeley.edu

Introduction

The basic idea of melt zone refining is that you have a large bar of material with impurities.  You then past a liquid region down the length of the bar.  Because the impurities have a higher solubility in liquid than in the solid, as the liquid region moves the length of the bar the impurities become trapped in the liquid and are moved toward the end of the bar.  By repeatedly passing this narrow melted region the impurities are eventually pulled to one end of the bar.  After enough passes a final distribution is reached.  The shape of this region is exponential and the values depend on the initial concentration in the bar, the width of the melt region (relative to the total bar length), and the segregation coefficient.  Below I step through how to calculate the defect distribution during this process.  

Parameters

Initial, uniform, concentration of defects in bar, in units .  

Segregation coefficient,

Geometry of bar: length=100cm, area=π

Heater length, 3cm

Number of subdivisions of total bar length, this needs to be large enough that the number of steps that are liquid (meltsteps) are small compared to the total number of steps (nsteps)

Number of subdivisions in melt region.  The heater length must be chosen such that the meltsteps parameter is an integer.

Volume per step

Basis of Calculation

Zone Melt Purification

Define the initial concentration

The heater will step along the bar, one step at a time starting at step=1.  

The number of impurities in the melted region are given by the sum of the impurities in each melted region.  The impurity in a region is the concentration of impurities in the region times the volume of the region.  

The concentration in the liquid region is the number of impurities in the liquid region divided by the volume of the liquid region

Write this to the concentration profile

When the liquid freezes impurities will be ejected into the liquid.  This will increase the concentration of impurities in the liquid.  

The number of impurities absorbed into the solid is given by the concentration in the solid times the volume of the newly frozen region

The number of impurities in the diminished liquid region is the number of initial number of impurities minus those absorbed by the solid

The concentration in the liquid is the number of impurities divided the smaller liquid region.

Write this to the concentration profile

The liquid region is smaller than it is defined to be.  The first event in the next step involves melting the next subregion to the right of the bar.  Because the bar has a lower concentration of defects than the liquid this melt will act to dilute the liquid.

Repeat the above steps until the heater reaches the edge of the bar.  Once at the edge of the bar the normal freezing occurs

Normal Freeze

Normal freezing occurs at the end of the bar.  In normal freezing the solidification front moves forward one step at a time, ejecting solid into the liquid.  No new solid is melted to dilute the liquid so as the final liquid solidifies the concentration goes to infinity.  Since this bar is discrete, the final step is finite, but none the less, the curvature shows a trend toward infinity.

The impurity in the liquid is given by

The concentration in the liquid is then

The concentration of the frozen solid will be

and will have a total number of impurities

The new number of impurities in the liquid is

giving a concentration of

Record these numbers

Repeat until there are no longer steps to freeze except the last step

Here we see the concentration of defects after a single melt zone pass.  

Results

Zone melt can be preformed multiple times, each pass nudging the bar closer to the ultimate distribution predicted by Pfeund.  Here we shall show the distribution of defects as a function of pass number.  Note that unlike the above plots, the results below show the log of the concentration along the bar length.

Initial Concentration Distribution

Pass 0

Pass 1

Pass 5

Pass 10

Pass 50

Pass 100

Pass 200

Discussion

Here I want to comment on the validity of the method used above.  While one can imagine that other more orthodox numerical or even symbolic methods are available they will be tricky to employ because of the transition from zone melt purification to normal freezing at the far right boundary.  I have reasonable confidence in the method used here because stepping through yields results that match intuition.  In addition I calculated the net defect count as a function of pass.  Because there are no sources or sinks for defects the total number must be preserved.  I found that the method above results in no net gain or loss of defects (to within the numerical accuracy of Mathematica).