I started with a ship full of max-level veterans, and I'm just a complete noob.

Consciousness continued to sink along the golden edge of the Tongtian Record. The rotational angular velocity on the parchment gradually disappeared, replaced by an old laboratory with a water tank, a disc, and a velocimeter.

The drive shaft along the wall passes through the bracket, driving the metal disc immersed in the liquid to rotate. The dyeing liquid near the disc spreads outward, while the liquid further away flows in along the axis, forming curved flow lines.

A scholar in a dark, worn suit stood by the sink, his palm against the transmission seat. Only after the disc's rotation speed stabilized did he turn to look at the single-crystal growth record that Jiang Ming had spread out.

"You think of the seed crystal as a rotating thin rod, so you always try to explain the changes in the melt by the stirring speed. But what really controls the interfacial transport is the bottom surface of the seed crystal."

Jiang Ming walked to the water tank and watched the dyeing liquid flow from the center of the disc to the edge. He then drew the relative positions of the crucible, seed crystal, and solid-liquid interface on the glass plate.

"The seed crystal has a finite base radius, and its edge is connected to a meniscus. It differs from an infinitely rotating plane. Can this solution still be used directly?"

"When the radius is much larger than the boundary layer thickness, the central region can be approximated by the rotating disk solution, and the edge effect needs to be left in the experimental correction."

The pioneers picked up chalk and wrote down the angular velocity, kinematic viscosity, and boundary layer thickness on a glass plate. They then listed the radial velocity, circumferential velocity, and axial replenishment near the disk as three components.

The rotating disk drags the liquid close to the disk surface. After the liquid gains circumferential velocity, it flows outward due to centrifugal force. In order to make up for the flow, the melt above the disk surface can only be pressed axially towards the solid-liquid interface.

"Theodore von Kármán addressed this problem in 1921. He used similar variables to transform the axisymmetric partial differential equation into an ordinary differential equation, and the scale of the velocity boundary layer is determined by this ratio."

The chalk fell on the glass plate, writing that δ is proportional to the square root of ν divided by ω. Jiang Ming stared at the formula, converting twelve per minute into angular velocity, and then substituting the kinematic viscosity of the germanium melt into it.

"As the rotational speed increases, the velocity boundary layer narrows, the axial flow is enhanced, and the concentration layer near the interface also becomes thinner."

"You're on the right track, but the velocity boundary layer and the concentration boundary layer are still separated by the Schmidt number. What you need is impurity diffusion, not just momentum transfer."

The pioneer erased an intermediate section, introduced the diffusion coefficient D into a similar variable, and finally wrote that the concentration boundary layer thickness is related to one-third of D and the square root of ν divided by ω.

Jiang Ming connected this relationship to the segregation model left over from last night, and the blank in the formula finally had a quantity that could be controlled by the equipment parameters.

"The equilibrium segregation coefficient k₀ is determined by the properties of the impurities and germanium. The pulling speed V and diffusion coefficient D can also be obtained from the process and literature. As long as δ can be calculated, the effective segregation coefficient can be found in the operating parameters."

Pioneer rearranged the formula written by Jiang Ming, keeping k₀ in the denominator, and multiplied one minus k₀ by the exponent of negative Vδ divided by D.

keff equals k₀ divided by k₀ plus one minus k₀ multiplied by exp(negative Vδ) divided by D.

When the concentration boundary layer is thick, impurities tend to accumulate in front of the interface, and the proportion of impurities actually accepted by the solid phase will deviate from the equilibrium state, and the resistivity difference of the single crystal along the length direction will also increase.

As the rotational speed increases, δ decreases with the reciprocal of the square root of the angular velocity. Impurities accumulated near the interface are carried away by the radial flow, and keff gradually approaches k₀.

Jiang Ming wrote the parameters of the first qualified single crystal next to the formula: the pulling speed was 0.8 millimeters per minute, the rotation speed was 12 revolutions per minute, and the measured resistivity at the head and tail differed by about 30%.

"If the rotation speed is doubled, the reduction in the boundary layer is limited. The extent to which the difference between the beginning and end can be reduced depends on the diffusion coefficients of phosphorus, arsenic, and antimony."

"Calculate them separately, don't combine several impurities into an average. Resistivity measurement only gives you the overall result, and the specific impurity composition still needs further analysis."

Jiang Ming nodded, divided the three types of common impurities into three rows, and listed the equilibrium segregation coefficient, diffusion coefficient and expected contribution range after each row.

Pioneer raised his hand and turned the speed control wheel next to the water tank. The speed of the disc gradually increased, and the originally smooth dyeing lines began to sway, with tiny vortices forming near the edge of the disc.

"You also need to set an upper limit on the rotational speed. The laminar solution only holds true within a certain Reynolds number range. Beyond the critical region, the interface temperature and concentration will change with the eddy current."

Jiang Ming immediately wrote down the rotating Reynolds number, Re, which is equal to ω multiplied by the square of R and then divided by ν, where R is the effective radius of the current seed crystal and the solid-liquid interface.

"Can the critical value be given directly?"

"The experimental setup, the free surface, and the crucible sidewalls can all alter the critical range. You can only estimate it based on the current dimensions and the viscosity of the germanium melt, and then confirm it through crystal pulling experiments."

The pioneer wrote the range of experience next to the formula, and then pointed with his finger to the first failure section recorded by Jiang Ming. That section of milky white grains corresponded to an interface change caused by a mechanical jam.

"If you rotate too slowly, the concentration boundary layer will be thick and the segregation deviation will be large. If you rotate too fast, the flow instability will damage the solid-liquid interface. What you need to find is the process window between the two limits."

The laboratory's water tank and disc began to fade. Jiang Ming seized the last moments to copy the finite radius correction, the Schmidt number relation, and the critical Reynolds number into his kraft paper notebook.

When consciousness returned to the secure compartment, the wick in the kerosene lamp had burned out, the pencil on the table was still pressed next to δ, and the space to be investigated next to the diffusion coefficient D left by Fang Xudong was still empty.

Jiang Ming sharpened his pencil again, wrote the Keff formula in its entirety in the blank space, and then found the equilibrium condensation coefficients of phosphorus, arsenic, and antimony from the literature excerpts brought by Lin Lanying.

The diffusion coefficients are temporarily based on temperature conversion values ​​from Soviet journal abstracts, with the temperature conditions indicated next to each set of numbers to avoid directly mixing data from different melt temperatures.

He first substituted the process parameters of twelve revolutions per minute and calculated that the effective phosphorus segregation coefficient was close to 0.08, which could be connected with the direction of change of the second single crystal.

After the rotation speed is increased to 25 revolutions per minute, the concentration boundary layer narrows significantly, the keff crosses 0.11, and the resistivity difference between the head and tail is expected to be reduced to about 17%.

When the rotation speed reaches 28 to 30 revolutions per minute, the keff of phosphorus approaches 0.12. Based on the existing crystal length and charge weight, the difference between the beginning and end will fall within 15%.

Jiang Ming continued to push the angular velocity upwards, and the rotational Reynolds number quickly approached the empirical critical region. Currently, the seed crystal radius, crucible size, and germanium melt viscosity together held the upper limit.

Thirty-five revolutions per minute is already close to the boundary. Going any higher would push the laminar flow model into the failure zone. Any temperature fluctuations caused by turbulence could re-induce polycrystalline nuclei.

He set up three axes on the graph paper: rotational speed, pulling speed, and effective segregation coefficient, and marked the calculation results of 12 to 35 revolutions and pulling speed from 0.6 to 1 millimeter point by point.

The surface gradually flattens after 25 revolutions, and the benefits of continuing to increase the speed begin to decrease, while getting closer and closer to the critical Reynolds number.

Jiang Ming eventually achieved a rotation speed of 28 revolutions per minute while maintaining a lifting speed of 0.8 millimeters per minute. This set of parameters can both compress the concentration boundary layer and leave room for mechanical fluctuations and viscosity errors.

The first page describes the theoretical derivation, from the velocity boundary layer of the rotating disk to the concentration boundary layer, and then to the relationship between the effective segregation coefficient of BPS.

The second page only contains the operating parameters, expected results, and stopping conditions. If polygonal ridges appear on the meniscus, or if temperature fluctuations and rotation speed form a fixed cycle, immediately reduce the rotation speed and stop lifting.

As for the source of the derivation, he noted in the footnote the chapter on rotating fluids in Qian Xuesen's lecture notes on mechanics, and added the reverse derivation process of the zone purification and the second single crystal data.

The sound of a key hitting a metal ring came from outside the door. Fang Xudong pushed the door open and came in, carrying the notebook he had left in the cubicle the day before.

He walked to the blackboard, followed the flow lines of the rotating disk to see the Keff curve, and stopped next to the diffusion coefficient D, picking up the chalk to add a question mark.

Is this value mentioned in the literature?

"The Soviet Union has a set of measured values ​​for the diffusion coefficient of phosphorus in germanium melt. I asked Da Liu to check the journal in the archives."

Fang Xudong nodded, changed the question mark to a check mark, and copied the temperature conditions on the blackboard into the notebook so that Da Liu could exclude data that did not meet the conditions when checking the journal.

"Twenty-eight revolutions, more than twice as high as the second furnace, the transmission mechanism needs to be re-recorded for idle rotation."

"First, test for fifteen minutes and record the gear cycle, axial movement, and motor temperature rise. Thirty-five revolutions is the upper limit for calculation, and the equipment knobs also need to be limited."

Fang Xudong turned to the blank page, circled numbers twenty-eight and thirty-five respectively, and then pulled away the records of furnace number three that were bound on the corner of the table.

From the other end of the corridor, one could hear Liu pushing a document cart. The wheels rolled over the cement seams, and several Soviet periodicals were crushed under the canvas straps.

Fang Xudong stepped aside to clear the doorway and glanced back at the two pages of derivation.

"Wang Shouwu will be here tomorrow. If he asks you about this formula, you'll have to write it out for him from the beginning."

Jiang Ming put the two pages of paper into his canvas bag, picked up Qian Xuesen's black lecture notes, and followed the document cart to the outer room.