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

The chalk spread out to both sides along the solid-liquid interface. Jiang Ming marked the narrow band as the concentration boundary layer δ, and wrote the pulling speed V and rotational angular velocity ω next to the seed crystal.

"The equilibrium segregation coefficient can only indicate the concentration ratio of impurities between the solid and liquid phases when the interface reaches equilibrium. In actual crystal pulling, there is still a transport process involved."

He wrote keff on the right side of the diagram, with the numerator being k₀ and the denominator consisting of k₀ and a term minus k₀ multiplied by an exponent.

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

Wang Shouwu paused with his pen resting on the paper, then flipped through his notebook and found a page of English paper abstracts that he had copied two years ago.

The top of the page lists the names of Burton, Prim, and Slichter, and the year of publication is 1953. Below that, the relationship between the effective segregation coefficient and the concentration boundary layer is also listed.

The abstract ends with a Chinese annotation stating that there is a lack of reliable measurement methods for the δ value, and that existing equipment conditions make it difficult to convert the model into process parameters.

Wang Shouwu looked up at the blackboard again. Jiang Ming had already broken down the exponential terms and explained the roles of the lifting speed, diffusion coefficient, and boundary layer thickness.

"V can be controlled by the lifting mechanism, and D can be obtained from the literature according to the type of impurity. What really hinders the calculation is δ, which changes with the melt flow."

"How do you plan to ask for δ?"

After asking his question, Wang Shouwu turned to a new page in his notebook and put his pen down again.

Jiang Ming drew a horizontal disc next to the crucible diagram. The arrows on the disc rotated circumferentially, while the streamlines close to the disc spread outward from the center.

"The bottom surface of the seed crystal can be approximated as a finite radius rotating disk immersed in germanium melt. The central region is first solved by von Kármán rotating disk laminar flow, and the edge effects are left to experimental correction."

The chalk starts with the continuity equation under axisymmetric conditions, and then lists the three velocity components: radial, circumferential, and axial.

The rotating disk drags the nearby melt to obtain circumferential velocity, the radial outflow carries the liquid near the interface to the edge, and the axial replenishment sends the melt above to the solid-liquid interface.

Jiang Ming recombined the radial and axial coordinates and introduced similar variables that vary with the square root of ω divided by ν, thus transforming the original partial differential equations into ordinary differential equations.

Huang Kun leaned against the recording table, his gaze following the similar variables. He was familiar with lattice and solid theory, and only recorded the conclusions and measurable parameters for this fluid derivation.

The transformation process on the blackboard continued downwards, and Jiang Ming's chalk always followed the same derivation path, only moving to the right side after filling half the blackboard.

The scale of the velocity boundary layer lies on the square root of ν divided by ω, and momentum diffusion is linked to impurity diffusion through the Schmidt number.

The final concentration boundary layer relationship is written next to the Keff formula, where δ equals 1.6 times D raised to the power of one-third, then multiplied by ν and divided by the square root of ω.

Wang Shouwu drew a horizontal line under the words "The Von Kármán Rotating Disk" in his notebook, and then copied the entire delta relationship next to the abstract of the previous paper.

After two years of work, he finally connected an adjustable rotational speed to a classical fluid dynamics model, thus gaining an operational entry point for the BPS relationship, which was originally only used to explain experimental results.

Lin Lanying stared at ω in the formula, then flipped open the second process card and pushed the record of twelve revolutions per minute to the bottom of the blackboard.

"As the rotational speed increases, δ decreases, the accumulation of impurities near the interface decreases, and keff moves closer to the equilibrium value."

"It will also change the temperature field of the melt, so a critical speed limit is required."

Jiang Ming wrote the rotating Reynolds number below the blackboard, Re equals ω multiplied by the square of R divided by ν, and included the current seed crystal radius and germanium melt viscosity in the calculation.

"Within the laminar flow range, the boundary layer relationship can be used for process estimation. As we approach the critical region, radial flow will fluctuate, and new crystal nuclei may also be generated at the solid-liquid interface."

Upon hearing this, Lao Sun took out the idle time record of the single crystal furnace transmission mechanism from his toolbox. It showed that the two speeds, 28 and 35, had each run for 15 minutes.

Jiang Ming took the record, first converting twelve per minute into angular velocity, and then substituting it into the kinematic viscosity of germanium melt and the diffusion coefficient of phosphorus.

The δ calculated on the blackboard corresponds to a thicker concentration layer, and the effective segregation coefficient of phosphorus is close to 0.08. Based on the existing charge weight and crystal length, the resistivity difference between the head and tail is about 30%.

Fang Xudong held up the measured curve of the second single crystal next to the blackboard. The calculated value and the curve drop were in the same range, but the deviation at the end was still slightly higher than that of the model.

Jiang Ming continued to increase the rotation speed to 28 revolutions per minute. The change in angular velocity caused δ to shrink. After substituting back into the BPS relationship, the keff of phosphorus rose to around 0.12.

"Based on the current crystal pulling length, the resistivity difference between the beginning and end is expected to be reduced to about 14%. The contribution directions of arsenic and antimony are the same, and the magnitude will need to be calculated separately according to their respective diffusion coefficients."

Wang Shouwu flipped back to his summary page, checked the k₀ used by Jiang Ming, and recalculated the exponent term.

What is the viscosity and temperature range of the germanium melt you used?

"The values ​​around 945 degrees are calculated from adjacent temperature data. If the process temperature deviates from the specified range, this set of results is invalid."

"Should we use the seed crystal radius or the outer edge of the meniscus?"

"First, determine the effective solid-liquid interface radius. Then, include the correction caused by the outer edge of the meniscus in the error range. For the third crystal pulling, we need to recalculate using the measured diameter."

Wang Shouwu wrote both radii on the margin of the page, then pointed to the Reynolds number.

"How is the maximum speed of 35 revolutions per minute determined?"

"Based on the current effective radius and viscosity, 35 revolutions is close to the empirical critical range. The crucible sidewall of the simple furnace is even closer, and the actual instability may occur earlier. Therefore, the experimental parameter is only 28 revolutions."

As Jiang Ming wrote the last line in the chalk, he circled the 28 revolutions, the 0.8 millimeters lifting speed, and the expected 14% drop together.

Wang Shouwu stared at the set of numbers. His hand, which had been turning the pages of the pen, stopped at the edge of the notebook. After a while, he closed the notebook and clipped the pen to the spine.

His assistant was still holding the stack of purification reports from the fourth round. Wang Shouwu reached out and took them, but then casually placed the entire stack of reports flat on the recording table of furnace number three.

The data, originally used to compare the temperature control and purification results of 502, has now become experimental data that can be directly used in the next calculation.

Huang Kun stood up from the corner of the table and pointed to the process parameters circled on the blackboard.

"Are you planning to use 28 revolutions per minute for your next monocrystalline silicon wafer?"

"Maintain a lifting speed of 0.8 millimeters per minute. First, change the rotation speed separately. If the head-to-tail pressure difference is within the predicted range, the relationship of the rotating boundary layer can be established."

Jiang Ming remarked the part of the tail that was higher than the theoretical line.

"If the deviation is still higher than the prediction, the carbon impurities contributed by the crucible wall must be treated separately, and the charge weight, contact area and melt holding time must also be re-disassembled."

Wang Shouwu put the thin notebook into his jacket pocket, but his palm was still pressing down on the opening of the pocket, as if he were checking every condition he had just recorded.

Which set of literature values ​​for the diffusion coefficient D did you use?

Jiang Ming reported the name of the Soviet journal, the year of publication, the measurement temperature, and the measured value of phosphorus in germanium melt, and also explained that Da Liu had already checked the temperature conversion in the early morning.

Wang Shouwu took out the notebook again, wrote down the data on the back cover, and added the archive index number next to the pen.

Wu Hanzhang walked to the record table and handed the third crystal pulling process card to Huang Kun. The twenty-eight turns on the card had already been verified by Lao Sun.

"The furnace is in the factory. We'll load it this afternoon. If Researcher Wang still wants to see the interface, he can come with us."

Wang Shouwu glanced at the 14% mark on the blackboard, then looked at the single crystal in the cork box, which was 30% smaller at the beginning and end.

"I'll go check it out."