Beyond Conversion Factors: A New Approach to MFC Calibration

A new technique allows mass flow controllers to be accurately calibrated for all types of gases using only an inert gas.

Joseph Dille, Brooks Instrument, Hatfield, Pa. -- Semiconductor International, 7/1/1998

All thermal mass flow instruments work by sensing a temperature difference caused by gas flow and inferring a flow from the temperature difference. The ideal equation for a thermal mass flow sensor is as follows:

(Eq. 1)

Where V_o is the output signal of the mass flow instrument, DT is the temperature difference observed in the sensor, A is a proportionality constant, Q is the heat input to the sensor, C_p is the specific heat of the gas and m is the gas mass flow. Since it is customary to report gas mass flow in standard volumetric units, the molar specific heat should be used in Equation 1.

Traditional CFs are derived from this ideal relationship as shown below: 

(Eq. 2)

Where the subscripts 1 and 2 indicate different gases. Assuming V_o1 = V_o2, the above equation reduces to the following:

	(Eq. 3)
	(Eq. 4)

The quantity C_p1/C_p2 is commonly referred to as the CF.

The CF method has allowed mass flow instrument manufacturers and users to get in the ballpark of the correct flow. If the calibration process is performed consistently, the CF method will give consistent results from instrument to instrument for a given gas, pressure and full scale flow. Unfortunately, the ideal relationship has two fundamental flaws: The specific heat for a gas can be a strong function of temperature and pressure, and the output of the mass flow instrument is influenced by more than the specific heat of the gas.

Fig. 1. Traditionally, MFCs are calibrated by measuring results with a gas such as nitrogen and then applying a conversion factor.

Fig. 2. The results of tests where a variety of MFC designs were calibrated on nitrogen, and then their actual performance on helium was measured.

Fig. 3. Although argon is theoretically a good surrogate for helium, the actual performance can be quite poor.3

It would seem a simple matter to remove the first flaw by using specific heat values that are temperature-  and pressure-dependent. In practice, this is quite difficult. About 35% of the gases used in MFCs do not have the specific heat published as a function of temperature and pressure, C_p(T,P). About 25% of the total do not have the specific heat published as a function of temperature, C_p(T). Some of the more exotic gases that are used in processes, particularly by the semiconductor industry, have not been studied to determine their specific heat. Even  when published data exist, different databases can calculate specific heats that differ by up to 10% for the same set of conditions. An error of this magnitude would cause a corresponding 10% error in flow measurement based on Equation 4.

Ideally, the output of the mass flow instrument is only affected by specific heat. In reality, it is impossible to design a MFC that is completely ideal. Real mass flow instruments are also plagued by 'Reynolds Number' effects, which are a function of the meter geometry, gas density and viscosity. To incorporate a Reynolds correction into the mass flow conversion would require the modeling of the existing design and accurate values of the density and viscosity for all of the gases as a function of temperature and pressure. This would require extensive testing to create the model, and good values for the temperature-dependent viscosity and density properties for each gas. The gas properties as a function of temperature and pressure are not available in the literature for many gases.

Some in the thermal mass flow industry have tried to improve the specific heat-based CF by using an additional 'molecular structure factor'¹ based on the structure of the molecule.

These factors are an attempt to incorporate some of the non-ideal effects in a real mass flow instrument. However, the usefulness of these factors is questionable, since they do not incorporate any information particular to the mass flow design. In fact, SEMATECH even questions their usefulness: 'These numbers are of murky origin: They represent an empirical 'fudge factor' that incorporates second-order effects caused by such things as viscosity, thermal conductivity and the temperature dependence of specific heat.'¹

Performance using traditional methods

The CF is usually incorporated into a mass flow instrument during the calibration process. This is simply done by multiplying the results of the calibration data and presenting it as the calibration report, as shown in Figure 1.

Fig. 4. Several MFCs calibrated for use on WF6 using an inert gas were tested on WF6.4

The traditional method is easy to implement and makes it simple for a manufacturer to calibrate an instrument to the customer's requirements. To do this, the manufacturer sets the calibration flow rate goal to the customer's required flow, divided by the CF between the customer's gas and the calibration gas. In the example shown in Figure 1, the instrument is  calibrated for 100/1.40 or 71.4 sccm of nitrogen, where 1.40 is the CF from nitrogen to helium. The results of the calibration are then multiplied by the CF to get the data for the calibration report.

CFs do get the user in the ballpark for the process gas, but they are not as accurate as the above example would have you believe. To quantify this effect, several MFCs of different designs and of different full scale flows were calibrated on nitrogen, and then their actual performance on helium was measured.² The results of this test are shown in Figure 2. The letters A through E indicate the different designs, and the size indicates the nitrogen equivalent full scale flow of the instrument.

It is easy to see that by using traditional flow methods, one can be in error by more than 20% of the expected flow. The amount of error is influenced by MFC design and full scale flow. The error can be quite nonlinear. These problems make it particularly difficult to dial in a process, since single CF cannot give accurate results. Process-to-process deviations because of the inadequacies of traditional CFs are a daily experience in most fabs.

Fig. 5. The instruments used to generate the data in Figure 3 were recalibrated on nitrogen and then had a new function applied for helium. The results are shown.5

One possible way of improving performance on a process gas is to calibrate the mass flow instrument on a gas with a similar specific heat. By carefully selecting a non-toxic gas with a similar specific heat, one will only make a small correction with the CF. The idea is that with only a small correction, there will only be a small error. This technique is called surrogate gas calibration and is fairly common among mass flow instrument manufacturers. While popular, it does have some negative aspects. The most important drawback is that no two gases will have completely identical thermophysical properties. Also, many of the common surrogate gases are chlorinated fluorocarbon compounds (CFCs), which have been linked to the depletion of the ozone layer. The surrogate gases are also complex molecules where the thermal properties are a strong function of temperature and pressure. This can lead to a lack of repeatability from instrument to instrument if the calibration conditions change.

To quantify the performance of using surrogate gases for calibration, argon was used as a calibration surrogate with helium as the process gas. Argon is theoretically a good surrogate for helium as they are both monoatomic, and the argon/helium CF is 0.994. The ideal surrogate CF is 1.000. The results shown in Figure 3 demonstrate that even though the argon is theoretically a good surrogate for helium, the actual performance is quite poor.

In practice, mass flow instruments that are manufactured for argon and helium are usually calibrated on these gases. This eliminates the use of CFs and their associated errors. The performance of mass flow instruments on helium and argon typically fall well within the manufacturers specifications. The above examples were provided to show that for even two inert, simple, monatomic gases, CFs do not work. Using one as the surrogate for the other did not work. Unless a surrogate gas is fully studied and found to exactly mimic the performance of the desired process gas in a mass flow meter, in all aspects it is a leap of faith to assume that it will reduce errors on process gas flows.

Fig. 6. The results of calibration for silane using the new calibration process.6

Fig. 7. The results of calibration for WF6 using the new calibration process.6

The problem with using traditional CF techniques is not confined to argon and helium. To illustrate this, several MFCs calibrated for use on tungsten hexafluoride (WF₆) using inert gas were tested on WF₆. These results are shown in Figure 4, which shows significant error on WF₆. Figure 4 also shows that using SF₆ as a surrogate gas for WF₆ does not improve the performance on WF₆.

In practice, CFs and surrogate gases can be eliminated by calibrating the mass flow instruments on the process gas. Helium and argon are both inert gases, with which it is easy to calibrate. The problem comes up when the process gas is toxic, pyrophoric or corrosive, like WF₆. Mass flow instruments should not be calibrated directly on hazardous gases because of the following:

• The calibration equipment is usually not safe to use with these gases
• Traces of the gas may remain that would contaminate the customer's process
• The instrument may be corroded or contaminated because of improper purging.

A new approach

A new method has been developed to meet the requirement of providing mass flow instruments that are accurate on the intended process gas, yet have not been contaminated by running the process gas during calibration. This method addresses the problem by breaking it into two parts. It is clear from the data previously presented that the ideal relationship shown in Equation 2 does not provide acceptable accuracy. An alternative relationship is shown as Equation 5. This more realistic relationship includes the effects of other gas properties beyond specific heat and adds a factor relating to the specific geometry of the instrument.

(Eq. 5)

Where f(r,k,v,G_c) is an additional function of the density (r), the thermal conductivity (k), the viscosity (v) and a geometric constant (G_c). G_c is a constant that relates to the specific design of instrument for that range of flow. Equation 5 can be rewritten by combining the terms that are associated with calibration of the instrument, A and Q, and combining them as Cal. The Cal function takes into account the part to part variations from instrument to instrument. The terms that are associated with the geometry and gas C₆ and f(r,k,v,G_c) can also be combined as a gas and geometric (CF) function. The GG function is specific to a gas and the instrument design for a given flow range. This is shown in Equation 6:

(Eq. 6)

The relationship governing gas conversion can now be written in terms of the calibration function (Cal) and the gas and geometry function as shown in Equation 7:

(Eq. 7)

Where the subscripts 1 and 2 indicate different gases.

Assuming V_o1 = Vo₂, the above equation reduces to Equation 8:

(Eq. 8)

With the conversion in this form, the gas-to-gas conversion is simply GG₁/GG₂, as the Cal function is not a function of the gas. One can then experimentally determine the function GG₁/GG₂ for specific gas pairs and instrument design geometries. In-struments calibrated on one gas can be used accurately on a second gas once the function GG₁/GG₂ is determined for the flow geometry.

To demonstrate the effectiveness of this process, the instruments used to generate the data in Figure 3 were recalibrated on nitrogen and then had the function GG_N2/GG_He applied for helium. These results are shown in Figure 5.

The process can also be applied to difficult processes gases. Figures 6 and 7 show the results of calibration for silane and WF₆, respectively, using the new process.

Conclusion

Mass flow instruments calibrated using traditional CFs can introduce process flow errors up to 30% of the expected flow. This error can cause a delay in process startup and wasted material. Using surrogate calibration gases for the actual process gas is not effective, since all the properties of the surrogate will not match the process gas. This method of calibration can also introduce large process errors, which process engineers have struggled with and have come to accept.

A new process developed by Brooks Instrument, called TruCal, determines the true relationship between the calibration gas and the process gas for a given design geometry. This process allows mass flow instruments to be calibrated on an inert gas, thus preserving the instrument's purity, operator safety and the environment, yet still delivering the same accuracy as if the instrument was calibrated on the actual process gas.

References

2. The data collected for Figures 2 and 3 were collected at Brooks Instrument.

3. Instrument ranges reported as nitrogen equivalent. Tests performed at Brooks Instrument.

4. All instruments calibrated for 500 sccm of WF₆. Design 'C' used nitrogen as the calibration gas. All other designs used SF₆ as a surrogate for WF₆. Data taken by W3 Corporation.

5. The data collected for Figure 5 were collected at Brooks Instrument.

6. Data taken by W3 Corporation. Instrument tested Brooks Instrument Model 6256 PureDigital Select.

Joseph C. Dille is a principal engineer for Brooks Instrument Semiconductor Group, specializing in flow and calibration. He has been active in SEMI Standards since the formation of the MFC Committee in 1986 and is now co-chair of the MFC task force. He has bachelor's and master's degrees in mechanical engineering from Lehigh University. Phone: (215) 362-3500 Fax: (215) 362-3745 E-mail: joe.dille@frco.com