Patent ID: 12188802

DETAILED DESCRIPTION

In view of the above issues,FIG.1shows a flow measurement system10according to one example of the present disclosure. The flow measurement system10comprises a plurality of mass flow controllers12arranged in parallel, each being configured to flow a respective one of a plurality of pure gases16. In this example, the first gas supply14asupplies a first gas16ato the first mass flow controller12a; the second gas supply14bsupplies a second gas16bto the second mass flow controller12b; and the third gas supply14csupplies a third gas16cto the third mass flow controller12c. The mass flow controllers12can include valves, pressure sensors, temperature sensors, and restrictors, the valves being controlled based on input from the pressure and temperature sensors of each mass flow controller12, to supply gases at preset target flow rates.

A mixing chamber18is provided downstream of the plurality of mass flow controllers12and configured to mix the plurality of pure gases16to produce a gas mixture. Alternatively, the mixing chamber18may be replaced by a mixing manifold.

A flow splitter20is provided downstream of the mixing chamber18with a plurality of channels22configured to flow the gas mixture. Each of the plurality of channels22a,22bis provided with at least a pressure sensor and a restrictor. The first channel22aand the second channel22bare provided with a first restrictor24aand a second restrictor24b, respectively.

In this example, the first upstream pressure sensor26ameasures an upstream pressure P1A upstream of the first restrictor24a, and the first downstream pressure sensor28aof the first channel22ameasures a downstream pressure P2A downstream of the first restrictor24a. The second upstream pressure sensor26bof the second channel22bmeasures an upstream pressure P1B upstream of the second restrictor24b, and the second downstream pressure sensor28bof the second channel22bmeasures a downstream pressure P2B downstream of the second restrictor24b.

The flow splitter20is provided with a temperature sensor30to measure a temperature T of the gas mixture. The processor38is configured to receive a plurality of temperature values from the temperature sensor30, and the plurality of flow rates for the plurality of channels of the flow splitter20are estimated using the temperature values from the temperature sensor30.

A flow rate sensor31can also be provided external to the flow splitter20to measure a flow rate of the gases. The flow rate sensor31may be another mass flow controller incorporating at least a restrictor, an orifice, or a heated capillary. The flow rate sensor31can alternatively be a chamber using the rate-of-rise method or the Pressure-Volume-Temperature and time (PVTt) method. The processor38is configured to receive a plurality of flow rates from the flow rate sensor31. The plurality of flow rates for the plurality of channels22of the flow splitter20are estimated using the plurality of flow rates from the flow rate sensor31.

Finally, a portion of the gas mixture from the first channel22ais delivered to a respective zone of a showerhead32, and a portion of the gas mixture from the second channel22bis delivered to a different respective zone of the showerhead32. The showerhead32is positioned within process chamber33in which a workpiece can be positioned to undergo a chemical process, such as chemical vapor deposition, vapor phase etching, etc., in the environment of the mixture gas. Alternatively, the portion of the gas mixture from the first channel22acan be delivered to the first process chamber33, while the portion of the gas mixture from the second channel22bcan be delivered to a second process chamber35different from the first process chamber33.

In this example, a gas box34contains the mass flow controllers12, the mixing chamber18, and the flow splitter20. However, it will be appreciated that in other examples, the gas box34can include additional components that are not depicted inFIG.1. The additional components can include tubing, valves, sensors, calibration instruments, and computers. The gas box34can also include more mass flow controllers and channels in the flow splitter than the example ofFIG.1.

The computing device36comprises a processor38and volatile memory40, which may be random access memory (RAM). In some implementations, the computing device36may be configured as a System on Module (SOM). The processor38may be a central processing unit (CPU), application-specific integrated circuit (ASIC), field-programmable gate array (FPGA), or other type of microprocessor, and may be a multi-core processor, for example.

The processor38is operatively coupled to non-volatile memory42which contains instructions44that, in response to execution by the processor38, cause the processor38to receive a plurality of flow rates48from the plurality of mass flow controllers12. In this example, a first flow rate48a(sccm1) is received from the first mass flow controller12a, a second flow rate48b(sccm2) is received from the second mass flow controller12b, and a third flow rate48c(sccm3) is received from the third mass flow controller12c.

The processor38then retrieves the plurality of physical properties50for the plurality of gases16flowing through the plurality of mass flow controllers12. The physical properties may include thermophysical properties, in some examples. The physical properties50may be retrieved from a database46stored on the non-volatile memory42. In this example, the first physical properties50aare retrieved for the first gas16aflowing through the first mass flow controller12a, the second physical properties50bare retrieved for the second gas16bflowing through the second mass flow controller12b, and the third physical properties50care retrieved for the third gas16cflowing through the third mass flow controller12c.

In this example, the physical properties include the virial coefficient B, molecular weight MW, viscosity eta, and heat capacity Cp of the gas. Of these, it will be appreciated that the virial coefficient, viscosity, and heat capacity are affected by temperature and thus are thermophysical properties. The first physical properties50ainclude the first virial coefficient B1, the first molecular weight MW1, the first viscosity eta1, and the first heat capacity Cp1of the first gas16a. The second physical properties50binclude the second virial coefficient B2, the second molecular weight MW2, the second viscosity eta2, and the second heat capacity Cp2of the second gas16b. The third physical properties50cinclude the third virial coefficient B3, the third molecular weight MW3, the third viscosity eta3, and the third heat capacity Cp3of the third gas16c. However, it will be appreciated that other combinations of physical properties of the gas may be alternatively used.

The processor38subsequently estimates a plurality of physical properties52for the gas mixture in the mixing chamber18based on the plurality of flow rates48and the plurality of physical properties50. In this example, the physical properties52include the mixture virial coefficient52a(BMix), the mixture molecular weight52b(MWMix), the mixture viscosity52c(etaMix), and the mixture heat capacity52d(CpMix). Alternatively, other combinations of physical properties may be estimated.

The mixture molecular weight52b(MWMix) and the mixture heat capacity52d(CpMix) may be computed from first principles using the plurality of flow rates48from the plurality of mass flow controllers12. The mixture virial coefficient52a(BMix) may be computed using an equation of state, which may be Tsonopoulos' method or Peng-Robinson's equation of state, for example. The mixture viscosity52c(etaMix) may be computed with a mixture-viscosity method, which may be Wilke's method or Reichenberg's method, for example. Additionally or alternatively, the mixture virial coefficient52a(BMix) and the mixture viscosity52c(etaMix) may be computed using conventional fluid equations of state.

The processor38subsequently receives a plurality of pressure values56from the pressure sensors26,28of the plurality of channels22, and estimates a plurality of flow parameters for the plurality of channels22of the flow splitter20using the mathematical flow model54, the plurality of physical properties52for the gas mixture in the mixing chamber18, the plurality of pressure values56from the pressure sensors26,28of the plurality of channels22, and the temperature values60from the temperature sensor30. The processor38further estimates a plurality of flow splits58(sccmA, sccmB) using the plurality of estimated flow parameters for the plurality of channels22of the flow splitter20. The mathematical flow model54may be retrieved from the database46stored on the non-volatile memory42.

In this example, the plurality of flow splits58through the plurality of channels22are estimated using the mathematical flow model54, which may be represented as F=f(G, MWMix, BMix, MAC, CpMix, etaMix, P1, P2, T), where F is flow rate (either sccmA or sccmB), f( . . . ) is a function which depends on the variables inside the parenthesis, G is the set of geometrical parameters describing the restrictor24a,24b, and MAC is a momentum accommodation coefficient, which is typically 0.9 to 1.

The function f may be a function for micro-tubes (for restrictors24which include micro-tubes), or a function for micro-channels (for restrictors24which include micro-channels). One example of function f for micro-channels is Arkilic's mathematical flow model54for micro-channels and ideal gases with MAC equal to one:

F=n⁢h3⁢w⁢P2224⁢etaX⁢LRT⁢(PR2⁢—⁢1+1⁢2⁢K⁢n⁡(PR-1))[Formula⁢1]

Here, F is in kmol/s, n is the number of micro-channels in the restrictor, h is the channel thickness, w is its width, L its length, etaX is the gas viscosity, R is the universal gas constant (8314 J/K-kmol), T is temperature in Kelvins, PR=P1/P2, and Kn is the Knudsen number. Kn may be computed with the equation:

K⁢n=0.5π⁢Cp/Cv⁢etaXc⁢D⁢2⁢h[Formula⁢2]

Here, Cp/Cv is the ratio of heat capacities, c is the sound speed in the gas, and D2is the density evaluated at P2and T. Alternatively, the flow rate F may be obtained using a computational fluid dynamics method, one-dimensional method, artificial neural networks, or a combination of the above.

The restrictors24a,24bmay be configured with multiple micro-channels, and G may be a set of geometrical parameters including number of channels, and thickness, width, and length of the channels. Alternatively, the restrictors24a,24bmay be configured with multiple micro-tubes, and G may be a set of geometrical parameters including number of micro-tubes, the length of the micro-tubes, and the diameter of the micro-tubes.

The processor38may control the flow splitter20in accordance with the estimated plurality of flow splits58. In this example, the flow splitter20controls the split ratios sccmA/(sccmA+sccmB) and sccmB/(sccmA+sccmB), or more simply the ratio sccmA/sccmB, where sccmA is the first channel flow rate of the first channel22a, and sccmB is the second channel flow rate of the second channel22b.

The first channel flow rate (sccmA) of the first channel22aor the second channel flow rate (sccmB) of the second channel22bmay be expressed by the formula F=f(G, MWMix, BMix, MAC, CpMix, etaMix, P1, P2, T), where f( . . . ) is a function depending on variables corresponding to the gases inside the channel: geometrical parameters G, the mixture molecular weight52b(MWMix), the mixture virial coefficient52a(BMix), momentum accommodation coefficient (MAC), the mixture heat capacity52d(CpMix), the mixture viscosity52c(etaMix), the upstream pressure P1, the downstream pressure P2, and gas temperature60(T). This formula may be expressed as F=F′ F″, where F′=f″(S), F″=f′(G, MWMix, BMix, MAC, CpMix, etaMix, P1, P2, T), F′ is an empirical function dependent on a set of parameters S, and F′ is a polynomial function or a hyperbolic function. The parameters S can include any combination of parameters among the ones inside the parenthesis of F″ ( . . . ). F′ and F″ may be polynomials with a suitable number of terms. For example, F′ may be a 5thorder polynomial as follows:
F′=a0+a1S+a2S2+a3S3+a4S4+a5S5[Formula 3]

Here, a0, a1, a2, a3, a4 and a5 are constants. F′ may also be a function using hyperbolic functions such as:
F′=b1−b2tanh(b3S)  [Formula 4]

Here, b1, b2 and b3 are constants. F″ may be obtained with a conventional formula using a viscosity-inference method. Alternatively, the flow rate F may be obtained using a computational fluid dynamics method, one-dimensional method, artificial neural networks, or a combination of the above.

Referring toFIG.2, a first method100is described for estimating a plurality of flow splits for a plurality of channels of a flow splitter using a flow measurement system. As discussed above, the flow measurement system typically includes a plurality of mass flow controllers arranged in parallel, each being configured to flow a respective one of a plurality of pure gases, a mixing chamber provided downstream of the plurality of mass flow controllers and configured to mix the plurality of pure gases to produce a gas mixture, and a flow splitter provided downstream of the mixing chamber with a plurality of channels configured to flow the gas mixture. Each of the plurality of channels is typically provided with a pressure sensor. The system further typically includes non-volatile memory storing a mathematical model representing a flow of gas through the restrictor, and a processor. Alternatively, first method100may be used with other suitable hardware.

Thus, the following description of first method100is provided with reference to the software and hardware components described above and shown inFIG.1. It will be appreciated that first method100also may be performed in other contexts using other suitable hardware and software components.

At step102, a plurality of flow rates are received from a plurality of mass flow controllers. At step104, a plurality of physical properties for the plurality of pure gases flowing through the plurality of mass flow controllers are retrieved. At step106, a plurality of physical properties are estimated for the gas mixture in the mixing chamber. At step108, a plurality of pressure values are received from the pressure sensors of the plurality of channels.

At step110, a plurality of flow parameters are estimated for the plurality of channels of the flow splitter using the mathematical model, the plurality of physical properties for the gas mixture, and the plurality of pressure values from the pressure sensors of the plurality of channels.

At step112, a plurality of flow splits are estimated using the plurality of flow parameters for the plurality of channels of the flow splitter. At step114, the processor is caused to control the flow splitter in accordance with the estimated plurality of flow splits.

Referring toFIG.3, a second method200is described for obtaining a viscosity value of a desired process gas using a flow measurement system comprising a mass flow controller provided with a restrictor, flow rate sensor, a pressure sensor, and a temperature sensor, non-volatile memory storing a mathematical model representing a flow of gas through the restrictor, and a processor. The second method200may be used to determine a viscosity of at least one high-uncertainty gas among the plurality of pure gases flowing through the plurality of mass flow controllers in step104of the first method100.

The following description of second method200is provided with reference to the software and hardware components described above and shown inFIG.1. It will be appreciated that second method200also may be performed in other contexts using other suitable hardware and software components.

At step202, a plurality of flow rate values, a plurality of pressure values, and a plurality of temperature values from the mass flow controller are received for a reference gas, for a low-uncertainty-viscosity gas, and for a desired gas by flowing the reference gas, the low-uncertainty-viscosity gas, and the desired gas through the restrictor, respectively. The desired gas may be a mixture of gases.

At step204, a plurality of physical properties and a plurality of flow parameters are estimated for each of the reference gas, the low-uncertainty-viscosity gas, and the desired gas using the plurality of flow rates, the plurality of pressure values, and the plurality of temperature values. The physical properties can include at least one of virial coefficient, molecular weight, viscosity, and heat capacity, and the plurality of flow parameters can include at least a flow rate or a Mach number.

At step206, a plurality of viscosity values of the low-uncertainty-viscosity gas are determined using the mathematical model, the plurality of physical properties, and the plurality of flow parameters. The mathematical model may be at least one of a computational fluid-dynamics model, a one-dimensional method, or an artificial neural network. The viscosity may be estimated using at least one of published values or Lucas' method or Chung's method, and the virial coefficient may be estimated using at least one of Tsonopoulos' method or Peng-Robinson's equation of state.

At step208, a set of viscosity values of the low-uncertainty-viscosity gas is selected from the plurality of viscosity values of the low-uncertainty-viscosity gas according to a set of constraints.

At step210, a difference in viscosity is determined between a known viscosity of the low-uncertainty-viscosity gas and an average of the selected set of viscosity values.

At step212, the difference in viscosity is reduced by repeating the steps of selecting the set of viscosity values of the low-uncertainty-viscosity gas and determining the difference in viscosity. The Nelder-Mead algorithm may be used to repeat the steps of selecting the set of viscosity values of the low-uncertainty-viscosity gas and determining the difference in viscosity.

At step214, an optimal set of constraints is established by reducing the difference in viscosity.

At step216, a plurality of viscosity values of the desired gas is determined using the mathematical model, the plurality of physical properties, and the plurality of flow parameters.

At step218, a set of viscosity values of the desired gas is selected from the plurality of viscosity values according to the optimal set of constraints.

At step220, the flow of gas through the restrictor is controlled in accordance with the selected set of viscosity values of the desired gas.

Referring toFIG.4, a third method300is described for determining a high-uncertainty-viscosity of at least one of a plurality of pure gases. The third method300may be used to determine a viscosity of at least one high-uncertainty gas among the plurality of pure gases flowing through the plurality of mass flow controllers in step104of the first method100.

The following description of a third method300is provided with reference to the software and hardware components described above and shown inFIG.1. It will be appreciated that the third method300also may be performed in other contexts using other suitable hardware and software components.

The third method300comprises an experimental portion (steps302to310), a first calculation portion (steps312to330), and a second calculation portion (steps332to348).

Referring to the experimental portion (steps302to310), at step302, a gas supply supplies a reference gas to a first channel having a restrictor. One example of a reference gas is nitrogen. The restrictor can alternatively be configured as a laminar-flow element.

At step304, an upstream pressure (P1) upstream of the restrictor is measured using an upstream pressure sensor, a downstream pressure (P2) downstream of the restrictor is measured using a downstream pressure sensor, and a gas temperature (T) of the reference gas is measured using a temperature sensor.

At step306, a molar flow rate (F0exp) of the reference gas flowing out of the mass flow controller is measured with a flow rate sensor external to the first channel. The uncertainty of the flow rate sensor is preferably lower than the desired viscosity uncertainty.

At step308, steps302to306are iterated for the process gas with low-uncertainty viscosity and the process gas with high-uncertainty viscosity. Thus, the iteration of steps302to306involves receiving a plurality of flow rate values, a plurality of pressure values, and a plurality of temperature values from the mass flow controller for a reference gas, for a low-uncertainty-viscosity gas, and for the desired gas (process gas with high-uncertainty viscosity) by flowing the reference gas, the low-uncertainty-viscosity gas, and the desired gas through a restrictor, respectively.

Examples of process gases with low-uncertainty viscosities include argon (Ar), sulfur hexafluoride (SF6), and xenon (Xe). It will be appreciated that the molar flow rates of the process gas with low-uncertainty viscosity and the process gas with high-uncertainty viscosity are preferably measured under similar upstream pressures (P1), similar downstream pressures (P2), and similar gas temperatures (T) using the same restrictor with the same set of geometrical parameters (G).

At step310, the experimental gas ratios GRLexp and GRHexp are calculated for each of the process gas with low-uncertainty viscosity and the process gas with high-uncertainty viscosity, respectively. The experimental gas ratio GRLexp for the process gas with low-uncertainty viscosity is calculated by the formula GRLexp=FXLexp/F0exp, where FXLexp is the experimental molar flow rate of the process gas with the low-uncertainty viscosity, and F0exp is the experimental molar flow rate of the reference gas. The experimental gas ratio GRHexp for the process gas with high-uncertainty viscosity is calculated by the formula GRLexp=FXHexp/F0exp, where FXHexp is the experimental molar flow rate of the process gas with the high-uncertainty viscosity, and F0exp is the experimental molar flow rate of the reference gas. Hence, the experimental gas ratios GRLexp and GRHexp are functions of the upstream pressure (P1), downstream pressure (P2), and the gas temperature (T).

Referring to the first calculation portion (steps312to330), at step312, a low-uncertainty viscosity etaLX1is obtained of the process gas with a known low-uncertainty viscosity. Alternatively, the low-uncertainty viscosity etaLX1may be estimated using a conventional method.

At step314, a set of geometrical parameters G are obtained of the restrictor. For example, for a multiple-micro-channel restrictor, this set includes the number of channels, and the thickness, width, and length of the channels. For a multiple-micro-tube restrictor, this set includes the number of tubes, the length of the tubes, and the diameter of the tubes.

At step316, a plurality of physical properties are estimated for the reference gas and the process gas with the low-uncertainty viscosity. The physical properties may include thermophysical properties. Examples of these physical properties include molecular weight MW, virial coefficient B, momentum accommodation coefficient MAC, heat capacity Cp, and the viscosity eta. Other combinations of properties may be used.

At step318, a plurality of flow parameters are estimated for the process gas with the low-uncertainty viscosity using the various gas properties estimated in step316. Examples of these parameters include the Mach number Ma, and the Knudsen number Kn. Ma is the mass flow rate divided by the product of the following: average gas density in the restrictor, the sound speed in the gas, the restrictor flow path cross-sectional area (e.g., thickness times width in a micro-channel, pi times squared radius in a micro-tube), and the restrictor total number of flow paths. Kn is the ratio of the gas mean free path and a restrictor length scale, such as the channel thickness in a multiple-micro-channel restrictor.

At step320, the reference gas molar flow rate F0, and the molar flow rate FXL for the process gas with low-uncertainty viscosity are calculated using a mathematical flow model describing the flow in the restrictor. The mathematical flow model may be stored in a database in non-volatile memory. In the mathematical flow model, F0may be expressed as F0=f(G, MW0, B0, MAC, Cp0, etaX0, P1, P2, T), and FXL may be expressed as FXL=f(G, MWL, BL, MAC, CpL, etaXL, P1, P2, T).

G is the set of geometrical parameters of the restrictor, MW0and MWL are the molecular weights of the reference gas and the low-uncertainty-viscosity gas respectively, B0and BL are the virial coefficients of the reference gas and the low-uncertainty-viscosity gas respectively, Cp0and CPL are the heat capacities of the reference gas and the low-uncertainty-viscosity gas respectively, etaX0are etaXL are the viscosities of the reference gas and the low-uncertainty-viscosity gas respectively, P1is the upstream pressure upstream of the restrictor, P2is the downstream pressure downstream of the restrictor, and T is the temperature of the gas.

At step322, the gas ratio, GRLcalc is calculated using the flow rates calculated in step320. The gas ratio GRLcalc may be calculated using the formula GRLcalc=FXL/F0.

At step324, steps320and322are iterated using different values of etaLX to reduce the difference between the calculated gas ratio GRLcalc and the experimental gas ratio GRLexp of the process gas with low-uncertainty viscosity obtained at step310. This may be done, for example, using the Nelder-Mead algorithm. Thus, a plurality of viscosity values etaLX of the low-uncertainty-viscosity gas are determined.

At step326, a calculated value etaLX2corresponding to a set of constraints is selected among the different values of etaLX. In other words, a set of viscosity values of the low-uncertainty-viscosity gas is selected from the plurality of viscosity values of the low-uncertainty-viscosity gas according to a set of constraints. An example of this set is Ma<Ma*; Kn<Kn*; F0exp>F*; FXLexp>F*; P1<P*; and P2<P*. Here, the quantities with an asterisk are predetermined values. Then, an average etaLX3of the selected values of etaLX is calculated.

At step328, step326is repeated using different sets of constraints to reduce the difference between the average etaLX3and the known low-uncertainty gas viscosity etaLX1obtained in step312. The different sets of constraints may be retrieved from a database stored in non-volatile memory. In this step, a difference in viscosity between a known viscosity of the low-uncertainty-viscosity gas and an average of the selected set of viscosity values is determined, and the difference in viscosity is reduced by repeating the steps of selecting the set of viscosity values of the low-uncertainty-viscosity gas and determining the difference in viscosity.

At step330, an optimal set of constraints is established by reducing the difference between the average etaLX3and the known low-uncertainty gas viscosity etaLX1obtained in step312. For the above example, this step finds optimal values of Ma*, Kn*, F*, and P* to reduce uncertainty.

Referring to the second calculation portion (steps332to348), at step332, a high-uncertainty viscosity etaHX1is obtained of the process gas with a known high-uncertainty viscosity. Alternatively, the high-uncertainty viscosity etaHX1may be estimated using conventional methods.

At step334, a set of geometrical parameters G of the restrictor are obtained, similarly to step314.

At step336, a plurality of physical properties are estimated for the reference gas and the desired gas (process gas with the high-uncertainty viscosity), similarly to step316.

At step338, a plurality of flow parameters are estimated for the process gas with the high-uncertainty viscosity using the various gas properties estimated in step336.

At step340, the molar flow rates for the reference gas, F0, and the process gas FXH with the high-uncertainty viscosity are calculated using the mathematical model describing the flow in the restrictor. In the mathematical flow model, F0may be expressed as F0=f(G, MW0, B0, MAC, Cp0, etaX0, P1, P2, T), and FXH may be expressed as FXH=f(G, MWH, BH, MAC, CpH, etaXH, P1, P2, T). MWH is the molecular weight of the high-uncertainty-viscosity gas, BH is the virial coefficient of the high-uncertainty-viscosity gas, CpH is the heat capacity of the high-uncertainty-viscosity gas, and etaXH is the viscosity of the high-uncertainty-viscosity gas.

At step342, the gas ratio, GRHcalc is calculated using the flow rates calculated in step320. The gas ratio GRHcalc may be calculated using the formula GRHcalc=FXH/F0.

At step344, steps340and342are iterated using different values of etaHX to reduce the difference between the calculated gas ratio GRHcalc and the experimental gas ratio GRHexp of the process gas with high-uncertainty viscosity obtained at step310. This may be done, for example, using the Nelder-Mead algorithm. Thus, a plurality of viscosity values of the desired gas (high-uncertainty-viscosity gas) are determined using the mathematical model, the plurality of physical properties, and the plurality of flow parameters.

At step346, a calculated value etaHX2corresponding to the optimal set of constraints obtained at step330is selected among the different values of etaHX. In other words, a set of viscosity values of the desired gas (high-uncertainty-viscosity gas) is selected from the plurality of viscosity values of the high-uncertainty-viscosity gas according to the optimal set of constraints. An example of this set of constraints is Ma<Ma*; Kn<Kn*; F0exp>F*; FXHexp>F*; P1<P*; and P2<P*. Here, the quantities with an asterisk are values that were estimated for the process gas with the low-uncertainty viscosity in the first calculation portion (steps312to330). Then, an average etaHX3is taken over the selected values of etaHX.

At step348, the viscosity of the process gas with the high-uncertainty viscosity is determined based on the average etaHX3.

The third method300may be used to obtain viscosities at different temperatures, when the viscosity at one temperature is known. For example, when the viscosity of a process gas is known to be etaX25at 25° C., the third method300can be used to determine the viscosity etaX35of the process gas at 35° C. For this example, the reference process gas is the process gas at 25° C. Since the set of constraints is already known, there is no need to conduct the first calculation portion, or the steps for determining the optimal set of constraints for the low-uncertainty process gas. Therefore, in the experimental portion of the third method300, the reference gas is the process gas at 25° C., and steps302,304, and306are iterated for the process gas at 35° C.

Subsequently, the steps of the second calculation portion (steps332to348) are performed by using the process gas at 25° C. as the reference gas, and the viscosity of the process gas at 35° C. is determined by solving for the high-uncertainty gas viscosity at step348. This approach may be repeated for other temperatures.

The second method200and the third method300may be used in a semiconductor manufacturing process in a variety of applications, including the calibration of the mass flow controllers before installation, the re-calibration of the mass flow controllers during re-installation or maintenance, and the measurement of gas viscosity during semiconductor manufacturing operations.

Referring toFIG.5, the results of a Monte-Carlo analysis of flow-split (sccmA/sccmB) uncertainties are shown. These results are for values of total flow rate, P1, P2and T typical of flow splitters, and for 18 mixtures typical of semiconductor-manufacturing applications. For this purpose, a mathematical flow model F with F′ was used. The input uncertainties (mass flow controller sccm; eta and Cp for each pure gas; etaMix and CpMix; G; P1, P2and T) were varied according to two scenarios: one was a high-uncertainty scenario, shown with squares inFIG.5; and the other was a low-uncertainty scenario, shown with circles inFIG.5.

The horizontal axis ofFIG.5is a logarithmic scale indicating the molecular weight MWmix of the gas mixture. The vertical axis is the ratio of the uncertainty of a conventional method, indicated as U(N2-assump), which uses the assumption that all of the gases behave like nitrogen, and the uncertainty of the first method100, U(Method).

InFIG.5, values of U(N2-assump)/U(Method) are larger than one, indicating that the first method100reduces uncertainty compared to the conventional method. For example, when U(N2-assump)/U(Method)=2, the first method100reduces uncertainty by a factor of 2.FIG.5indicates that U(N2-assump)/U(Method) is close to one for various mixtures with molecular weight close to the nitrogen molecular weight of 28 kg/kmol. For these mixtures, the prior-art method is sufficient to achieve accurate flow splits. However,FIG.5also shows that for several mixtures with molecular weights larger or smaller than the nitrogen molecular weight of 28 kg/kmol, the first method100reduces uncertainty compared to the conventional method. This reduction is pronounced in mixtures that are significantly heavier or lighter than nitrogen. Therefore, compared to conventional methods, the first method100offers a way to achieve accurate flow splits for a broader range of mixtures.

FIG.6shows flow-split errors with reference to experimental data when using the prior-art method versus the first method100. The prior-art method, marked as N2-assump inFIG.6, uses the assumption that all of the gases behave like nitrogen. Two error metrics are shown: mean absolute errors (MAE) and root-mean-square errors (RMS). The process gases that were studied inFIG.6were two instances of tetrafluoroethane (C2H2F4), two instances of hexafluoro-1,3,-butadiene (C4F6), octafluorocyclobutane (C4F8), two instances of sulfur hexafluoride (SF6), one methane-helium (CH4-He) mixture, two ethane-helium (C2H4-He) mixtures, and two oxygen-helium (O2-He) mixtures. Results are for pressures on the high end of those typical of flow splitters.FIG.6shows that the error is less with the first method100than conventional methods (except for one instance of MAE). Moreover,FIG.6indicates that the error reduction with the first method100may be significant, in particular in octafluorocyclobutane (C4F8), hexafluoro-1,3,-butadiene (C4F6), one instance of sulfur hexafluoride (SF6), and two oxygen-helium (O2-He) mixtures.

FIG.7shows for argon (Ar) and sulfur hexafluoride (SF6) the ratio of the viscosity obtained with the third method300(etaX2) and a reference viscosity (etaX1) plotted against Mach number (Ma). Data satisfying constraints for Ma*, Kn*, F* and P* are indicated with dark squares, while the data not satisfying the constraints are indicated with white squares. Ideally, all data would stay at etaX2/etaX1=1. If all the data points were used, the deviation of the measured viscosity with respect to the reference viscosity would be up to 4%, which would be within many published viscosity values. However, by enforcing the above set of constraints, this deviation is about 0.1%. This is an enhancement of an order of magnitude.

Referring toFIGS.8A, Table 1 shows mean absolute errors (MAE) and root-mean-squared (RMS) values between viscosities determined for each of the process gases at 25° C. using the third method300, and reference zero-density viscosity values at 25° C. that were published for each of the process gases. The process gases that were studied in Table 1 were argon (Ar), ethane (C2H6), tetrafluoroethane (C2H2F4), krypton (Kr), neon (Ne), sulfur hexafluoride (SF6), and xenon (Xe). As shown in Table 1, the MAE values and RMS values were all less than 1%.

Similarly, Table 2 shows MAE values and RMS values between viscosities determined for each of the process gases at 35° C. using the third method300, and reference zero-density viscosity values at 35° C. that were published for each of the process gases. The process gases that were studied in Table 2 were argon (Ar), ethane (C2H6), krypton (Kr), sulfur hexafluoride (SF6), and xenon (Xe). As shown in Table 2, the MAE values and RMS values were all less than 1%.

Referring toFIG.8B, Table 3 shows high-uncertainty-viscosities and their RMS values determined for each of the process gases at 25° C. using the third method300. The process gases that were studied in Table 3 were hexafluoroisobutene (C4H2F6) and 1,1,3,3,3-pentafluoropropene (C3HF5). As shown in Table 3, the RMS values were all less than 1%.

Table 4 shows viscosity ratios at different temperatures and their RMS values determined for each of the high-uncertainty-viscosity process gases using the third method300. The process gases that were studied in Table 4 were hexafluoroisobutene (C4H2F6) and 1,1,3,3,3-pentafluoropropene (C3HF5). The viscosity ratios were calculated by dividing the viscosity determined at 35° C. by the viscosity determined at 25° C. As shown in Table 4, the RMS values were all less than 1%.

It will be appreciated that the results in Tables 1 to 4 were obtained using the mathematical flow model for micro-channels and ideal gases with MAC equal to one.

Referring toFIG.8B, Table 5 shows two sets of MAE values. The left column shows a first set of MAE values between viscosities determined for each of the process gases using the third method300and a mathematical flow model for restrictors with micro-channels, and reference zero-density viscosity values that were published for each of the process gases. The right column shows a second set of MAE values between viscosities determined for each of the process gases using the third method300and a computational fluid dynamics method, and reference zero-density viscosity values that were published for each of the process gases. The process gases that were studied in Table 5 were argon (Ar), ethane (C2H6), neon (Ne), and sulfur hexafluoride (SF6) As shown in Table 5, only a relatively small difference was found between the two approaches for determining viscosities.

It will be appreciated that the results in Tables 1 to 5 were obtained by enforcing the following set of constraints for all the process gases: Ma<Ma*=0.03; Kn<Kn*=0.03; FN2exp>F*=0.1 sccm; FXexp>F*.

In accordance with the present disclosure, flow rates of gas mixtures can be estimated accurately, and flow splits of flow splitters can be obtained more accurately for gas mixtures.

In some embodiments, the methods and processes described herein may be tied to a computing system of one or more computing devices. In particular, such methods and processes may be implemented as a computer-application program or service, an application-programming interface (API), a library, and/or other computer-program product.

FIG.9schematically shows a non-limiting embodiment of a computing system400that can enact one or more of the processes described above. Computing system400is shown in simplified form. Computing system400may embody the computing device36described above and illustrated inFIG.1. Computing system400may take the form of one or more personal computers, server computers, tablet computers, home-entertainment computers, network computing devices, gaming devices, mobile computing devices, mobile communication devices (e.g., smart phone), and/or other computing devices, and wearable computing devices such as smart wristwatches and head mounted augmented reality devices.

Computing system400includes a logic processor402volatile memory404, and a non-volatile storage device406. Computing system400may optionally include a display subsystem408, input subsystem410, communication subsystem412, and/or other components not shown inFIG.9.

Logic processor402includes one or more physical devices configured to execute instructions. For example, the logic processor may be configured to execute instructions that are part of one or more applications, programs, routines, libraries, objects, components, data structures, or other logical constructs. Such instructions may be implemented to perform a task, implement a data type, transform the state of one or more components, achieve a technical effect, or otherwise arrive at a desired result.

The logic processor may include one or more physical processors (hardware) configured to execute software instructions. Additionally or alternatively, the logic processor may include one or more hardware logic circuits or firmware devices configured to execute hardware-implemented logic or firmware instructions. Processors of the logic processor402may be single-core or multi-core, and the instructions executed thereon may be configured for sequential, parallel, and/or distributed processing. Individual components of the logic processor optionally may be distributed among two or more separate devices, which may be remotely located and/or configured for coordinated processing. Aspects of the logic processor may be virtualized and executed by remotely accessible, networked computing devices configured in a cloud-computing configuration. In such a case, these virtualized aspects are run on different physical logic processors of various different machines, it will be understood.

Non-volatile storage device406includes one or more physical devices configured to hold instructions executable by the logic processors to implement the methods and processes described herein. When such methods and processes are implemented, the state of non-volatile storage device406may be transformed—e.g., to hold different data.

Non-volatile storage device406may include physical devices that are removable and/or built-in. Non-volatile storage device406may include optical memory (e.g., CD, DVD, HD-DVD, Blu-Ray Disc, etc.), semiconductor memory (e.g., ROM, EPROM, EEPROM, FLASH memory, etc.), and/or magnetic memory (e.g., hard-disk drive, floppy-disk drive, tape drive, MRAM, etc.), or other mass storage device technology. Non-volatile storage device406may include nonvolatile, dynamic, static, read/write, read-only, sequential-access, location-addressable, file-addressable, and/or content-addressable devices. It will be appreciated that non-volatile storage device406is configured to hold instructions even when power is cut to the non-volatile storage device406.

Volatile memory404may include physical devices that include random access memory. Volatile memory404is typically utilized by logic processor402to temporarily store information during processing of software instructions. It will be appreciated that volatile memory404typically does not continue to store instructions when power is cut to the volatile memory404.

Aspects of logic processor402, volatile memory404, and non-volatile storage device406may be integrated together into one or more hardware-logic components. Such hardware-logic components may include field-programmable gate arrays (FPGAs), program- and application-specific integrated circuits (PASIC/ASICs), program- and application-specific standard products (PSSP/ASSPs), system-on-a-chip (SOC), and complex programmable logic devices (CPLDs), for example.

The terms “module,” “program,” and “engine” may be used to describe an aspect of computing system400typically implemented in software by a processor to perform a particular function using portions of volatile memory, which function involves transformative processing that specially configures the processor to perform the function. Thus, a module, program, or engine may be instantiated via logic processor402executing instructions held by non-volatile storage device406, using portions of volatile memory404. It will be understood that different modules, programs, and/or engines may be instantiated from the same application, service, code block, object, library, routine, API, function, etc. Likewise, the same module, program, and/or engine may be instantiated by different applications, services, code blocks, objects, routines, APIs, functions, etc. The terms “module,” “program,” and “engine” may encompass individual or groups of executable files, data files, libraries, drivers, scripts, database records, etc.

When included, display subsystem408may be used to present a visual representation of data held by non-volatile storage device406. The visual representation may take the form of a graphical user interface (GUI). As the herein described methods and processes change the data held by the non-volatile storage device, and thus transform the state of the non-volatile storage device, the state of display subsystem408may likewise be transformed to visually represent changes in the underlying data. Display subsystem408may include one or more display devices utilizing virtually any type of technology. Such display devices may be combined with logic processor402, volatile memory404, and/or non-volatile storage device406in a shared enclosure, or such display devices may be peripheral display devices.

When included, input subsystem410may comprise or interface with one or more user-input devices such as a keyboard, mouse, touch screen, or game controller. In some embodiments, the input subsystem may comprise or interface with selected natural user input (NUI) componentry. Such componentry may be integrated or peripheral, and the transduction and/or processing of input actions may be handled on- or off-board. Example NUI componentry may include a microphone for speech and/or voice recognition; an infrared, color, stereoscopic, and/or depth camera for machine vision and/or gesture recognition; a head tracker, eye tracker, accelerometer, and/or gyroscope for motion detection and/or intent recognition; as well as electric-field sensing componentry for assessing brain activity; and/or any other suitable sensor.

When included, communication subsystem412may be configured to communicatively couple various computing devices described herein with each other, and with other devices. Communication subsystem412may include wired and/or wireless communication devices compatible with one or more different communication protocols. As non-limiting examples, the communication subsystem may be configured for communication via a wireless telephone network, or a wired or wireless local- or wide-area network, such as a HDMI over Wi-Fi connection. In some embodiments, the communication subsystem may allow computing system400to send and/or receive messages to and/or from other devices via a network such as the Internet.

The subject disclosure includes all novel and non-obvious combinations and subcombinations of the various features and techniques disclosed herein. The various features and techniques disclosed herein are not necessarily required of all examples of the subject disclosure. Furthermore, the various features and techniques disclosed herein may define patentable subject matter apart from the disclosed examples and may find utility in other implementations not expressly disclosed herein.

It will be appreciated that “and/or” as used herein refers to the logical disjunction operation, and thus A and/or B has the following truth table.

ABA and/or BTTTTFTFTTFFF

To the extent that terms “includes,” “including,” “has,” “contains,” and variants thereof are used herein, such terms are intended to be inclusive in a manner similar to the term “comprises” as an open transition word without precluding any additional or other elements.