## Abstract

Thermionic converters have potential as an energy conversion technology for high-temperature space and terrestrial applications using concentrated solar, nuclear reaction, and combustion processes as the heat source. Recent studies have generated experimental performance data for narrow-gap thermionic energy conversion devices. This investigation explores the use of genetic algorithms to fit existing data with physics-inspired model equations. The resulting model equations can be used for performance prediction for system design optimization or to explore parametric effects on performance. The model equations incorporate Richardson’s law for current density, including both the saturated and Boltzmann regimes, with appropriate relations for power delivered to the external load. The transition regime is characterized using two separate models, each accounting for nonuniformity in emission surfaces and irregularities in the manufacturing process. The trained models enable performance prediction of small-gap thermionic energy conversion devices. In this study, data were fitted for two different prototype designs. The prototype test data and postulated values for the work functions and a transition regime parameter are substituted into physics-inspired model equations, yielding performance models with three adjustable constants. Optimized values of these constants are determined using a genetic algorithm to best fit the experimentally determined performance data for prototype thermionic conversion devices tested in earlier studies. This approach is demonstrated to fit the performance data to within 9%. This methodology also allows the user to back-infer the effective work function values, which were found in this study to be consistent with independent measurement.

## 1 Introduction

In this equation, $Ath$ is the theoretical Richardson constant, $TE$ is the emitting electrode temperature in Kelvin, $q$ is electron charge in Coulombs, $\varphi $ is the work function of the emitting electrode in eV, and $k$ is Boltzmann’s constant.

Heating an electrode encourages more electrons to leave the surface of the emitter. These electrons are accelerated toward a collector electrode that is maintained at a lower temperature than the emitter. This process is completed in a circuit with a load resistance that enables electrons to travel externally from the collector to the emitter [2]. The work function, $\varphi $, represents the energy required to remove an electron from the surface of the electrode. This thermionic energy conversion device directly yields electrical energy without requiring any moving parts, providing an advantage over mechanical heat engines [2]. The high-temperature operation and heat rejection, coupled with the low-pressure operation with high power density, gives the thermionic energy conversion device a distinct advantage in space applications. However, the challenge is designing a device that combats issues such as inhibiting space charge while producing sufficient power output. A space charge region develops when there is an accumulation of electrons in the interelectrode space between the emitter and collector. Space charge accumulates in the interelectrode space of diodes operating at high temperatures with high current emission. This accumulation of charges impacts the output current for the device by reducing the number of electrons, thus reaching the collector and reflecting emitted electrons with insufficient kinetic energy back to the emitter. [3]. Under these conditions, the physics of the space charge region can be modeled using Langmuir’s space charge theory [4]. However, the output current is limited by space charge under these operating conditions, reducing the efficiency of the device.

One approach to limiting the space charge between the emitter and collector plates in a device is to design for a small gap between electrode plates. If the interelectrode gap is sufficiently small, the inhibiting space charge effect can be significantly reduced. Several experiments have been conducted using this approach where small-gap thermionic prototypes were constructed and tested [5–11]. The prototype data used in this paper to design the performance model is obtained from a similar small-gap vacuum thermionic energy conversion device, having a gap between electrodes on the order of 15 $\mu $ m [1]. The emitter and collector substrate materials are sapphire and quartz, respectively. Each substrate then includes a chromium layer followed by a tungsten layer. The low work function coatings on the emitter and collector are stated as being BaCO$3$:SrCO$3$:CaCO$3$ [1]. This surface coating is described as patchy, rather than uniform [1].

One problem with using physics models for thermionic energy conversion is the uncertainty associated with work function measurements [12]. Not only is it critical to account for the temperature dependence of work functions for thermionic emitters, but effects such as surface texturing and different coating techniques can alter the device performance such that an effective work function for thermionic emission is not equivalent to the measured work function using alternative techniques [4]. These factors can be problematic for thermionic diode modeling due to the inconsistency between a measured work function value and the effective value exhibited by the device. On the other hand, a first-principle approach to modeling work functions is often computed at 0 K. Not only is this insufficient for a material operating at high temperature but also it is computationally expensive and insufficient to model patchy surface coating applications and surface texturing. To solve this problem and develop an accurate performance-predictive model tuned to a specific thermionic device, a genetic algorithm can be used to back-infer the effective values from experimental performance data. Using this approach, the final performance-predictive model is catered specifically to the diode and its electrode coatings.

By using the data provided for the thermionic energy conversion prototype, a genetic algorithm was used to determine and optimize unknown variables in postulated model equations to determine a performance-predictive model. This activity was split into two main parts: defining voltage data as a function of load resistance and temperature and determining a model for output power density using work function values and model parameters determined by the data. Both parts took advantage of the optimization capabilities of genetic algorithms.

## 2 Genetic Algorithm

The genetic algorithm used in this work follows the notion of natural selection in which the fittest individuals in a population survive. Figure 1 includes a diagram that provides an outline of the data population used in this study [13]. The $N$ rows indicate the DNA vectors that are filled with individual gene variables that change as the algorithm iterates through several thousand generations to produce the final optimized result. In the algorithm, an array of genes is generated by adding random perturbations to the baseline initial values to populate a set of organisms. In this study, the genes include unknown variables needed to complete a postulated model and physics parameters such as emitter and collector work function values. These genes are specific to the problem being optimized and will be specified in the following relevant sections.

Following the generation of an initial population, each organism is randomly paired with an experimental data point, a predicted value is computed using the model specified for the system of interest, and the error is calculated for that point. The error calculation at this point is the determination of fitness for that organism and its set of genes. As shown in Fig. 2, an error result for an organism that is above a predetermined limit means that set of genes does not survive the selection process and is deleted [13]. Otherwise, that organism or solution survives the initial selection process and is randomly paired with another organism that survived the selection process and allowed to mate. This mating process produces an offspring with some genetic material from each parent and adds additional random mutations to the process. The size of the population matrix must be kept constant in this algorithm, so mating continues until the original population is restored.

The algorithm then repeats the pairing and re-population process, reevaluating the mean relative error. This process continues to iterate through the set number of generations and returns the gene values that produced results with the lowest error.

Figure 3 demonstrates how the average values for each of the genes change with each iteration. The dotted yellow line represents the mean relative error of the model using the genes determined in that generation. The remaining variables indicated in the legend represent the value of the genes determined for each generation, which correlate with the associated error value. This figure is representative of the 3-regime performance-predictive model run for the 1170 K emitter temperature that will be described in detail in Sec. 4.2. In this case, the performance parameter $z$ is the output power density $J$. The tunable constants $n1$, $n2$, and $n3$ are $\varphi E$, $\varphi C$, and $\alpha $, and the $\beta $ parameters are the emission parameters such as $T$ and $Ath$. The $x$-axis represents the generation, or iteration, of the genetic algorithm showing the resulting error for a given set of constants. The algorithm completed 7000 iterations and saved the value for the combination of the three genes that produced the minimum error.

For this algorithm, the optimized gene values that produce a minimum mean relative error are selected. Note that these are not the values computed on the final iteration, rather they were found on a previous generation. This figure shows how the first iteration’s guesses for the three parameters produced approximately 99% error. As the algorithm proceeded through iterations, the values for the three parameters evolved, with the algorithm saving the optimized parameter values producing a total mean relative error of less than 8%. After 1000 iterations, the genetic algorithm bounces around this minimum relative error with each iteration due to the random gene mutations imposed on each generation. The error consistently returns to the optimized values due to the selection process of the genetic algorithm, which produces the noise shown in later generations.

## 3 Voltage Data Preparation

The performance equations discussed in later sections require knowledge of the output voltage to determine the value of the output power density for the thermionic device. The work in this section is data preparation for use later in the performance-predictive model. The intention is to transform the voltage term in these equations into a form that is more useful for the design process using data provided in the prototype source paper.

A four-gene genetic algorithm relating voltage and resistance was first attempted in Sec. 3.1, which computes a separate expression for data at each temperature. An attempt was then made to determine a single, 12-gene temperature-dependent expression, and the details of which are available in Appendix A. However, the 11% error associated with this fit was deemed unacceptably high when compared to the temperature-independent approach in Sec. 3.1, which produced fits with an error less than 1%. The four-gene equations developed at each of the four operating temperatures were used to determine coefficients for a Lagrange interpolating polynomial in Sec. 3.2, yielding a single temperature-dependent equation with a lower error than the 12-gene model. The result of this effort is a function that outputs voltage with given inputs of load resistance and emitter temperature.

### 3.1 Four-Gene Genetic Algorithm Relating Voltage and Resistance.

*A, B, C*, and

*D*. These variables were input into the algorithm such that they provided the genes to create the population set of organisms for the optimization effort as described in Sec. 2. The data were separated by temperature such that the genes $A$, $B$, $C$, and $D$ were optimized for the model at each of the four temperatures available in the prototype data. The mean relative error produced by optimizing these genes at each temperature compares the predicted value by the model to the data provided for the prototype. This calculated error follows the form of Eq. (2) and ranges from 0.06% to 0.32% for emitter temperatures of 1170 K and 870 K, respectively. To aid convergence, the data were normalized by the median in each category. The temperature data were divided by the median temperature, 1020 K, and the load resistance was divided by its median value. The voltage data were provided 0 V and 1 V, so no normalization was necessary.

Figure 4 shows the optimized variables for the dataset at each of the four tested temperatures. These are the genes determined using the genetic algorithm and the normalized data. Each iteration of the algorithm produced optimized variables that exhibited different trends with temperature. The most common behavior showed a quadratic temperature dependence for each of the four variables. While the curve fit for each set of temperature data is useful, the point is to have a general expression to provide the output voltage, provided a load resistance and emitter temperature.

This four-gene model was transformed into a 12-gene model to develop a general temperature-dependent expression. Each of the adjustable constants, *A, B, C* and *D*, were assumed to have a quadratic dependence. For example, $A$ was expanded to $A1T2+A2T+A3$ where $A1$, $A2$, and $A3$ are the adjustable constants and $T$ is the emitter temperature. The relative error using this temperature-dependent model is shown in Appendix A to be 11%. Comparing these results to the $<1%$ error of the four-gene model, the 12-gene model was determined to be inadequate for using in the performance-predictive model presented in Sec. 4.

### 3.2 Lagrange Interpolation Solution for Temperature-Dependent Voltage Determination.

## 4 Performance Prediction

The goal for this work was to develop a performance-predictive modeling capability for the thermionic energy converter using a genetic algorithm approach. To do this, the relationship developed for voltage in previous sections was used with known physics for thermionic emission to develop a model predicting power density. The first approach used a two-regime model, which assumed that the thermionic energy conversion device operated in two modes. After obtaining the results for this model, a modification of the model to accommodate a third transition region between the two regimes was developed to improve the fit. The final model outputs the thermionic diode output power density for a given value of emitter temperature, voltage, emitter work function, and collector work function. The values of the work function inputs are back-inferred from the data using a genetic algorithm.

### 4.1 Operation in Two Regimes.

The Boltzmann region characterized by Eq. (8) applies when $V>\varphi E\u2212\varphi C$. In this case, there is a retarding potential developed between the electrodes with the violation of the inequality [2]. Because of this, the electrons must have additional energy to overcome the potential barrier to reach the electrode in accordance with the retarding potential $VR=V\u2212(\varphi E\u2212\varphi C)$.

Using these two expressions to characterize the two regimes of operation for the device, a genetic algorithm was used to develop an optimized expression to characterize the data. In Eqs. (7) and (8), the only components that are not constants or already defined by the data are the emitter and collector work functions. In the genetic algorithm used to determine the optimized form of these equations, the work functions are the genes being modified with each iteration of the model. In this algorithm, two approaches were attempted. First, the emitter work function was assumed to be linearly temperature dependent of the form $\varphi E=a1+a2T$ and the collector temperature was assumed to be held constant, leading to a constant value for the collector work function, $\varphi C=a3$. A second iteration assumed the emitter work function to have a quadratic temperature dependence. However, this form did not improve the model and did not provide any improvement in the relative error of the predicted value to the prototype data as compared to the linearly dependent model.

The results of this optimization for the data at $TE$ of 1070 K is shown in Fig. 6. In this figure, the saturated and Boltzmann regimes appear to perform well on each side of the transition point where the regime changes. However, there is an area of high error in the transition region between the two regimes where the two predictive curves diverge from the trend in the data. The transition region between the saturated and Boltzmann regimes is the region of highest power output for the device and is the most critical for accurate modeling. This high error prompted a reevaluation of the model used to characterize the thermionic energy conversion process for this prototype.

### 4.2 Operation in Three Regimes.

The error in the transition region of the previous section prompted a reevaluation of the model equations. The transition regime is handled as follows. As noted previously, the intersection of the two-regime model relations occurs at $V=\Delta V=\varphi E\u2212\varphi C$.

$Psat$ and $PBoltz$ are the output power densities in the saturated and Boltzmann regimes, respectively.

The output power transition of interest here is a consequence of the current flux transition between the temperature-limited regime and the Boltzmann regime. The transition between the temperature-limited regime and the Langmuir-Child regime in thermionic devices has been previously studied by Longo [14]. As stated by Longo, the physics causing the gradual transition between regimes has been studied by various individuals, but no satisfactory physical explanation has been determined [15–17]. Longo suggests that the rounded transition region may be attributed to a variation in surface behavior, roughness, or a nonuniform surface coating on the electrodes providing varying work function values across the surface area [14].

With this specification of the model, the varying parameters determined by the genetic algorithm include $\varphi E$, $\varphi C$, and $\alpha $. In contrast to the model of Sec. 4.1, the work functions in this section have an assumed form of temperature dependence. As a result, the genetic algorithm uses Eq. (12) with three variables to determine an optimized model for the data. The data were split with two-thirds serving as training data and one-third reserved for model validation. The training process includes running the algorithm as described in Sec. 2. The validation process used the optimized gene values for $\varphi E$, $\varphi C$, and $\alpha $ and Eq. (12) to compute the output power. The computed output power was then compared to the validation data to determine how well the model performs on data outside the training set. This process was repeated separately for each temperature, and the result of this modeling effort is summarized in Table 1. This table indicates the optimized values of $\varphi E$, $\varphi C$, and $\alpha $ for each set of temperature data and the associated mean relative error. The combined error was calculated for each data set in addition to a separate error calculation for the training and validation datasets.

$TE$ (K) | 1170 | 1070 | 970 | 870 |
---|---|---|---|---|

$\varphi E$ (eV) | 2.4723 | 2.2646 | 2.0787 | 1.8758 |

$\varphi C$ (eV) | 1.8505 | 1.8000 | 1.7937 | 1.5111 |

$\alpha $ | 0.5622 | 0.4924 | 0.4144 | 0.3209 |

Training error | 0.0920 | 0.0475 | 0.0428 | 0.0642 |

Validation error | 0.0801 | 0.0495 | 0.0559 | 0.0519 |

Combined error | 0.0879 | 0.0482 | 0.0476 | 0.0599 |

$TE$ (K) | 1170 | 1070 | 970 | 870 |
---|---|---|---|---|

$\varphi E$ (eV) | 2.4723 | 2.2646 | 2.0787 | 1.8758 |

$\varphi C$ (eV) | 1.8505 | 1.8000 | 1.7937 | 1.5111 |

$\alpha $ | 0.5622 | 0.4924 | 0.4144 | 0.3209 |

Training error | 0.0920 | 0.0475 | 0.0428 | 0.0642 |

Validation error | 0.0801 | 0.0495 | 0.0559 | 0.0519 |

Combined error | 0.0879 | 0.0482 | 0.0476 | 0.0599 |

Figure 7 plots the results for all data sets, where the $x$-axis is voltage and the $y$-axis is output power density. Each curve represents Eq. (12), with optimized constants for each of the four sets of temperature data. These plots also indicate the training data and the validation data, with the curve marking the predicted power value for the associated voltage. This figure illustrates how the modified model is well behaved within the range of voltage data, with a combined error below 0.09 for all emitter temperatures.

An alternative presentation of the model performance is shown in Fig. 8. This figure plots the predicted versus measured output power density. The $y=x$ line on the figure indicates the model fitness, with a perfect fit having data points directly on the line. The mean relative error for the model compared to the total data available for the prototype is 0.067, which is a significant improvement to the 0.20 error of the two-regime model.

Figure 9 shows the effect of temperature and voltage on the output power density. In this figure, the color of the surface represents output power density, with yellow representing a high power density and dark blue representing a low power density. This figure shows that high temperature and moderate voltage produces the highest power density. The high temperature of the emitter produces a greater current density as more electrons are emitted from the electrode material [2]. However, the retarding potential defined in Sec. 4.1 negatively impacts the acceleration of the electrons to the collector as the output voltage increases beyond the threshold value of $V=\varphi E\u2212\varphi C$ [2]. This effect creates the mound on the surface at moderate voltages near this threshold value at high temperatures. Effectively, this figure shows how output power continues to increase with the increasing temperature. The theoretical maximum is indicated by the two-regime model of Sec. 4.1. However, the modified three-regime model represents the performance capabilities of a real device.

In addition to the modified Longo model, we considered an alternative and restated the model for predicting power $Pout$ from the two-regime model using Eqs. (9) and (10) to a three-regime model (Eq. (13)):

The total error produced by each model is indicated in Table 2. The error values produced by Eq. (13) generally shows a slightly lower total error for the four emitter temperatures. However, the difference in error is not significant between the two models. Both models produced a maximum combined error of less than 0.09.

Not only do both models produce similar mean relative error values when comparing the model prediction of output power density with the output power density data but also both models agree in the values of effective work functions for both the emitter and collector at all emitter temperatures. These values are presented in Table 3, where the values determined using the genetic algorithm agree between the two models. These computed values are then compared to those indicated by the source of the prototype data used in this work and are represented in the rows labeled literature value [1]. Note that while there is a temperature-dependent literature value of the emitter work function, the collector work function appears strictly constant with temperature. Because the literature do not explicitly measure the collector temperature or the collector work function at each operating temperature, we do not have that information available for comparison. However, our methodology provides a pathway to account for the temperature dependence of the collector if data regarding the collector temperature are available. The data available to us did not make this possible so we explored how the model performs with the assumption that the collector is maintained at a fixed temperature. Future work with data for measured collector temperatures can explore its effects on performance with this type of model.

$TE$ (K) | 1170 | 1070 | 970 | 870 | |
---|---|---|---|---|---|

$\varphi E$ (eV) | Literature | 1.7356 | 1.6949 | 1.6543 | 1.6136 |

Eq. (12) | 2.4723 | 2.2646 | 2.0787 | 1.8758 | |

Eq. (13) | 2.4666 | 2.2602 | 2.1083 | 1.9127 | |

% Difference (12,13) | 0.2308 | 0.1945 | 1.4139 | 1.9480 | |

$\varphi C$ (eV) | Literature | 1.3149 at T = 500 K_{C} | |||

Eq. (12) | 1.8505 | 1.8000 | 1.7937 | 1.5111 | |

Eq. (13) | 1.7917 | 1.7847 | 1.8077 | 1.5502 | |

% Difference (12,13) | 3.2288 | 0.4622 | 0.7775 | 2.5545 |

$TE$ (K) | 1170 | 1070 | 970 | 870 | |
---|---|---|---|---|---|

$\varphi E$ (eV) | Literature | 1.7356 | 1.6949 | 1.6543 | 1.6136 |

Eq. (12) | 2.4723 | 2.2646 | 2.0787 | 1.8758 | |

Eq. (13) | 2.4666 | 2.2602 | 2.1083 | 1.9127 | |

% Difference (12,13) | 0.2308 | 0.1945 | 1.4139 | 1.9480 | |

$\varphi C$ (eV) | Literature | 1.3149 at T = 500 K_{C} | |||

Eq. (12) | 1.8505 | 1.8000 | 1.7937 | 1.5111 | |

Eq. (13) | 1.7917 | 1.7847 | 1.8077 | 1.5502 | |

% Difference (12,13) | 3.2288 | 0.4622 | 0.7775 | 2.5545 |

The percent difference in work function values between the two models are also included in Table 3. Note that the maximum percent difference between work function values is 3.23%, indicating a good agreement between the two models. This agreement indicates that the work function determination is not dependent on the type of model we use. We used two different models in Eqs. (12) and (13) and determined the same effective work functions using the genetic algorithm.

The effective work functions determined using the genetic algorithm for the two three-regime models are consistently higher than the value determined in the source report for this thermionic device. However, the computed values follow the same trend in temperature dependence as assumed in the article, with higher emitter temperatures producing an increased work function. The computed values of the emitter work function are between 0.3 and 0.7 eV higher than the measured values for the electrodes in the prototype. However, recent research into work function trends indicate experimental measurements of work function values having an uncertainty on the order of 0.5 eV [12].

### 4.3 Potential for Back Emission.

After completing our performance prediction modeling effort, we checked the assumptions used in the modeling approach for sources of error. The key assumption we made in our modeling approach included that the collector electrode is sufficiently cold such that it will not emit electrons and produce a back emission in the thermionic device.

To perform this check, we computed the back emission to forward-emission fraction to be 0.0152. In other words, the back emission was calculated to be at most 1.52% of the forward emission. We then concluded that neglecting the back emission term in our model to be a valid assumption for this device. The details of the calculations performed are available in Appendix C.

## 5 Extrapolating to High Temperatures

In addition to the modeling work within the range of the test data, we used the performance-predictive model to extrapolate to high temperatures. The device in this study was operated at relatively low emitter temperatures, with a maximum at 1170 K. Frequently, thermionic experiments choose to operate the devices at emitter temperatures near 2000 K to take advantage of the high-temperature heat rejection and increased current density associated with the increased emitter temperature. To understand how the device studied in this article performs at higher operating temperatures, each of the parameters in the performance-predictive model needed to be transformed into a temperature-dependent function. Figure 10 shows the work functions determined by the genetic algorithm and the associated curve fits used in the extrapolation to high temperatures. The temperature-dependent work functions shown in Fig. 10 are necessary for extrapolating the parameters used in the performance-predictive models to high temperatures.

The third parameter required to complete the performance-predictive model is $\alpha $. The alpha parameter required to complete the model of Eq. (12) was similarly fit with a quadratic expression as shown in Fig. 11. As shown in the figure, the dimensionless alpha parameter varies between 0.3 and 0.6. With the temperature-dependent fits for $\varphi E$, $\varphi C$, and $\alpha $, the output power density performance prediction at high-temperature operation for the device was performed.

Using the stated curve fits for the three parameters allowed for the model to predict output power density at high emitter temperatures. The result of these efforts up to $TE$ of 2500 K is shown in Fig. 12. This figure exhibits the same trend as shown in Fig. 9. These results demonstrate that not only the model is well behaved within the range of the data tested for the device but also the model also appears to extrapolate well to high emitter-operating temperatures. However, caution should be used with this approach when extrapolating data for a thermionic diode with no space charge to higher temperatures. With a significant increase in temperature, the emitted current density may push the thermionic device into a space charge limited operating mode. The model presented in this article is not meant to predict performance in devices operating with significant space charge.

## 6 Applying Method to Alternative System

To conclude this study, we used data from a 2020 publication for an alternative microgap thermionic energy converter [9]. The same optimization process for the modified Longo model was used to produce the results shown in Fig. 13. The top plot shows the output power density versus the output voltage for the device and the model. The bottom figure plots the measured versus predicted power density with a $y=x$ line of perfect fit.

The optimized effective emitter and collector work functions are $1.7504$ eV and $1.3854$ eV, respectively, with the optimized parameter $\alpha =0.6816$. Prior comparison suggests that $\alpha $ has a weakly nonlinear trend with temperature. The optimized $\alpha $ parameter for these new data is consistent with the trend found in the results of the study discussed in Secs. 4.2 and 5. The modeling effort for this new thermionic system produced a mean relative error of 0.056 when comparing the model prediction to the data. This effort demonstrates the model’s predictive capability with a thermionic system different from the one used for the original model formulation.

## 7 Conclusion

In this study, two three-regime models were developed to predict performance for a thermionic energy conversion device using physics knowledge of thermionic emission and genetic algorithm optimization techniques. These models require an input of both load resistance and emitter temperature to determine the output power density for the thermionic prototype studied. This predictive capability was developed using test data for the device to determine effective work function values. The successful demonstration of this predictive capability indicates that this modeling approach can provide an effective tool for thermionic energy conversion device design if sufficient test data are available. This work also demonstrates an approach for exploring the effects of various work function values on the performance of thermionic energy conversion devices. The three-regime models can be trained for different electrode materials and then used to predict the device performance using the material properties for those electrodes. The effects of including materials with different work functions and electrode geometries can then be explored using the model in this work.

This article also presents a methodology for extracting work function information from performance data. In thermionic emission applications, a low work function material is critical for designing an effective device. However, experimental measurements of work functions have an uncertainty that could be 25% of their value. The agreement between work function values determined using both three-regime models demonstrates that these effective values are not dependent on the type of model used for performance prediction. The equations defining both three-regime models are distinctly different and produced the same result. Not only did we determine the same work functions but we also determined nearly the same transition prediction between the two regimes regardless of the model used. The trends in the model prediction of the transition between the saturated and Boltzmann regimes are consistent for both model equations.

Regarding the uncertainty in the performance data, we suggest the following interpretation of our model. If the uncertainty of the data is known, the fit of the model to the data is expected to be comparable to the uncertainty in the data, and any predictions of the resulting model will be interpreted to have an expected uncertainty at least equal to that in the data. If the uncertainty in the data is not provided by the source, we suggest that the scatter of the data around the fitted model prediction be used as an estimate of the data uncertainty. This is only an estimate but does provide a means for assessing the expected uncertainty in any predictions of the fitted model. The purpose of this article is to explore the use of this methodology. It is best applied and interpreted when uncertainty of the data is known, but when it is not known, scatter of the data around the fitted model prediction can provide a path to apply the model with some useful assessment of the uncertainty in its predictions.

This work demonstrates that the use of genetic algorithm optimization tools enables an effective data-driven determination of material work functions. Finally, the form of the three-regime models illuminates the effect of the transition region between the saturated and Boltzmann regimes. As shown in the two-regime model for the prototype studied in this article, the measured performance curve did not follow the ideal behavior given by the physics-defined saturated and Boltzmann regimes, where a sharp transition would be expected between the two. However, as speculated by Longo for the transition between saturated and Langmuir-Child regimes, the nonuniform electrode coating may be causing electrons to be emitted from the surface at different rates, forming a current density curve for the device that is an average of several current density curves over the entire surface. The transition region then characterizes inconsistencies with the emitter surface coating and provides a more accurate performance prediction for thermionic energy conversion devices.

## Acknowledgment

This work was supported by a NASA Space Technology Graduate Research Opportunity (Grant No. 80NSSC22K1220) and the endowment of the A. Richard Newton Chair in Engineering at UC Berkeley. Part of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

- $k$ =
Boltzmann’s constant (m $2$ kg s $\u22122$ K $\u22121$)

- $q$ =
electron charge (C)

- $I$ =
current (A)

- $R$ =
load resistance ($\Omega $)

- $T$ =
temperature (K)

- $V$ =
voltage (V)

- $Ath$ =
Richardson’s constant (A cm $\u22122$ K $\u22122$)

- $J0$ =
saturated current density (A cm $\u22122$)

- $TC$ =
collector temperature (K)

- $TE$ =
emitter temperature (K)

- $VR$ =
retarding potential (V)

- $\varphi $ =
work function (eV)

- $\varphi C$ =
collector work function (eV)

- $\varphi E$ =
emitter work function (eV)

## Appendix A: 12-Gene Solution for Temperature-Dependent Voltage Determination

*A, B, C*, and

*D*. Equation (A1) is the generic temperature-dependent, 12-gene model.

In this expression, the variables such as $A$ are transformed into $A=A1T2+A2T+A3$, creating 12 total genes per organism as shown in Table 4. Note that the temperature $T$ in Eq. (A1) refers to the emitter temperature. Table 4 lists the final optimized solution for the genes.

Constant | Optimized solution |
---|---|

$A1$ | 6.2278 |

$A2$ | $\u22127.8693$ |

$A3$ | 3.0401 |

$B1$ | $\u22120.6922$ |

$B2$ | 0.9741 |

$B3$ | $\u22120.4753$ |

$C1$ | 6.2916 |

$C2$ | $\u221214.760$ |

$C3$ | 10.660 |

$D1$ | $\u22121.6136$ |

$D2$ | 3.121 |

$D3$ | $\u22122.5307$ |

Constant | Optimized solution |
---|---|

$A1$ | 6.2278 |

$A2$ | $\u22127.8693$ |

$A3$ | 3.0401 |

$B1$ | $\u22120.6922$ |

$B2$ | 0.9741 |

$B3$ | $\u22120.4753$ |

$C1$ | 6.2916 |

$C2$ | $\u221214.760$ |

$C3$ | 10.660 |

$D1$ | $\u22121.6136$ |

$D2$ | 3.121 |

$D3$ | $\u22122.5307$ |

Several iterations of the genetic algorithm were performed with different program parameters, including varying the number of generations and adjusting the fraction of the median threshold retained in each iteration from 0.4 up to 0.9. The most effective setting proved to be 0.6 for these data. The algorithm was run several times with different random perturbations included in the initial DNA vectors, with the optimized constants shown in Table 4 producing the minimum mean relative error of 11.39%. This is a larger error compared to the results for this equation when fit separately to the data for each temperature as shown in Secs. 3.1 and 3.2, where the error is less than 1%.

The predicted voltage is plotted against the available output voltage data in Fig. 14. Data lying on the $y=x$ line indicate a perfect fit for the model to the data, and the deviations from this line illustrate the error within the model prediction. This 12-gene genetic algorithm initially appeared to produce a good fit to the data while providing a single solution for all temperature ranges. However, these results caused additional problems when used in the performance-predictive model as the error in the voltage calculations caused a disproportionately large error in later performance modeling. To solve this problem and determine a general equation that applies at all temperatures, a Lagrange interpolating polynomial was implemented with the coefficient values computed using the four-gene genetic algorithm from Sec. 3.1.

## Appendix B: Alternative Three-Regime Model

We then performed the same optimization and validation procedure as for the modified Longo model using the algorithm from Sec. 2. The results were nearly identical to those produced using the modified Longo model, with the optimized genes and mean relative error values listed in Table 5.

$TE$ (K) | 1170 | 1070 | 970 | 870 |
---|---|---|---|---|

$\varphi E$ (eV) | 2.4667 | 2.2602 | 2.1083 | 1.9127 |

$\varphi C$ (eV) | 1.7917 | 1.7847 | 1.8077 | 1.5502 |

$\eta $ | 2.4229 | 2.5404 | 1.8723 | 2.9042 |

Training error | 0.0398 | 0.0390 | 0.0303 | 0.0409 |

Validation error | 0.1611 | 0.0605 | 0.0312 | 0.0320 |

Combined error | 0.0812 | 0.0461 | 0.0306 | 0.0378 |

$TE$ (K) | 1170 | 1070 | 970 | 870 |
---|---|---|---|---|

$\varphi E$ (eV) | 2.4667 | 2.2602 | 2.1083 | 1.9127 |

$\varphi C$ (eV) | 1.7917 | 1.7847 | 1.8077 | 1.5502 |

$\eta $ | 2.4229 | 2.5404 | 1.8723 | 2.9042 |

Training error | 0.0398 | 0.0390 | 0.0303 | 0.0409 |

Validation error | 0.1611 | 0.0605 | 0.0312 | 0.0320 |

Combined error | 0.0812 | 0.0461 | 0.0306 | 0.0378 |

## Appendix C: Back-Emission Calculations

There is a possibility for back emission when the collector temperature is sufficiently high. However, for the device considered in this article, the back emission is not a significant source of error. To determine if back emission was a source for error in our model, we used the available information for collector temperature gathered when the prototype we are modeling underwent a power consumption mode or Schottky mode test. During this test, the emitter and collector electrode temperature pairings were indicated to be the values listed in Table 6. Because of the lack of information on collector temperature thermionic energy conversion operating mode, a proportion of the collector-to-emitter temperature was used for back emission calculations to provide realistic collector temperatures. The proportion used is also indicated in Table 6 as $\theta =TCTE$. The maximum value of $\theta $ is 0.6585, and the minimum value of $\theta $ is 0.5982. These $\theta $ values are the bounding limits for the back emission calculations.

$TE(K)$ | $TC(K)$ | $\theta =TC/TE$ |
---|---|---|

1120 | 670 | 0.5982 |

1020 | 625 | 0.6127 |

920 | 580 | 0.6304 |

820 | 540 | 0.6585 |

$TE(K)$ | $TC(K)$ | $\theta =TC/TE$ |
---|---|---|

1120 | 670 | 0.5982 |

1020 | 625 | 0.6127 |

920 | 580 | 0.6304 |

820 | 540 | 0.6585 |

The results of computing these back emission values for each emitter temperature is shown in Table 7. Note that the $\theta =1$ column represents the forward emission current density.

Current density (J) (A/cm $2$) | |||
---|---|---|---|

$TE$ (K) | $\theta =0.6585$ | $\theta =0.5982$ | $\theta =1$ |

$1170$ | $0.2782\xd710\u22125$ | $0.5602\xd710\u22124$ | 0.0037 |

$1070$ | $0.0331\xd710\u22125$ | $0.0796\xd710\u22124$ | 0.0030 |

$970$ | $0.0011\xd710\u22125$ | $0.0035\xd710\u22124$ | 0.0018 |

$870$ | $0.0076\xd710\u22125$ | $0.0201\xd710\u22124$ | 0.0012 |

Current density (J) (A/cm $2$) | |||
---|---|---|---|

$TE$ (K) | $\theta =0.6585$ | $\theta =0.5982$ | $\theta =1$ |

$1170$ | $0.2782\xd710\u22125$ | $0.5602\xd710\u22124$ | 0.0037 |

$1070$ | $0.0331\xd710\u22125$ | $0.0796\xd710\u22124$ | 0.0030 |

$970$ | $0.0011\xd710\u22125$ | $0.0035\xd710\u22124$ | 0.0018 |

$870$ | $0.0076\xd710\u22125$ | $0.0201\xd710\u22124$ | 0.0012 |

The values determined in Table 7 were then used to compare the magnitude of back emission to the forward emission current density at each temperature. The results are included in Table 8, where the back emission current density is at most 1.52% of the forward emission at an emitter temperature of 1170 K. Because back emission is between 0.0006% and 1.52% of the forward emission for our device, we concluded that back-emission is not a significant source of error for this study.

Back emission fraction | ||
---|---|---|

$TE$ (K) | $J\theta =0.6585/J\theta =1$ | $J\theta =0.5982/J\theta =1$ |

$1170$ | $0.000755$ | $0.0152$ |

$1070$ | $0.000112$ | $0.0027$ |

$970$ | $0.000006$ | $0.0002$ |

$870$ | $0.000061$ | $0.0016$ |

Back emission fraction | ||
---|---|---|

$TE$ (K) | $J\theta =0.6585/J\theta =1$ | $J\theta =0.5982/J\theta =1$ |

$1170$ | $0.000755$ | $0.0152$ |

$1070$ | $0.000112$ | $0.0027$ |

$970$ | $0.000006$ | $0.0002$ |

$870$ | $0.000061$ | $0.0016$ |